GeLoMa 2016 Conference Program VIII International Meeting on Lorentzian Geometry Málaga, Spain GeLoMa 2016 Schedule at a glance 9:00 - 10:00 Mini course 10:00 - 11:00 Invited speakers 11:30 - 12:30 Invited speakers Afternoon Contributed talks Best Poster Award The Academia Malagueña de Ciencias supports a prize for the Best Poster, according to the decision of the Scientific Committee. The award, including a commemorative gift, will be presented after the Friday morning session, just before lunch time. 1 Málaga, Spain VIII International Meeting on Lorentzian Geometry 2 VIII International Meeting on Lorentzian Geometry Málaga, Spain Contents Welcome 5 Instructions for Participants 6 List of Participants 7 Invited speakers Felix Finster. Linear stability of the non-extreme Kerr black hole . . . . . . . . . . . José Luis Jaramillo. Understanding isolated system dynamics in General Relativity . Philippe G. LeFloch. The Global Nonlinear Stability of Minkowski Spacetime for Self-Gravitating Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pawel Nurowski. How the green light was given for gravitational wave search (report on a joint work with C. Denson Hill) . . . . . . . . . . . . . . . . . . . . . . . . Tim-Torben Paetz. Conformal methods in general relativity . . . . . . . . . . . . . . Rafael M. Rubio. Calabi-Bernstein type problems in Lorentzian Geometry . . . . . . Miguel Sánchez. Generalized Fermat principle and Zermelo navigation: a link between Lorentzian and Generalized Finslerian Geometries . . . . . . . . . . . . . . . . . Kotaro Yamada. Type changes of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 11 12 13 14 15 16 17 18 Mini course 19 Gregory J. Galloway. On the geometry and topology of initial data sets in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 General interest talk 23 José M.M. Senovilla. Ondas gravitacionales: el amanecer de una nueva era . . . . . . 27 Contributed talks Masashi Yasumoto. Trivalent maximal surfaces in Minkowski 3-space . . . . . . . . . Roland Steinbauer. The singularity theorems in low regularity . . . . . . . . . . . . . Benjamı́n Olea. Induced Riemannian structures on null hypersurfaces . . . . . . . . . Stephanie Alexander. Convex functions in space-time geometry . . . . . . . . . . . . . Rafael López. Existence of spacelike graphs of constant mean curvature in the steady state space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giovanni Calvaruso. Recents results on the oscillator spacetime . . . . . . . . . . . . Marco Castrillon López. Low dimensional Lie groups with cyclic Lorentzian metrics . Miguel Ortega. Translating solitons, semi-Riemannian manifolds and Lie groups . . . Lorenzo Nicolodi. Lawson correspondence and Laguerre deformation . . . . . . . . . . Melani Graf. Volume comparison for C 1,1 -metrics . . . . . . . . . . . . . . . . . . . . Miguel Ángel Javaloyes. An overview about Finsler spacetimes and Penrose´s singularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clemens Sämann. On causality in low regularity . . . . . . . . . . . . . . . . . . . . . Alma Luisa Albujer Brotons. Some uniqueness results for the solution to the Hr = Hl surface equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 29 31 32 33 35 36 38 39 40 41 42 43 44 45 Málaga, Spain VIII International Meeting on Lorentzian Geometry Eraldo Almeida Lima Júnior. Uniqueness of complete maximal surfaces in certain Lorentzian warped products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ivan P. Costa e Silva. Ridigity of geodesic completeness in gravitational wave spacetimes Nastassja Cipriani. Umbilical properties of spacelike co-dimension two submanifolds . Jonatan Herrera. Conformal group actions and their influence on the causal boundary Jose Carlos Dı́az Ramos. Cohomogeneity one actions on Minkowski spaces . . . . . . 46 48 49 50 51 Posters Adriana Araujo Cintra. The Björling problem for minimal timelike surfaces in a Lorentzian 3-dimensional Lie group . . . . . . . . . . . . . . . . . . . . . . . . . Marco Antonio Lázaro. Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime . . . . . . . Marilena Moruz. Translation and homothetical surfaces in Lorentzian space with constant curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Álvaro Pampano. Elastica constrained problem in hypersurfaces of Lorentzian space forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yoshinori Machida. Geometric structures of Hamilton-Jacobi equations associated with metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Takahashi Masatomo. Evolutes of curves in the Lorentz—Minkowski plane . . . . . . Goo Ishikawa. Null frontal singular surfaces in Lorentzian 3-spaces . . . . . . . . . . Aleksy Tralle. On Clifford-Klein forms . . . . . . . . . . . . . . . . . . . . . . . . . . Nieves Álamo Antúnez. Cosmological horizons: a simple model to clarify some common misconceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Verónica López Cánovas. Marginally trapped submanifolds in generalized RobertsonWalker spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Esma Demir. Homothetic motions and surfaces with constant curvatures in Lorentzian 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junichi Mukuno. On the fundamental group of a complete globally hyperbolic Lorentzian manifold with a lower bound for the curvature tensor . . . . . . . . . . . . . . . Cristina Draper. ∇-Einstein connections on Lorentzian spheres . . . . . . . . . . . . . Fábio Reis. On the quadric CMC spacelike hypersurfaces in Lorentzian space forms . Magdalena Caballero Campos. Geometric properties of surfaces with the same mean curvature in R3 and L3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antonio Wilson Rodrigues da Cunha. Higher order mean curvature estimates for complete hypersurfaces into horoballs . . . . . . . . . . . . . . . . . . . . . . . . Luis A. Aké. Some properties of glogally hyperbolic spacetimes with timelike boundary José Antonio S. Pelegrı́n. Constant mean curvature spacelike hypersurfaces in spacetimes admitting a parallel lightlike vector field . . . . . . . . . . . . . . . . . . . Juan J. Salamanca. φ-minimal graphs in certain manifolds with density . . . . . . . . Daniel de la Fuente. Completeness of certain accelerated relativistic trajectories . . . Tuncer Ogulcan. Surfaces with a common isophote curve in Minkowski 3-space . . . . Osman Ates. A curve whose position vector lies on the orthogonal complement of its any Frenet vector in Minkowski n-space . . . . . . . . . . . . . . . . . . . . . . . Özgür Keskin. Normal Fermi-Walker derivative in Minkowski 3-space . . . . . . . . . Erdem Kocakusakli. A quaternionic representation for canal surfaces in Minkowski 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xavier Valle-Regueiro. Lorentzian generalized quasi-Einstein metrics and static structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Notes 81 4 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 79 80 VIII International Meeting on Lorentzian Geometry Málaga, Spain GeLoMa 2016 Welcome It is our great pleasure to welcome you to the VIII International Meeting on Lorentzian Geometry in Málaga, Spain. Scientific Committee Luis J. Alias (Universidad de Murcia, Spain) Anna Maria Candela (Università Degli Studi Di Bari, Italy) † Sergio Dain (Universidad Nacional de Córdoba, CONICET, Argentina) Eduardo Garcı́a Rı́o (Universidad de Santiago, Spain) Alfonso Romero (Universidad de Granada, Spain) (coordinator) José M.M. Senovilla (Universidad del Paı́s Vasco, Spain) Masaaki Umehara (Tokyo Institute of Technology, Japan) † Sadly, Sergio Dain passed away on February 24th, 2016. His loss is deeply regretted. Organizing Committee Miguel Atencia (Universidad de Málaga, Spain) Marı́a A. Cañadas-Pinedo (Universidad de Málaga, Spain) Cristina Draper (Universidad de Málaga, Spain) José Luis Flores (Universidad de Málaga, Spain) Manuel Gutiérrez (Universidad de Málaga, Spain) Francisco Palomo (Universidad de Málaga, Spain) Technical Secretariat Viajes El Corte Inglés [email protected] Sponsors Vicerrectorado de Investigación y Transferencia Departamento de Álgebra, Geometrı́a y Topologı́a Departamento de Matemática Aplicada Universidad de Málaga National Research Project MTM2013-47828-C2-1-P Gobierno de España National Research Project MTM2013-47828-C2-2-P Ministerio de Economı́a y Competitividad National Research Project MTM2013-41768-P European Union Academia Malagueña de Ciencias Sociedad Malagueña de Astronomı́a Metro de Málaga 5 Málaga, Spain VIII International Meeting on Lorentzian Geometry Instructions for Participants Internet access • Wireless network: Congresos – Open any web site in a browser – In the identification page that will appear, enter: ∗ Username: geloma ∗ Password: Lorentz2016 • Alternatively, you can use your eduroam account on the wireless network eduroam. • There are some available terminals in the computer rooms. You will have to register in the Virtual Campus on these computers, providing your personal data. • Contact the Registration Desk or ask someone of the Organization Committee for help. Instructions for Contributors • The room will have a digital projector and a Windows computer, but you are advised to bring your own laptop to avoid configuration issues. • Both speakers and chairs should arrive 10 minutes before the session starts, to check the equipment works. • Please stick to the schedule. • Posters should be displayed since the beginning of the Conference, and authors should stand by their posters during the Poster Session. • Poster boards will be installed in the hall of the ground floor, with room for up to A0 posters. • Contact the Registration Desk or ask someone of the Organization Committee for help. • Please, wear your badge visibly, specially at lunches. Social program • • • • • • Date: Thursday 22nd. Meeting point for both activities: AC Hotel Malaga Palacio, Street Cortina del Muelle, 1 Walking visit to Málaga: 17:00 h. Dinner: 20:30 (then we walk to the nearby restaurant). Be aware of any updates by checking the signboard next to the Registration Desk. Please, wear your badge visibly at all social activities and lunches. Transport • The conference bag includes a couple of Metro tickets, each charged with two trips. Do not dispose of the cards, they can be recharged with additional trips without paying again for the physical support (see http://metromalaga.es/en/tickets-and-fares/). 6 VIII International Meeting on Lorentzian Geometry Málaga, Spain List of Participants Name Affiliation Amir Aazami Luis Alberto Aké Hau Alma Luisa Albujer Brotons Stephanie Alexander Luis J. Alı́as Adriana Araujo Cintra Miguel Atencia Osman Ates Rosella Bartolo Magdalena Caballero Giovanni Calvaruso Anna Marı́a Candela Marı́a A. Cañadas-Pinedo Erasmo Caponio Marco Castrillón López Nastassja Cipriani Daniel de la Fuente Benito Esma Demir Çetin José Carlos Dı́az Ramos Cristina Draper Fontanals Faik Nejat Ekmekci Ángel Ferrández Felix Finster Mathias Fischer José Luis Flores Gregory Galloway Eduardo Garcı́a Rı́o Antonio Garvı́n Wolfgang Globke Ismail Gok Melanie Graf Manuel Gutiérrez Steve Harris Jonatan Herrera Halyson Irene Baltazar Goo Ishikawa José Luis Jaramillo Miguel Ángel Javaloyes Özgur Keskin Erdem Kocakusakli Philippe G. Lefloch Clark University, USA Universidad de Málaga, Spain Universidad de Córdoba, Spain University of Illinois at Urbana-Champaign, USA Universidad de Murcia, Spain Universidade Federal de Goiás, Brazil Universidad de Málaga, Spain Ankara University, Turkey Politecnico di Bari, Italy Universidad de Córdoba, Spain University of Salento, Italy Università degli Studi di Bari Aldo Moro, Italy Universidad de Málaga, Spain Politecnico di Bari, Italy ICMAT & Universidad Complutense de Madrid, Spain KU Leuven, Belgium & Universidad del Paı́s Vasco, Spain Universidad de Granada, Spain Nevşehir Hacı Bektaş Veli Üniversitesi, Turkey Universidade de Santiago de Compostela, Spain Universidad de Málaga, Spain Ankara University, Turkey Universidad de Murcia, Spain Universität Regensburg. Germany Universität Greifswald, Germany Universidad de Málaga, Spain University of Miami, USA Universidade de Santiago de Compostela, Spain Universidad de Málaga, Spain University of Adelaide, Australia Ankara University, Turkey University of Vienna, Austria Universidad de Málaga, Spain Saint Louis University, USA Universidade Federal de Santa Catarina, Brazil Universidade Federal do Piauı́, Brazil Hokkaido University, Japan Université de Bourgogne Franche-Comté, France Universidad de Granada, Spain Erciyes Üniversitesi, Turkey Ankara University, Turkey Université Pierre et Marie Curie, France 7 Málaga, Spain Name VIII International Meeting on Lorentzian Geometry Affiliation Eraldo Lima Universidade Federal da Paraı́ba, Brazil Rafael López Universidad de Granada, Spain Verónica López Cánovas Universidad de Murcia, Spain Yoshinori Machida Numazu College of Technology, Japan Abdelmalek Mohammed Université Badji Mokhtar, Annaba, Algeria Junichi Mukuno Nagoya University, Japan Lorenzo Nicolodi Università degli Studi di Parma , Italy Pawel Nurowski University of Warsaw. Poland Tuncer Ogulcan Ankara University, Turkey Benjamı́n Olea Universidad de Málaga, Spain Miguel Ortega Universidad de Granada, Spain Tim-Torben Paetz University of Vienna, Austria Francisco José Palomo Universidad de Málaga, Spain Álvaro Pampano Universidad del Pais Vasco, Spain Iván Pontual Universidade Federal de Santa Catarina, Brazil Fábio Reis dos Santos Universidade Federal de Campina Grande, Brazil Antonio Wilson Rodrigues da Cunha Universidade Federal do Piauı́, Brazil José Rojo Universidad CEU San Pablo, Spain Alfonso Romero Universidad de Granada, Spain Rafael M. Rubio Universidad de Córdoba, Spain Clemens Saemann University of Vienna, Austria Juan Jesús Salamanca Jurado Universidad de Granada, Spain Miguel Sánchez Caja Universidad de Granada, Spain Jose Antonio Sánchez Pelegrı́n Universidad de Granada, Spain Ignacio Sánchez Rodrı́guez Universidad de Granada, Spain José M.M. Senovilla Universidad del Pais Vasco, Spain Roland Steinbauer University of Vienna, Austria Masatomo Takahashi Muroran Institute of Technology, Japan Aleksy Tralle University of Warmia and Mazury, Olsztyn, Poland Javier Valle Regueiro Universidade de Santiago de Compostela, Spain Gabriela Wanderley Universidade Federal da Paraı́ba, Brazil Kotaro Yamada Tokyo Institute of Technology, Japan Masashi Yasumoto Kobe University, Japan Yusuf Yayli Ankara University, Turkey 8 VIII International Meeting on Lorentzian Geometry Invited speakers 9 Málaga, Spain Málaga, Spain VIII International Meeting on Lorentzian Geometry 10 VIII International Meeting on Lorentzian Geometry Málaga, Spain Linear stability of the non-extreme Kerr black hole Felix Finster Universität Regensburg, Germany ABSTRACT After a general introduction to black holes and wave equations in black hole geometries, I will report on recent results on the linear stability of Kerr black holes under perturbations of general spin. This is joint work with Joel Smoller. 11 Málaga, Spain VIII International Meeting on Lorentzian Geometry Understanding isolated system dynamics in General Relativity José Luis Jaramillo Institut de Mathématiques de Bourgogne, France ABSTRACT The work by Sergio Dain in mathematical relativity provides a brilliant chapter in the theoretical developments of General Relativity over the last two decades. Sergio left us last February, at the age of 46. The relevance of his contributions in different areas of the theory shape him as an outstanding relativist and a crucial figure among the researchers of his generation. This talk aims at presenting a perspective on his work, focusing on those aspects related to the dynamics of isolated systems in General Relativity. This topic offers a natural frame to illustrate the richness, soundness and astonishing inner consistency of his variate contributions. It also underlines a key feature of Sergio as a researcher: his most singular combination of exceptional geometrical and analytical skills with a deep intuition and understanding of the physics modelled by the theory. Lust for Understanding, fresh enthusiasm and elegance of spirit, distinguishing imprints of our colleague and friend. 12 VIII International Meeting on Lorentzian Geometry Málaga, Spain The Global Nonlinear Stability of Minkowski Spacetime for Self-Gravitating Matter Philippe G. LeFloch Université Paris 6, France ABSTRACT This lecture will review recent results on self-gravitating massive matter, modeled by the Einstein equations of general relativity. A new vector field method, the Hyperboloidal Foliation Method developed in collaboration with Y. Ma (Xian), is now available in order to establish global-in-time results for the class of systems of coupled wave-Klein-Gordon equations posed on a curved spacetime. This method has recently been used to investigate the global dynamics of massive matter fields. 13 Málaga, Spain VIII International Meeting on Lorentzian Geometry How the green light was given for gravitational wave search (report on a joint work with C. Denson Hill) Pawel Nurowski Uniwersytet Warszawski, Poland ABSTRACT The recent detection of gravitational waves by the LIGO/VIRGO team is an incredibly impressive achievement of experimental physics. It is also a tremendous success of the theory of General Relativity. It confirms the existence of black holes; shows that binary black holes exist; that they may collide and that during the merging process gravitational waves are produced. These are all predictions of General Relativity theory in its fully nonlinear regime. The existence of gravitational waves was predicted by Albert Einstein in 1916 within the framework of linearized Einstein theory. Contrary to common belief, even the very definition of a gravitational wave in the fully nonlinear Einstein theory was provided only after Einstein’s death. Actually, Einstein had arguments against the existence of nonlinear gravitational waves (they were erroneous but he did not accept this), which virtually stopped development of the subject until the mid 1950s. This is what we refer to as the Red Light for gravitational waves research. In the following years, the theme was picked up again and studied vigorously by various experts, mainly Herman Bondi, Felix Pirani, Ivor Robinson and Andrzej Trautman, where the theoretical obstacles concerning gravitational wave existence were successfully overcome, thus giving the ‘Green Light’ for experimentalists to start designing detectors, culminating in the recent LIGO/VIRGO discovery. In this lecture we tell the story of this theoretical breakthrough. Particular attention will be given to the fundamental 1958 papers of Trautman, which seem to be lesser known outside the circle of General Relativity experts. 14 VIII International Meeting on Lorentzian Geometry Málaga, Spain Conformal methods in general relativity Tim-Torben Paetz Universität Wien, Austria ABSTRACT Friedrich’s conformal field equations substitute Einstein’s field equations in conformally rescaled vacuum spacetimes. They provide an extremely powerful tool to construct (semi-)global solutions to the vacuum equations and to control the asymptotic behavior of the gravitational field. In this talk I will review the conformal field equations and discuss some striking results obtained via these equations for vacuum spacetimes with both positive and vanishing cosmological constant. Although a main focus will be on the construction of spacetimes which admit a smooth null infinity, the issue of constructing spacetimes with smooth timelike and spacelike infinity (using the cylinder representation for the latter one) will be addressed as well. 15 Málaga, Spain VIII International Meeting on Lorentzian Geometry Calabi-Bernstein type problems in Lorentzian Geometry Rafael M. Rubio Universidad de Córdoba, Spain ABSTRACT The Calabi-Bernstein theorem states that the only entire solutions to the maximal hypersurface equation in the Lorentz-Minkowski spacetime are the spacelike affine hyperplanes. The present work review some of the classical and recent proofs of the theorem for the two dimensional case, as well as several extensions for Lorentzian warped products and other relevant spacetimes. On the other hand the problem of uniqueness of complete maximal hypersurfaces is analized under the perspective of some new results. 16 VIII International Meeting on Lorentzian Geometry Málaga, Spain Generalized Fermat principle and Zermelo navigation: a link between Lorentzian and Generalized Finslerian Geometries Miguel Sánchez Universidad de Granada, Spain ABSTRACT Recently, a link between the geometric properties of some classes of Lorentzian and Finslerian manifolds has been developed, including the correspondence between the conformal geometry of stationary spacetimes and the geometry of (Finslerian) Randers spaces, as well as between several notions of convexity and boundares in both fields. In particular, a connection between two classical variational problems have been found: 1. Relativistic Fermat’s principle: among the lightlike curves joining a point and an observer γ0 , pregeodesics are the critical points of the arrival proper time functional. 2. Zermelo’s navigation: the fastest path between two points for the movement of a plane in windy air or a ship on a current, are geodesics for a certain Finslerian metric F constructed by using a Riemannian metric gR and a vector field W such that gR (W, W ) < 1. Such a link allows to extend and solve both problems beyond the classical scope, namely, when γ0 is not timelike and when the wind/current W is not mild (that is, the restriction gR (W, W ) < 1 is not imposed). Moreover, the latter problem requires an extension of Finslerian Geometry with a natural interpretation from the classical Lorentzian viewpoint. Along this talk, based on joint work with E. Caponio and M.A. Javaloyes (arxiv: 1407.5494), a review on the topic will be provided. 17 Málaga, Spain VIII International Meeting on Lorentzian Geometry Type changes of spacelike maximal surfaces in Minkowski 3-space to timelike surfaces Kotaro Yamada Tokyo Institute of Technology, Japan ABSTRACT A certain class of spacelike maximal surfaces in Lorentz-Minkowski 3-space can be extended to timelike zero-mean-curvature surface. We explain such a phenomena and construct new examples. In particular, several examples of entire zero-mean-curvature graphs are introduced. Such a phenomena does not occur for minimal surfaces in Euclidean 3-space because Bernstein’s theorem. 18 VIII International Meeting on Lorentzian Geometry Mini course 19 Málaga, Spain Málaga, Spain VIII International Meeting on Lorentzian Geometry 20 VIII International Meeting on Lorentzian Geometry Málaga, Spain On the geometry and topology of initial data sets in General Relativity Gregory J. Galloway University of Miami, USA ABSTRACT An initial data set in spacetime consists of a spacelike hypersurface V , together with its its induced (Riemannian) metric h and its second fundamental form K. After a brief introduction to Lorentzian manifolds and Lorentzian causality, we will study some topics of recent interest related to the geometry and topology of initial data sets. In particular, we will consider the topology of black holes in higher dimensional gravity, inspired by certain developments in string theory and issues related to black hole uniqueness. We shall also discuss recent work on the geometry and topology of the region of space exterior to all black holes, which is closely connected to the notion of topological censorship. Many of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry. 21 Málaga, Spain VIII International Meeting on Lorentzian Geometry 22 VIII International Meeting on Lorentzian Geometry General interest talk 23 Málaga, Spain Málaga, Spain VIII International Meeting on Lorentzian Geometry 24 Málaga, Spain VIII International Meeting on Lorentzian Geometry 26 VIII International Meeting on Lorentzian Geometry Málaga, Spain Ondas gravitacionales: el amanecer de una nueva era José M.M. Senovilla Universidad del Paı́s Vasco, Spain ABSTRACT Después de una larga espera de decenas de años estamos de enhorabuena: el 14 de Septiembre de 2015 una colaboración cientı́fica internacional (LIGO/VIRGO) logró detectar, en sus interferómetros más avanzados en funcionamiento, lo que llevábamos esperando con tanta ansia: una onda gravitatoria. La primera de la historia. Poco después, el 26 de Diciembre, se detectó la segunda. El logro cientı́fico-técnico es imponente: medir una variación de longitud equivalente a la del tamaño del radio atómico en la distancia entre la Tierra y el Sol. ¿Se puede medir eso? ¡Se ha hecho! Simultáneamente, hemos podido observar, por partida doble y directamente, un sistema binario de agujeros negros. ¡Sensacional! Lo mejor, con seguridad, está por llegar. Estos hechos excepcionales e históricos demuestran que la humanidad se ha dotado de un nuevo “sentido” para observar el Universo, una nueva ventana por la que escudriñar lo que hay ahı́ fuera. Hasta ahora éramos insensibles a la radiación gravitatoria, a partir de ahora ya podemos “gravi-sentir” el Universo. La nueva era de la “Astronomı́a por gravedad” ha empezado. En esta conferencia se explicará, de forma amena y asequible, qué es una onda gravitatoria, cómo y con qué aparatos se mide, la información que porta, sus diferencias con otras ondas cotidianas. Veremos ası́ que todo lo que a partir de ahora se descubrirá superará todas nuestras expectativas, cambiará nuestra cosmovisión radicalmente, nos aportará sorpresas impensables. Los cielos están henchidos de “luceros gravitatorios”, ignotos hasta ahora, inenarrables. Podremos conocerlos y estudiarlos. Aprender acerca de, y comprender, el firmamento. Todo lo que existe, sea visible o invisible, gravita. Podremos por ello observar, e indagar, todo el Universo, sus más recónditos rincones y hasta su origen. Explicaremos, en definitiva, por qué nos encontramos en los albores de una nueva etapa para la humanidad, un momento único y apasionante. ¡No se lo pierdan! 27 Málaga, Spain VIII International Meeting on Lorentzian Geometry 28 VIII International Meeting on Lorentzian Geometry Contributed talks 29 Málaga, Spain Málaga, Spain VIII International Meeting on Lorentzian Geometry 30 VIII International Meeting on Lorentzian Geometry Málaga, Spain Trivalent maximal surfaces in Minkowski 3-space Masashi Yasumoto∗ Department of Mathematics, Faculty of Science, Kobe University [email protected] ABSTRACT In this talk we introduce trivalent maximal surfaces in Minkowski 3-space based on the combination of two ideas below. In particular, we see that such trivalent maximal surfaces admit associated families and they also have variational properties and certain singularities. Interesting examples are given by Schramm’s orthogonal circle patterns. Lam [1] investigated two types of trivalent surfaces in Euclidean 3-space R3 with vanishing mean curvature, which are called trivalent minimal surfaces in R3 . The advantage of this discretization is that we can treat both integrable geometric aspects of such minimal surfaces in R3 and their variational properties, which generalize many previous works. On the other hand, Yasumoto [3] described quadrilateral surfaces (or, discrete surfaces) in R2,1 with mean curvature identically 0, which are called discrete maximal surfaces in R2,1 . Unlike the case of discrete surfaces in R3 with vanishing mean curvature, it is shown that discrete maximal surfaces in R2,1 generally have singularities, which occurs only in the case of discrete constant mean curvature surfaces in Lorentzian spaceforms. This talk is based on joint work with Wai Yeung Lam (TU Berlin) [2]. R2,1 References [1] W.Y. Lam, Discrete minimal surfaces: critical points of the area functional from integrable systems, preprint, arXiv:1510.08788v2. [2] W.Y. Lam and M. Yasumoto, Trivalent maximal surfaces in Minkowski space, in preparation. [3] M. Yasumoto, Discrete maximal surfaces with singularities in Minkowski space, Differential Geometry and its Applications 43 (2015), 130-154. ∗ Current address: Institut für Mathematik, Universität Tübingen 31 Málaga, Spain VIII International Meeting on Lorentzian Geometry The Singularity Theorems in Low Regularity Roland Steinbauer Faculty of Mathematics, University of Vienna, Austria [email protected] ABSTRACT Classically the singularity theorems of General Relativity are proven for smooth, actually C 2 -metrics. However, the physically most reasonable and at the same time conceptionally most natural regularity class for the statement of the Theorems is C 1,1 , i.e., the first order derivatives of the metric being locally Lipschitz continuous. In this talk we present the recent proofs of both the Hawking and the Penrose singularity theorem in C 1,1 ([3, 4]) based on regularisation techniques adapted to the causal structure of the spacetime ([1, 2]). Finally we provide an outlook to further lines of research in this area. References [1] M. Kunzinger, R. Steinbauer, M. Stojković. The exponential map of a C 1,1 metric. Diff. Geom. Appl, 34, 2014. [2] M. Kunzinger, R. Steinbauer, M. Stojković, J. Vickers. A regularisation approach to causality theory for C 1,1 -Lorentzian metrics. Gen. Rel. Grav., 46, 2014. [3] M. Kunzinger, R. Steinbauer, M. Stojković, J. Vickers. Hawking’s singularity theorem for C 1,1 -metrics. Class. Quant. Grav. 32, 2015. [4] M. Kunzinger, R. Steinbauer, J. Vickers. The Penrose singularity theorem in regularity C 1,1 . Class. Quant. Grav. 32, 2015. 1 32 VIII International Meeting on Lorentzian Geometry Málaga, Spain Induced Riemannian structures on null hypersurfaces Manuel Gutiérrez‡ and Benjamín Olea∗ ‡ Departamento ∗ de Álgebra, Geometrı́a y Topologı́a. Universidad de Málaga Presenting author. Departamento de Matemática Aplicada Universidad de Málaga. [email protected] ABSTRACT A hypersurface in a Lorentzian manifold is null if the induced metric tensor is degenerate on it. These hypersurfaces do not have a Riemannian counterpart, so they are interesting by their own, both geometrically and physically. Null hypersurfaces can not be treated as spacelike or timelike hypersurfaces, since an orthogonal projection can not be defined on them. So, neither the second fundamental form or the induced connection can be constructed in the usual way and specific techniques were developed for this. One of the most usual (but not the unique) is to fix a geometric data formed by a null section and a screen distribution on the null hypersurface. This allows to define an induced connection and a null second fundamental form, which gives the expected information on the extrinsic geometry. However, the induced connection does not arise necessarily from a metric and is clear that it is not an appropriate tool to study intrinsic geometric properties. Moreover, both the null section and the screen distribution are fixed arbitrarily and independently and it is not clear how to choose them in order to have a reasonable coupling between the properties of the null hypersuperface and the ambient space. Alternatively, we show a technique to construct a Riemannian metric ge on a null hypersurface L. It is based on the arbitrary choice of a transverse vector field, called rigging field, from which we construct a null section, which we call rigged field and a screen distribution. The improvement over the above technique is twofold: first, the geometric data depends only on the choice of a unique object, the rigging field. Secondly, we introduce a Riemannian structure coupled with it, which is used to study the null hypersurface. Those structures are not natural in the sense that they depend on the choice of the rigging field, but the flexibility to choose it turns this limitation into an advantage, allowing us to use valuable information on the ambient space, for example in the presence of symmetries. We relate extrinsic properties of the null hypersurface to the properties of the Riemannian manifold (L, ge) and we establish some formulas linking the curvature of the ambient manifold and the curvature of (L, ge). This allows us to obtain some new results on null hypersuperfaces. For example, we use the Bochner technique to show a curvature condition which implies that a compact totally umbilic null hypersurface must be totally geodesic. We also show that the induced Riemannian metric ge in a totally umbilic null hypersurface is locally a twisted product, which can be a warped or direct product depending on the properties of the ambient space and the rigging field. This is used to prove that the first conjugate point of a null geodesic contained in a totally umbilic null cone has maximum multiplicity. Finally, we adapt the main ideas to null submanifods of arbitrary codimension, which allows us to apply Gauss-Bonnet theorem to compact null surfaces. 33 Málaga, Spain VIII International Meeting on Lorentzian Geometry References [1] M. Gutiérrez and B. Olea, Totally umbilic hypersurfaces in generalized Robertson-Walker spaces, Differ. Geom. Appl. 42 (2015) 15-30. [1] M. Gutiérrez and B. Olea, Induced Riemann structures on null hypersurfaces, to appear in Math. Nachr. DOI 10.1002/mana.201400355 [2] B. Olea, Canonical variation of a Lorentzian metric, J. Math. Anal. Appl. 419 (2014) 156-171. 2 34 VIII International Meeting on Lorentzian Geometry Málaga, Spain Convex functions in space-time geometry Stephanie B. Alexander University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] ABSTRACT This is joint work with William Karr. We study classically convex and “space-time convex” functions (introduced in [1]) in space-time geometry. Using classically convex functions, we give a geometric-topological approach to geodesic connectedness of space-times [2]. For instance: A null-disprisoning space-time is geodesically connected if it supports a proper, nonnegative strictly convex function whose critical set is a point. Timelike strictly convex hypersurfaces of Minkowski space are geodesically connected. We also give a criterion for the existence of a convex function on a semiRiemannian manifold, and compare our results to those obtained by analytic methods (see [3]). Turning to space-time convex functions [4], we prove they are fundamentally related to curvature bounds of the form R ≤ K (i.e. space-like sectional curvatures ≤ K, timelike sectional curvatures ≥ K). These bounds were introduced and applied in [5] and characterized geometrically in [6]. The connection with curvature bounds allows us to identify many space-times that support spacetime convex functions. Such functions rule out, for example, closed spacelike geodesics and closed marginally inner and outer trapped surfaces [1]. References [1] G. Gibbons, A. Ishibashi, Convex functions and spacetime geometry, Classical Quantum Gravity 18 (2001), no. 21, 4607 -4627. [2] S. Alexander, W. Karr, Convex functions and geodesic connectedness of spacetimes, http://arxiv.org/abs/1510.04228v2. [3] A. Candela, M. Sanchez, Geodesics in semi-Riemannian manifolds: Geometric properties and variational tools. In Recent Developments in PseudoRiemannian Geometry, 359-418, ESI Lect. Math. Phys. Eur. Math. Soc. Publ. House, Zürich, 2008. [4] S. Alexander, W. Karr, Space-time convex functions and curvature, in preparation. [5] L. Andersson, R. Howard, Comparison and rigidity theorems in semiRiemannian geometry, Comm. Anal. Geom. 6 (1998), 819-877. [6] S. Alexander, R. Bishop, Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds, Comm. Anal. Geom. 16 (2008), 251-282. 35 Málaga, Spain VIII International Meeting on Lorentzian Geometry Existence of spacelike graphs of constant mean curvature in the steady state space Rafael López Departamento de Geometrı́a y Topologı́a Instituto de Matemáticas (IEMath-GR) Universidad de Granada 18071 Granada, Spain [email protected] ABSTRACT The steady state space H3 is a model for the universe proposed by Bondi, Gold and Hoyle which is homogeneous and isotropic [1, 3, 4]. From the mathematical viewpoint, H3 corresponds with the Lorentzian analogue to the upper half-space model of hyperbolic space. In this talk we consider spacelike surfaces in H3 with constant mean curvature (cmc). We study the Dirichlet problem for the prescribed mean curvature equation on a domain of a slice, that is, finding a solution u ∈ C 2 (Ω) ∩ C 0 (Ω) of Du = div √1−|Du| 2 |Du| < 1 u=h 2 u H+√ 1 1−|Du|2 in Ω ⊂ R2 in Ω along ∂Ω, where H ∈ R and h ∈ R+ . This problem was firstly considered by Montiel in [6] for convex domains and values H < −1; see also [2] for the existence of radially symmetric spacelike cmc surfaces in H3 with different techniques. Under suitable conditions on the convexity of Ω, we prove the existence of spacelike cmc graphs with −1 ≤ H < 0 [5]. This extends the range of H in the results obtained in [6]. The techniques employed are the maximum principle and the methods to establish a priori C 1 estimates for a cmc graph by comparing with hyperbolic planes as barriers. References [1] H. Bondi, T. Gold, On the generation of magnetism by fluid motion, Monthly Not. Roy. Astr. Soc. 110 (1950) 607–611. [2] D. de la Fuente, A. Romero, P. J. Torres, Radial solutions of the Dirichlet problem for the prescribed mean curvature equation in a RobertsonWalker spacetime. Adv. Nonlinear Stud. 15 (2015), 171–182. [3] S. W. Hawking, G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge, 1973. 36 VIII International Meeting on Lorentzian Geometry Málaga, Spain [4] F. Hoyle, A new model for the expanding universe, Monthly Not. Roy. Astr. Soc. 108 (1948) 372–382. [5] R. López, Spacelike graphs of prescribed mean curvature in the steady state space, preprint. [6] S. Montiel, Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces, J. Math. Soc. Japan 55 (2003), 915– 938. 2 37 Málaga, Spain VIII International Meeting on Lorentzian Geometry Recent results on the oscillator spacetimes Giovanni Calvaruso Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Lecce, Italy. [email protected] ABSTRACT The oscillator group is a four-dimensional connected, simply connected Lie group, whose Lie algebra coincides with the one generated by the differential operators, acting on functions of one variable, associated to the harmonic oscillator problem. After its introduction [R.F. Streater, CMP 1967], the oscillator group has been extended to a one parameter family Gµ (µ > 0) and proved several times to be an interesting object to study both in differential geometry and in mathematical physics. The four-dimensional oscillator group is a well known homogeneous spacetime [R. Duran Diaz, P.M. Gadea and J.A. Oubiña, JMP 1999]. Its bi-invariant metric g0 was generalized to a one-parameter family ga , −1 < a < 1, of leftinvariant Lorentzian metrics. We shall illustrate the following recent results on the geometry of Lorentzian oscillator groups: Ricci solitons. A Ricci soliton is a pseudo-Riemannian manifold (M, g) admitting a smooth vector field X, such that LX g + % = λg, (1) where LX and % respectively denote the Lie derivative in the direction of X and the Ricci tensor and λ is a real number. In [1], we completely solved the system of partial differential equations, which translates (1) in a suitable set of global coordinates on (Gµ , ga ), proving that all these metrics are Ricci solitons (neither invariant, nor algebraic except in an extremely special case). Symmetries. If (M, g) denotes a Lorentzian manifold and T a tensor on (M, g), codifying some either mathematical or physical quantity, a symmetry of T is a one-parameter group of diffeomorphisms of (M, g), leaving T invariant. Hence, it corresponds to a vector field X satisfying LX T = 0. (Examples: Killing vector fields (T = g), curvature collineations (T =R), Ricci collineations (T =%), matter collineations (T = %− 21 τ g is the energy-momentum tensor)). In [2], we obtained a complete classification of symmetries of homogeneous spacetimes (Gµ , ga ), also pointing out the left-invariant examples. References [1] G. Calvaruso, Oscillator spacetimes are Ricci solitons, Nonlinear Analysis, to appear. DOI: 10.1016/j.na.2016.03.008. [2] G. Calvaruso and A. Zaeim, On the symmetries of the Lorentzian oscillator group, submitted. 38 VIII International Meeting on Lorentzian Geometry Málaga, Spain Low dimensional Lie groups with cyclic Lorentzian metrics M. Castrillón López ICMAT (CSIC, UAM, UC3M, UCM) & Dept. Geometrı́a y Topologı́a, Universidad Complutense de Madrid. [email protected] ABSTRACT Cyclic metrics on Lie groups can be considered as those metrics as far as possible from bi-invariant metrics. The name comes from the expression of the condition imposed on the corresponding homogeneous structure tensor. In the Riemannian framework, several rigidity results on cyclic metrics can be found in the literature. This talk will describe the counterpart for Lorentzian metrics on Lie groups of dimensions 3 and 4, which turn out to have a very different behaviour with respect to the metrics of positive signature. This is a joint work with G. Calvaruso. 1 39 Málaga, Spain VIII International Meeting on Lorentzian Geometry Translating Solitons, Semi-Riemannian Manifolds and Lie Groups Miguel Ortega IEMath-Granada, Departamento de Geometrı́a y Topologı́a, Facultad de Ciencias, Universidad de Granada, 18071 Granada (Spain) [email protected] ABSTRACT Famous solutions to the Mean Curvature Flow in Euclidean and Minkowski spaces are the translating solitons, which are submanifolds such that their mean ~ satisfy H ~ = v ⊥ , where v ∈ Rn is a fixed constant unit vector. curvature vector H For simpleness, it is very common to choose v = (1, 0, . . . , 0). These objects have been extensively studied. Now, let (M, g) be a semi-Riemannian manifold, and ∈ {1, −1} a constant. Given a map u : M → R, we say that its graph F : M → (M × R, g + dt2 ) ~ of F satisfies is a (vertical) translating soliton if the mean curvature vector H ~ = ∂ ⊥ . As a first result, when the graph is semi-Riemannian, we obtain the H t PDE that function u must satisfy. Next, we let a Lie group Σ act on M in such a way that the space of orbits M/Σ is diffeomorphic to an open interval I ⊂ R. In this way, the PDE can be tranformed in a ODE. We are able to obtain examples. Some of them were already known, like the rotationally symmetric ones in the Euclidean and Minkowski spaces, but others are new. This a joint work with M.-A. Lawn, Imperial College (UK). 1 40 VIII International Meeting on Lorentzian Geometry Málaga, Spain Lawson correspondence and Laguerre deformation Lorenzo Nicolodi Università degli Studi di Parma, Italy [email protected] ABSTRACT A surface f in Euclidean space R3 is called L-minimal if the first variation R of the surface integral (H 2 − K)/KdA vanishes, where H and K are the mean and Gauss curvatures. The class of L-minimal surfaces is preserved by the Laguerre group which is isomorphic to the restricted Poincaré group [1]. An L-minimal f arises in general as one of the two enveloping surfaces of its Laguerre Gauss map (middle sphere congruence), which defines a spacelike immersion in Minkowski 4-space R41 , away from umbilics and parabolic points. From the Laguerre Gauss map of a surface f , a Laguerre invariant quartic differential Q can be constructed, which is holomorphic if f is L-minimal [2]. A surface f is called generalized L-minimal if the differential Q constructed from its Laguerre Gauss map is holomorphic. We show that f is generalized L-minimal if and only if it is L-minimal or is L-isothermic and locally the T transform (Laguerre deformation) of an L-minimal isothermic surface. This is used to prove that: (1) a surface f is L-minimal isothermic if and only if its Laguerre Gauss map has zero mean curvature in some spacelike, timelike, or isotropic 3-plane of R41 ; and (2) the class of surfaces with holomorphic Q which are not L-minimal consists of surfaces whose Laguerre Gauss maps have constant mean curvature in some translate of hyperbolic 3-space or de Sitter 3space in R41 , or have mean curvature zero in some translate of a time-oriented lightcone in R41 . As an application, we show that various instances of the Lawson isometric correspondence can be viewed as special cases of the T -transformation of L-isothermic surfaces with holomorphic quartic differential. This is based on joint work with E. Musso [3]. References [1] W. Blaschke: Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie, B. 3. Berlin: J. Springer, 1929. [2] E. Musso, L. Nicolodi: A variational problem for surfaces in Laguerre geometry. Trans. Amer. Math. Soc. 348, 4321–4337 (1996). [3] E. Musso, L. Nicolodi: Holomorphic differentials and Laguerre deformation of surfaces. Math. Z., to appear, arXiv:1401.1776 [math.DG]. 1 41 Málaga, Spain VIII International Meeting on Lorentzian Geometry Volume comparison for C 1,1 -metrics Melanie Graf University of Vienna, Faculty of Mathematics. [email protected] ABSTRACT In recent years there has been increased interest in low regularity spacetimes due to a paper by Chruściel and Grant ([1]) showing that there exist nice approximating metrics respecting the causal structure. This sparked a variety of new results for C 1,1 -metrics, among them proofs that the exponential map is still a local bi-Lipschitz homeomorphism and that most results from causality theory as well as both the Hawking and the Penrose singularity theorem remain true in this regularity. In my talk I would like to present a new result in that direction ([2]), namely a generalization of a recent volume comparison theorem for smooth Lorentzian metrics (see [3]) to this regularity. To be more precise, we shall look at globally hyperbolic spacetimes with a C 1,1 -metric having timelike Ricci-curvature bounded from below and establish an upper bound on the volume of (future) balls above compact subsets of smooth, spacelike, acausal and future causally complete hypersurfaces with mean curvature bounded from above. Interestingly, the proof also requires a new result regarding the measurability of the Cut locus in this regularity – generalizing a well-known fact for smooth metrics. These volume estimates then allow us to give an alternative proof of Hawking’s singularity theorem in regularity C 1,1 . Additionally such comparison results open the door for rigidity theorems and may even show a way to turn the tables and define Ricci curvature bounds in terms of such estimates for metrics of even lower regularity. References [1] P. T. Chruściel and J. D. E. Grant, On Lorentzian causality with continuous metrics, Classical Quantum Gravity 29 (2012). [2] M. Graf, Volume comparison for C 1,1 -metrics, Ann. Global Anal. Geom. (2016), doi:10.1007/s10455-016-9508-2. [3] J.-H. Treude and J. D. E. Grant, Volume Comparison for hypersurfaces in Lorentzian manifolds and singularity theorems, Ann. Global Anal. Geom. 43 (2013), 233–241. 42 VIII International Meeting on Lorentzian Geometry Málaga, Spain An overview about Finsler spacetimes and Penrose’s Singularity Theorem Miguel Ángel Javaloyes Victoria Department of Mathematics, University of Murcia [email protected] ABSTRACT Finsler Geometry comes into play whenever there is anisotropy, which means that locally not all the directions are indistinguishable. The lack of anisotropy of the universe has been recently conjectured in the frame of the Extended Standard Model [3], which motivates the study of Finsler spacetimes. We will first discuss the different definitions of Finsler spacetimes and how they have been used throughout the years (see for example [2, Section 3]). Then we will show how it is possible to generalize Penrose’s Singularity Theorem to a very general class of Finsler spacetimes [1]. References [1] A. Aazami and M. A. Javaloyes ,Penrose’s singularity theorem in a Finsler spacetime, Classical Quantum Gravity, 33 (2016), 025003 (22 pages). [2] M. A. Javaloyes and M. Sánchez, Finsler metrics and relativistic spacetimes, Int. J. Geom. Methods Mod. Phys., 11 (2014), p. 1460032 (15 pages). [3] V. A. Kosteleckỳ, Riemann–Finsler geometry and Lorentz-violating kinematics, Physics Letters B, 701 (2011), pp. 137–143. 1 43 Málaga, Spain VIII International Meeting on Lorentzian Geometry On causality in low regularity Clemens Sämann Faculty of Mathematics, University of Vienna [email protected] ABSTRACT The solution theory for Einstein’s equations and physically relevant models of spacetimes (e.g. matched spacetimes, shock and impulsive waves, conical singularities, etc.) lead to metrics of regularity below C 1,1 . This class (i.e., the first derivative of the metric exists and is Lipschitz continuous) is the largest class, where the bulk of classical Lorentzian geometry remains valid. Consequently, developing Lorentzian geometry and causality with metrics below this threshold is a desirable goal for working with spacetimes of low regularity. In this talk we will discuss causality theory for spacetimes with continuous metrics. In particular, we discuss different equivalent notions of global hyperbolicity, the causal ladder and maximal causal curves. In fact, we show that global hyperbolicity implies the existence of maximal causal curves between any two causally related points (the Avez-Seifert theorem). Surprisingly, these maximal causal curves need not be piecewise C 1 . In [LY] it was proven that in a Riemannian manifold with α-Hölder continuous metric, α geodesics (minimizing curves) are C 1,β , where β = 2−α . Thus an analog of [LY] cannot hold in Lorentzian geometry. This is in part joint work with Michael Kunzinger. References [KS] Kunzinger M., Sämann C., Regularity of maximal curves, in preparation [LY] Lytchak A., Yaman A., On Hölder continuous Riemannian and Finsler metrics, Trans. Amer. Math. Soc. 358 7 2006 [S] Sämann C., Global Hyperbolicity for Spacetimes with Continuous Metrics, to appear in Ann. Henri Poincaré, 2016 1 44 VIII International Meeting on Lorentzian Geometry Málaga, Spain Some uniqueness results for the solutions to the HR = HL surface equation Alma L. Albujer Departamento de Matemáticas, Universidad de Córdoba, 14071 Córdoba, Spain [email protected] ABSTRACT Given a domain Ω ⊆ R2 , we consider the differential operator given by ! ! Du Du Q(u) = div p − div p , 1 − |Du|2 1 + |Du|2 where u ∈ C 2 (Ω), and D, div and | · | stand for the gradient, the divergence and the Euclidean norm on R2 . We are interested in studying the solutions to the equation Q(u) = 0, (1) satisfying |Du| < 1. Spacelike surfaces in the Lorentz-Minkowski space L3 can be endowed with two different Riemannian metrics, the metric induced by the Euclidean space R3 and the metric inherited from the Lorentz-Minkowski space L3 . Consequently, we can consider two different mean curvature functions on a spacelike surface, HR and HL . On the other hand, any spacelike surface can be locally described as a spacelike graph over a domain Ω ⊆ R2 . Let Σu be the spacelike graph determined by the function u. It is easy to check that if Σu satisfies HR = HL , then u is a solution of (1) with |Du| < 1. For this reason we will refer to (1) as the HR = HL surface equation. This equation is a quasilinear elliptic partial differential equation, everywhere except at those points at which Du vanishes, where the equation is parabolic. In this talk we will show some uniqueness results for entire solutions to the HR = HL surface equation, as well as for the Dirichlet problem related to it. The results presented in this talk are part of a joint work with Magdalena Caballero [1], and with Magdalena Caballero and Enrique Sánchez [2]. References [1] A. L. Albujer and M. Caballero, Geometric properties of surfaces with the same mean curvature in R3 and L3 , preprint. [2] A. L. Albujer, M. Caballero and E. Sánchez, Some uniqueness results for entire solutions to the HR = HL surface equation, preprint. 1 45 Málaga, Spain VIII International Meeting on Lorentzian Geometry Uniqueness of complete maximal surfaces in certain Lorentzian Warped products Eraldo Almeida Lima Júnior1 1 Profesor at Departamento de Matemática, Universidade Federal da Paraı́ba, João Pessoa, Paraı́ba, Brazil. 58059.900. eraldo[at]mat.ufpb.br ABSTRACT In this presentation we considered complete maximal surfaces in a Lorentzian manifold given by the warped product of the negative definite real line and a 2dimensional surface, such that the Gauss curvature is bounded from below, and the warping function is real-positive. We characterize such surfaces satisfying a comparison involving the height function and the shape operator as slices. These estimates are observed to be optimal as we show in examples. The results are in an homonym pre-print with Alfonso Romero and Adriano Medeiros see [17]. References [1] A.L. Albujer, New examples of entire maximal graphs in H2 ×R1 , Diff. Geom. Appl., 26 (2008), 456–462. [2] A.L. Albujer, L.J. Alı́as, Calabi-Bernstein results for maximal surfaces in Lorentzian product spacetimes, J. Geom. Physics, 59 (2009), 620–631. [3] A.L. Albujer, L.J. Alı́as, Parabolicity of maximal surfaces in Lorentzian product spacetimes, Math. Zeit., 267 (2011), 453–464 [4] A.L. Albujer, L.J. Alı́as, Calabi-Bernstein results and parabolicity of maximal surfaces in Lorentzian product spaces, Recent trends in Lorentzian geometry, Springer Proc. Math. Stat., 26 (2013), 49–85. [5] A.L. Albujer, F.E.C. Camargo H.F. de Lima, Complete spacetime hypersurfaces with constant mean curvature in −R × Hn , J. Math. Anal. Appl., 368 (2010), 650–657. [6] L.J. Alı́as, A.G. Colares, Uniqueness of spacetime hypersurfaces with constant higher order mean curvature in Generalized Robertson-Walker spacetimes, Math. Proc. Camb. Philos. Soc., 143 (2007), 703-729. [7] L. J. Alías and M. Dajczer, Constant Mean Curvature Hypersurfaces in Warped Product Spaces, Proc. of the Edinb. Math. Soc.(Series 2), 50 (2007), 511-526. 46 VIII International Meeting on Lorentzian Geometry Málaga, Spain [8] L.J. Alı́as, M. Dajczer J. Ripoll, A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. Glob. Anal. Geom., 31 (2007), 363–373. [9] L.J. Alı́as, A. Romero and M. Sanchez, Uniqueness of complete spacetime hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Rel. and Gravitation, 27 (1995), 71–84. [10] P. Bérard, R. Sa Earp, Examples of H-hypersurfaces in Hn × R and geometric applications, Mat. Contemp., 34 (2008), 19–51. [11] A. Borbély, A remark on the Omori-Yau maximum principle, Kuwait J. Sci. Engrg., 39 (2012), 45–56. [12] M. Caballero, A. Romero and R. M Rubio, New Calabi-Bernstein Results for Some Elliptic NonLinear Equations, Analysis and Applications, 11 (2013), 1350002 (13 pages). [13] I. Chavel, Eigenvalues in Riemannian Geometry, Pure Appl. Math. 115, Academic Press, New York, 1984. [14] J.M. Espinar H. Rosenberg, Complete constant mean curvature surfaces and Bernstein type theorems in M 2 × R, J. Diff. Geom., 82 (2009), 611–628. [15] J.M. Latorre A. Romero, Uniqueness of noncompact spacetime hypersurfaces of constant mean curvature in generalized Robertson-walker spacetimetimes, Geom. Dedicata, 93, (2002), 1–10. [16] H.F. de Lima, E.A. Lima Jr, Generalized maximum principles and the unicity of complete spacetime hypersurfaces immersed in a Lorentzian product spacetime, Beitr. Algebra Geom., 55 (2013), 59–75. [17] E. A. Lima JR, A. A. Medeiros, A. Romero, Uniqueness of complete maximal surfaces in certain Lorentzian Warped products, preprint [18] E. A. Lima JR, A. Romero, Uniqueness of complete maximal surfaces in certain Lorentzian product spacetimes to apper in Journal of Mathematical Analysis and Applications. [19] G. Li I. Salavessa, Graphic Bernstein results in curved pseudo-Riemannian manifolds,J. Geom. Phys., 59 (2009), 1306–1313. [20] S.Nishikawa, On Maximal spacetime Hypersurfaces in a Lorentzian Manifold, Nagoya, Math. J., 95, (1984), 117–124. [21] H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan, 19, (1967), 205–214. [22] B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London (1983). [23] H. Rosenberg, Minimal surfaces in M 2 × R, Illinois J. Math., 46 (2002), 1177–1195. [24] H. Rosenberg, F. Schulze J. Spruck, The half-spacetime property and entire positive minimal graphs in M × R, J. Diff. Geom., 95 (2013), 321–336. [25] S.T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201–228. 47 Málaga, Spain VIII International Meeting on Lorentzian Geometry Rigidity of geodesic completeness in gravitational wave spacetimes Ivan P. Costa e Silva∗ Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA. [email protected] ABSTRACT We will discuss a recent proof, carried out in collaboration with J.L. Flores and J. Herrera, of the following version of a conjecture by J. Ehlers and K. Kundt [1]: every strongly causal, autonomous, geodesically complete, Ricci-flat, 4-dimensional pp-wave spacetime is isometric to a Cahen-Wallach space. References J. Ehlers and K. Kundt, Exact solutions of the gravitational field equations, in Gravitation: an introduction to current research, L. Witten (ed.), J. Wiley & Sons, New York (1962) pp. 49-101. [1] Visiting Scholar. Permanent address: Department of Mathematics, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis-SC, Brazil. ∗ 1 48 VIII International Meeting on Lorentzian Geometry Málaga, Spain Umbilical properties of spacelike co-dimension two submanifolds Nastassja Cipriani1 , José M. M. Senovilla2 , and Joeri Van der Veken3 1 KU Leuven, Department of Mathematics, Celestijnenlaan 200B – Box 2400, BE-3001 Leuven, Belgium and Fı́sica Teórica, Universidad del Paı́s Vasco, Apartado 644, 48080 Bilbao, Spain [email protected] 2 Fı́sica Teórica, Universidad del Paı́s Vasco, Apartado 644, 48080 Bilbao, Spain [email protected] 3 KU Leuven, Department of Mathematics, Celestijnenlaan 200B – Box 2400, BE-3001 Leuven, Belgium [email protected] ABSTRACT For Riemannian submanifolds of a semi-Riemannian manifold, we introduce the concepts of total shear tensor and shear operators as the trace-free part of the corresponding second fundamental form and shape operators. The relationship between these quantities and the umbilical properties of the submanifold is shown. Several novel notions of umbilical submanifolds are then considered along with the classical concepts of totally umbilical and pseudo-umbilical submanifolds. Then we focus on the case of co-dimension 2, and we present necessary and sufficient conditions for the submanifold to be umbilical with respect to a normal direction. Moreover, we prove that the umbilical direction, if it exists, is unique —unless the submanifold is totally umbilical— and we give a formula to compute it explicitly. When the ambient manifold is Lorentzian we also provide a way of determining its causal character. We end the paper by illustrating our results on the Lorentzian geometry of the Kerr black hole. 1 49 Málaga, Spain VIII International Meeting on Lorentzian Geometry Conformal group actions and their influence on the causal boundary Jónatan Herrera Universidade Federal de Santa Catarina [email protected] ABSTRACT On this talk we will consider the problem of the computation of the ccompletion of an spacetime V where we have a conformal group G acting freely and properly discontinuously on it. Under such conditions, the quotient space M = V /G is again a Lorentz manifold with the induced metric, and then, we can define a principal covering projection π : V → M between both Lorentz manifolds. We will show that, under some mild conditions, previous projection extends to the corresponding completions at the point set, the chronological and the topological level. Concretely, we will give sufficient conditions to ensure that V /G and M are both, homeomorphic and chronologically isomorphic, being V and M the c-completions of V and M respectively. Finally, we will present some examples proving the applicability and optimality of our results. 1 50 VIII International Meeting on Lorentzian Geometry Málaga, Spain Cohomogeneity one actions on Minkowski spaces J. Carlos Dı́az-Ramos University of Santiago de Compostela [email protected] ABSTRACT This talk will be about isometric actions on Minkowski spaces under the assumption that there are orbits of codimension one. We do not assume that the action is proper, as it is usually done in the Riemannian setting. Thus, there are natural examples of isometric actions whose orbits are not closed or whose isotropy groups are noncompact. 1 51 Málaga, Spain VIII International Meeting on Lorentzian Geometry 52 VIII International Meeting on Lorentzian Geometry Posters 53 Málaga, Spain Málaga, Spain VIII International Meeting on Lorentzian Geometry 54 VIII International Meeting on Lorentzian Geometry Málaga, Spain The Björling problem for minimal timelike surfaces in a Lorentzian 3-dimensional Lie group Adriana Araujo Cintra ICET - Universidade Federal de Goiás - Regional Jataí - Goiás - Brazil [email protected] ABSTRACT The Weierstrass representation formula for minimal surfaces in R3 has been a fundamental tool for producing examples and proving general properties of such surfaces, since the surfaces can be parametrized by holomorphic data. In [4] the authors describe a general Weierstrass representation formula for simply connected minimal surfaces in an arbitrary Riemannian manifold. The classical Björling problem, proposed by Björling in 1844, asks for the construction of a minimal surface in R3 containing a given analytic curve β with a given analytic unit normal V along it. The problem was solved by H.A. Schwarz in 1890 by means of an integral formula in terms of β and V . We show how the Weierstrass representation formula can be used, if the ambient manifold is a 3-dimensional Lorentzian Lie group, in order to prove existence and uniqueness of the solution of the Björling problem. References [1] R.M.B. Chaves, M.P. Dussan, M. Magid, Björling problem for timelike surfaces in the Lorentz-Minkowski space, J. Math. Anal. Appl. 337 (2011), 481494. [2] A. A. Cintra, F. Mercuri and Irene I. Onnis,The Björling problem for minimal surfaces in a Lorentzian three-dimensional Lie group, Annali di Matematica Pura ed Applicata (2014), 1 - 16. [3] J.H. Lira, M. Melo, F. Mercuri, A Weierstrass representation for Minimal Surfaces in 3-Dimensional Manifolds, Results. Math. 60 (2011), 311-323. [4] F. Mercuri, S. Montaldo, P. Piu, Weierstrass representation formula of minimal surfaces in H3 and H2 × R, Acta Math. Sinica 22 (2006), 1603-1612. [5] F. Mercuri, Irene I. Onnis, On the Björling problem in a 3-dimensional Lie group, Illinois J. Math, 53 (2), (2009), 431-440. 1 55 Málaga, Spain VIII International Meeting on Lorentzian Geometry Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime Marco Antonio Lázaro Velásquez Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970, Campina Grande, Paraı́ba, Brazil [email protected], [email protected] ABSTRACT This work corresponds to the paper [1]. We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field V . In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field V , we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so-called generalized Robertson-Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett [2]. References [1] Henrique F. de Lima and Marco A. L. Velásquez, Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime, Mathematische Nachrichten 287, No. 11–12 (2014), 1223–1240. [2] J. Jellett, La surface dont la courbure moyenne est constant, J. Math. Pures Appl. 18 (1853), 163–167. 1 56 VIII International Meeting on Lorentzian Geometry Málaga, Spain Translation and homothetical surfaces in Lorentzian space with constant curvature Moruz Marilena Université de Valenciennes et du Hainaut Cambresis,LAMAV, Valenciennes, France [email protected] ABSTRACT In Lorentz-Minkowski space E13 a surface is said to be homothetical if it is the graph of the product of two functions, namely, z = f (x)g(y), where (x, y, z) are the standard coordinates of E13 . These surfaces were introduced in [3] in the problem of finding all minimal homothetical surfaces in E13 . In this talk we consider the problem of finding all non-degenerate (spacelike or timelike) homothetical surfaces with constant Gaussian curvature K. First, we prove that K must be zero ([1]). Then for K = 0, we give a complete description of the parametrizations of such surfaces. Then we also give a new proof that planes and helicoids are the only minimal homothetical surfaces in E13 ([2]). Finally, the corresponding problem is treated in Euclidean space. This is a joint work with Rafael López (University of Granada). References [1] R. López, M. Moruz, Translation and homothetical surfaces in Euclidean space with constant curvature, J. Korean Math. Soc. 52, (2015), 523–535. [2] I. Van de Woestyne, A new characterization of the helicoids, Geometry and topology of submanifolds, V (Leuven/Brussels, 1992), 267–273, World Sci. Publ., River Edge, NJ, 1993. [3] I. Van de Woestyne, Minimal homothetical hypersurfaces of a semi-Euclidean space, Results Math. 27 (1995), 333–342. 1 57 Málaga, Spain VIII International Meeting on Lorentzian Geometry Elastica Constrained Problem in Hypersurfaces of Lorentzian Space Forms Álvaro Pámpano Llarena Department of Mathematics, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Bilbao, Spain. [email protected] ABSTRACT A curve immersed in a pseudo-Riemannian manifold is called an elastic curve if it is a critical point of the bending energy [1]. The purpose of this talk is to study geodesics of hypersurfaces in a Lorentzian space form which are critical curves for the bending energy, but for variations constrained to lie on the hypersurface, the elastica constrained problem [3], [5]. First, the classification into three different types of critical geodesics for the constrained problem will be presented, in terms of their Frenet curvatures [2]. Finally, restricting ourselves to the flat Minkowski space L3 , surfaces which are foliated by critical geodesics of each type will be studied (and classified in two of these cases) [2]. Special emphasis will be put in the warped product metric of Hashimoto surfaces [4], which are foliated by critical geodesics of the third type [2]. References [1] L. Euler, ”Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes”, Sive Solutio Problematis Isoperimetrici Lattisimo Sensu Accepti, Bousquet, Lausannae et Genevae E 65A., O.O. Ser. I, vol 24, 1744. [2] O.J. Garay, A. Pámpano and C. Woo, ”Hypersurface Constrained Elasticae in Lorentzian Space Forms”, Advances in Mathematical Physics, vol. 2015, 2015. [3] O.J. Garay and M. Pauley, ”Critical curves for a Santaló problem in 3-space forms”, J. Math. Anal. Appl., vol. 398, pp. 80-99, 2013. [4] H. Hasimoto, ”Motion of a Vortex Filament and its Relation to Elastica”, J. Phy. Soc. Japan, vol. 31, pp. 293-294, 1971. [5] L.A. Santaló, ”Curvas sobre una superficie extremales de una función de la curvatura y torsión”, Abhndlungen der Hambrugische Universiteit, vol. 20, pp. 216-222, 1956. 1 58 VIII International Meeting on Lorentzian Geometry Málaga, Spain Geometric structures of Hamilton-Jacobi equations associated with metrics Goo Ishikawa∗ ; Yoshinori Machida† ∗ Hokkaido † Numazu University, Sapporo 060-0810, Japan. [email protected] College of Technology, Shizuoka 410-8501, Japan. [email protected] ABSTRACT In this presentation, we associate to an indefinite metric on a space X a Hamilton-Jacobi equation, and construct their complete solutions and general solutions from Legendre manifolds in the space Y of characteristic curves with a contact structure. We especially study the standard (1, 2)-metric (in the case B2 = C2 ), (1, 3)-metric and (2, 2)-metric (in the case D3 = A3 ), via geometric structures of double fibrations. We remark that the null direction bundle Z on X is regarded as an hypersurface of the 1-jet space J 1 (X), which defines a Hamilton-Jacobi equation intrinsically. Then the solutions are regarded as maximal integral submanifolds to the derived system of the canonical distribution on Z. By considering cone structures, we can discuss generalizations of our constructions to general Hamiltonians, second order PDEs, etc. 1 59 Málaga, Spain VIII International Meeting on Lorentzian Geometry Evolutes of curves in the Lorentz-Minkowski plane Masatomo Takahashi Muroran Institute of Technology, Muroran, Japan. [email protected] ABSTRACT We can use a moving frame, as in the case of regular plane curves in the Euclidean plane, in order to define the arc-length parameter and the Frenet formula for non-lightlike regular curves in the Lorentz-Minkowski plane. This leads naturally to a well defined evolute associated to non-lightlike regular curves without inflection points in the Lorentz-Minkowski plane (cf. [1]). However, at a lightlike point the curve shifts between a spacelike and a timelike region and the evolute cannot be defined by using this moving frame. We introduce an alternative frame, the lightcone frame, that will allow us to associate an evolute to regular curves without inflection points in the Lorentz-Minkowski plane. Moreover, under appropriate conditions, we shall also be able to obtain globally defined evolutes of regular curves with inflection points. We investigate here the geometric properties of the evolute at lightlike points and inflection points. This is a joint work with S. Izumiya and M.C. Romero Fuster ([2]). References [1] A. Saloom and F. Tari, Curves in the Minkowski plane and their contact with pseudo-circles, Geom. Dedicata. 159 (2012), 109–124. [2] S. Izumiya, M. C. Romero Fuster, M. Takahashi, Evolutes of curves in the Lorentz-Minkowski plane, Preprint. (2016). 1 60 VIII International Meeting on Lorentzian Geometry Málaga, Spain Null frontal singular surfaces in Lorentzian 3-spaces Goo Ishikawa∗ ; Yoshinori Machida† ; Masatomo Takahashi‡ ∗ Hokkaido University, Sapporo 060-0810, Japan. [email protected] † Numazu College of Technology, Shizuoka 410-8501, Japan. [email protected] ‡ Muroran Institute of Technology, Muroran 050-8585, Japan. [email protected] ABSTRACT Let X be a 3-dimensional Lorentzian manifold (with signature (1, 2)). A smooth map-germ ϕ : (R2 , 0) → X is called a null frontal surface or a null frontal in short if there exists a smooth lift ϕ e : (R2 , 0) → P T ∗ X = Gr(2, T X) of ϕ such that ϕ(t) e is a lightlike plane in Tϕ(t) X and ϕ∗ (Tt R2 ) ⊂ ϕ(t), e for 2 any t ∈ (R , 0). The notion of null frontals is a natural generalization of null immersions to singular surfaces. In this talk we present several classification results of singularities which arise in null frontals up to local diffeomorphisms and up to O(2, 3)-conformal transformations in the conformally flat case (cf. [1]). The classification is achieved by using the fact that null frontals are obtained as tangent surfaces to null curves in X, as well as associated varieties to Legendre curves in the space Y of null geodesics on X. We will mention also higher dimensional cases and related topics (cf. [2][3][4]). . References [1] G. Ishikawa, Y. Machida, M. Takahashi, Asymmetry in singularities of tangent surfaces in contact-cone Legendre-null duality, Journal of Singularities, 3 (2011), 126–143. [2] G. Ishikawa, Y. Machida, M. Takahashi, Geometry of D4 conformal triality and singularities of tangent surfaces, Journal of Singularities, 12 (2015), 27–52. [3] G. Ishikawa, Y. Machida, M. Takahashi, Singularities of tangent surfaces in Cartan’s split G2 -geometry, Asian Journal of Mathematics, 20–2, (2016), 353–382. [4] G. Ishikawa, Y. Machida, M. Takahashi, Dn -geometry and singularities of tangent surfaces, to appear in RIMS Kôkyûroku Bessatsu, (2016). http://eprints3.math.sci.hokudai.ac.jp/2353/ 1 61 Málaga, Spain VIII International Meeting on Lorentzian Geometry On Clifford-Klein forms Aleksy Tralle Department of Mathematics, University of Warmia and Mazury, Olsztyn, Poland [email protected] ABSTRACT Clifford-Klein forms are of significant interest to the relativity theory, since they may yield compactifications of homogeneous pseudo-Riemannian manifolds. However, the problem of their existence is far from being solved. Moreover, there is a general conjecture that they always have a special form (in the literature, the latter is called the Kobayashi Conjecture). We contribute to this conjecture showing that a large class of homogeneous spaces does not admit solvable Clifford-Klein forms. The latter generalizes a theorem of Benoist that there are no non-virtually abelian nilpotent Clifford-Klein forms (see [1] and references therein). References [1] M. Bocheński, A. Tralle, Clifford-Klein forms and a-hyperbolic rank, Internat. Math. Res. Notices IMRN, no. 15(2015), 6267-6285 1 62 VIII International Meeting on Lorentzian Geometry Málaga, Spain Cosmological Horizons: A simple model to clarify some common misconceptions. N. Alamoa) and C. Criadob) a) Dpto. Algebra, Geometria y Topologia, Universidad de Malaga, 29071 Malaga [email protected] b) Dpto. Fisica Aplicada I, Universidad de Malaga, 29071 Malaga, Spain c [email protected] ABSTRACT We use a rubber balloon model to give a simple explanation of the motion of photons and galaxies in an expanding and collapsing universe [1]. In particular, we study some misconceptions regarding the Hubble-sphere, the particlehorizon, the event-horizon, the optical-horizon, the neutrino-horizon, and the gravitational-wave-horizon. One of these confusions is the idea that we can not observe galaxies that have recessional velocities greater than the speed of light. References [1] C. Criado and N. Alamo, Round an expanding world: A simple model to illustrate the kinematical effects of the cosmological expansion, Am. J. Phys. 75, (2007) 331–335. 1 63 Málaga, Spain VIII International Meeting on Lorentzian Geometry Marginally trapped submanifolds in generalized Robertson-Walker spacetimes Verónica L. Cánovas Departamento de Matemáticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain [email protected] ABSTRACT The concept of trapped surfaces was originally formulated by Penrose for the case of 2-dimensional spacelike surfaces in 4-dimensional spacetimes in terms of the signs or the vanishing of the so-called null expansions. More generally, and following the standard terminology in relativity, a codimension two spacelike submanifold is said to be marginally trapped if its mean curvature vector field is lightlike. In this work we consider codimension two marginally trapped submanifolds in the family of general Robertson-Walker spacetimes. In particular we derive some rigidity results for this type of submanifolds which guarantee that, under appropiate hypothesis, the only ones are those contained in slices. We also derive some interesting non-existence results for weakly trapped submanifolds. In particular, we give applications to some cases of phisical relevance such as the Einstein-de Sitter spacetime and certain open regions of de Sitter spacetime, including the so called steady state spacetime. Our results will be an application of the (finite) maximum principle for closed manifolds and, more generally, of the weak maximum principle for stochastically complete manifolds. This is a joint work with Luis J. Alı́as and A. Gervasio Colares. References [1] L. J. Alías, V. L. Cánovas and A. G. Colares, Marginally trapped submanifolds in generalized Robertson-Walker spacetimes, Preprint (2016). 1 64 VIII International Meeting on Lorentzian Geometry Málaga, Spain Homothetic Motions and Surfaces With Constant Curvatures in Lorentz 3-Space Esma Demir Çetin1 , Yusuf Yaylı2 1 Department of Mathematics, Faculty of Science and Arts, Nevşehir Hacı Bektaş Veli University, Nevşehir, Turkey [email protected] 2 Department of Mathematics, Faculty of Science, Ankara University, Ankara [email protected] ABSTRACT In this work we search the surfaces with constant curvatures in Lorentz 3space, whose generating curve is a graph of a polynomial or Lorentzian circle under homothetic motion groups. Also we look for umbilic points and umbilic surfaces. In the first case we show that the degree of the polynomial is 0 or 1, that is the surfaces generated by graph of polynomials are ruled surfaces. But the surfaces, generated by a Lorentzian circle under homothetic motion groups, can not have constant or zero mean curvature. In the second case, we find some generalized umbilic surfaces. References [1] Beneki, C., Kaimakamis, G. and Papantoniou, B. J., Helicoidal Surfaces in Three Dimensional Minkowski Space., J. Math. Anal. Appl. 275, (2002), 586614. [2] Ji, F. and Hou, Z.H., A Kind of Helicoidal Surfaces in 3-Dimensional Minkowski Space, J. Math Anal., Appl. 304, (2005), 632-643. [3] Lopez, R., Differential geometry of curves and surfaces in Lorentz Minkowski space, http://arxiv.org/abs0810.3351 [4] Lopez, R. and Demir, E., Helicoidal Surfaces in Minkowski Space with Constant Mean Curvature and Constant Gauss Curvature, Cent. Eu. J. Math. 12(9), (2013), 1349-1361. [5] Sasahara, N. Spacelike Helicoidal Surfaces with Constant Mean Curvature in Minkowski 3-Space, Tokyo J. Math. 23, (2000), 477-502. [6] Tosun, M., Kucuk, A. and Gungor M. A., The homothetic motions in the Lorentz 3-space. Acta Mathematica Science 26B(4), (2006), 711-719. 1 65 Málaga, Spain VIII International Meeting on Lorentzian Geometry On the fundamental group of a complete globally hyperbolic Lorentzian manifold with a lower bound for the curvature tensor Junichi Mukuno Graduate School of Mathematics, Nagoya University. [email protected] ABSTRACT Calabi and Markus proved that any complete Lorentzian manifold with positive constant curvature of dimension greater than 2 has a finite fundamental group. Kobayashi asked whether the finiteness of the fundamental group still holds if we perturb the metric of positive constant curvature, and proposed a certain conjecture. We reformulate Kobayashi’s conjecture for the Lorentzian case under certain curvature constraints. Moreover we state and prove a theorem ([1]) on certain classes of Lorentzian products with the fiber compact, which is a partial solution of the reformulated conjecture. References [1] J. Mukuno On the fundamental group of a complete globally hyperbolic Lorentzian manifold with a lower bound for the curvature tensor, Differ. Geom. Appl., 41 (2015), 33–38. 1 66 VIII International Meeting on Lorentzian Geometry Málaga, Spain ∇-Einstein connections on Lorentzian spheres Cristina Draper Universidad de Málaga, Spain [email protected] ABSTRACT Every odd dimensional sphere can be endowed with a natural Lorentzian metric. On the contrary to the Riemannian case, these metrics do not satisfy the Einstein condition. Recently, the notion of ∇-Einstein manifold has been introduced in [2] as follows. A semi-Riemannian manifold (M, g) is said to be ∇-Einstein when there exists a metric affine connection ∇ on M with nonzero torsion tensor and sharing geodesics with the Levi-Civita connection such that the corresponding Ricci tensor of ∇ is proportional to the metric g. We provide an exhaustive list of the ∇-Einstein affine connections on the Lorentzian odd dimensional spheres which are invariant for some natural groups of isometries. In any case there exist such connections, and moreover, for some concrete dimensions the family of connections is a big variety. A nice survey on the relevance of the geometries with torsion can be found in [1]. The invariant ∇-Einstein affine connections on the Riemannian odd dimensional spheres have been studied in [3]. References [1] I. Agricola, Nonintegrable geometries, torsion, and holonomy, Handbook of pseudo-Riemannian geometry and supersymmetry, 277–346, IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, 2010. [2] I. Agricola and A.C. Ferreira, Einstein manifolds with skew-torsion, Q. J. Math. 65 (2014), 717–741. [3] C. Draper, A. Garvı́n and F.J. Palomo. Invariant affine connections on odd dimensional spheres. Ann. Global Anal. Geom. 49 (2016), 213–251. 1 67 Málaga, Spain VIII International Meeting on Lorentzian Geometry On the quadric CMC spacelike hypersurfaces in Lorentzian space forms Fábio Reis dos Santos Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970, Campina Grande, Paraı́ba, Brazil [email protected] ABSTRACT This work corresponds to the paper [2]. In 2008, Alı́as, Brasil and Perdomo [1] studied complete hypersurfaces immersed in the unit Euclidean sphere Sn+1 ⊂ Rn+2 , whose support functions with respect to a fixed nonzero vector of the Euclidean space Rn+2 are linearly related. In this setting, they showed that such a hypersurface having constant mean curvature must be either totally umbilical or isometric to a Clifford torus. Here, we deal with complete spacelike hypersurfaces immersed with constant mean curvature in a Lorentzian space form. Under the assumption that the support functions with respect to a fixed nonzero vector are linearly related, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of the ambient space. References [1] L.J. Alías, A. Brasil Jr. O. Perdomo, A characterization of quadric constant mean curvature hypersurfaces of spheres, J. Geom. Anal. 18 (2008), 687–703. [2] F.R. dos Santos, H.F. de Lima C.P. Aquino, On the quadric CMC spacelike hypersurfaces in Lorentzian space forms, to appear in Colloquium Mathematicum (2016). 1 68 VIII International Meeting on Lorentzian Geometry Málaga, Spain Geometric properties of surfaces with the same mean curvature in R3 and L3 Magdalena Caballero Departamento de Matemáticas, Universidad de Córdoba, 14071 Córdoba, Spain [email protected] ABSTRACT Spacelike surfaces in the Lorentz-Minkowski space L3 can be endowed with two different Riemannian metrics, the metric inherited from L3 and the one induced by the Euclidean metric of R3 . It is well known that the only surfaces with zero mean curvature with respect to both metrics are open pieces of the helicoid and of spacelike planes, [2]. We consider the general case of spacelike surfaces with the same mean curvature with respect to both metrics. Our central result states that those surfaces have non-positive Gaussian curvature in R3 , and if the mean curvature does not vanish at a point, then the surface is locally nonconvex at that point. As an application of this result, jointly with an argument on the existence of elliptic points, we present two geometric consequences for those surfaces, and a uniqueness result. This talk is based on a joint work with Alma L. Albujer [1]. References [1] A. L. Albujer and M. Caballero, Geometric properties of surfaces with the same mean curvature in R3 and L3 , preprint. [2] O. Kobayashi, Maximal Surfaces in the 3-Dimensional Minkowski Space L3 , Tokyo J. Math. Vol. 6, No. 2, 1983. 1 69 Málaga, Spain VIII International Meeting on Lorentzian Geometry HIGHER ORDER MEAN CURVATURE ESTIMATES FOR COMPLETE HYPERSURFACES INTO HOROBALLS Antonio Wilson Rodrigues da Cunha Universidade Federal do Piaui-Teresina-Brazil [email protected] ABSTRACT We consider properly immersed two-sided hypersurfaces ϕ : M → N such that ϕ(M ) is contained in a horoball of N , where N satisfies fairly weak curvature bounds and we prove higher order mean curvature estimates that are natural extensions of the estimates obtained by Alias, Dajczer and Rigoli in [1] and Albanese, Alias and Rigoli in [2]. We show that these ambient curvature bounds in the presence of the properness of ϕ guarantees that M satisfies a general version of the weak maximum principle established by Albanese, Alias and Rigoli in [2]. References [1] L. J. Alias, M. Dajczer and M. Rigoli, Higher order mean curvature estimates for bounded complete hypersurfaces, Nonlinear Anal., 84 (2013), 73—83. [2] G. Albanese, L. J. Alias and M. Rigoli, A general form of the weak maximum principle and some applications, Rev. Mat. Iberoam., 29 (2013), 1437–1476. 1 70 VIII International Meeting on Lorentzian Geometry Málaga, Spain Some properties of globally hyperbolic spacetimes with timelike boundary. Luis Alberto Aké Hau Universidad de Málaga [email protected] ABSTRACT In [1] the authors introduced spacetimes-with-timelike-boundary, that is manifolds with boundary such that the induced metric on the boundary is a Lorentzian metric. They extended the notions of strong causality and global hyperbolicity and showed that many results on spacetimes without boundary can be generalized to spacetimes with boundary. In our work we show further properties related with global hyperbolicity. References [1] G. Galloway and D.A. Solis, Global properties of asymptotically de Sitter and Anti de Sitter space-times, Ph.D. thesis, University of Miami, 2006. [2] L. Aké, J.L. Flores and M. Sánchez, Some properties of globally hyperbolic spacetimes with timelike boundary, In progress. 1 71 Málaga, Spain VIII International Meeting on Lorentzian Geometry Constant mean curvature spacelike hypersurfaces in spacetimes admitting a parallel lightlike vector field Author: José Antonio Sánchez Pelegrín, Universidad de Granada, e-mail: [email protected] Type of contribution: poster Abstract In this work, I study constant mean curvature spacelike hypersurfaces in spacetimes that have an infinitesimal symmetry given by a parallel lightlike vector field. In fact, these spacetimes can model radiation propagating at the speed of light . I mainly focus here on global properties of constant mean curvature spacelike hypersurfaces. So, I give a nonexistence result for nonzero constant mean curvature compact hypersurfaces as well as prove that every compact maximal hypersurface in a spacetime obeying the timelike convergence condition is totally geodesic. Furthermore, I also obtain a uniqueness theorem for complete (nonclosed) maximal surfaces in three dimensional spacetimes that also satisfy the previous energy condition. This is a proper generalization of the classical Calabi-Bernstein parametric theorem. References [1] H. Brinkmann, Einstein spaces which are mapped conformally on each other. Math. Ann., 94 (1925), 119-145. [2] J.A.S. Pelegrín, A. Romero and R.M. Rubio, On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field. Classical Quant. Grav., 33 (2016), 055003, 1-8 . 72 VIII International Meeting on Lorentzian Geometry φ-minimal Málaga, Spain graphs in certain manifolds with density Juan J. Salamanca Departamento de Geometría y Topología Universidad de Granada E-mail: [email protected] Abstract Given a manifold with density, the critical points of the weighted area functional are the φ-minimal hypersurfaces. Note that this notion generalizes properly the classical minimal hypersurfaces. We focus on the case the manifold is a warped product with density over a parabolic manifold. In this setting, the class of φ-minimal graphs is associated to a wide family of PDEs. Using geometrical techniques, we provide several uniqueness results for φ-minimal graphs. As application, uniqueness for new Moser-Bernstein type problems are shown. Classical minimal graphs are also considered as a special case. References [1] I.M.C. Salavessa and J.J. Salamanca, Uniqueness of φ-minimal hypersurfaces in warped product manifolds, J. Math. Anal. Appl., 422 (2015), 13761389. [2] A. Romero, R.M. Rubio and J.J. Salamanca, New examples of MoserBernstein problems for some nonlinear equations, (submitted). 73 Málaga, Spain VIII International Meeting on Lorentzian Geometry Completeness of certain accelerated relativistic trajectories Daniel De la Fuente Departamento de Matemática Aplicada (Universidad de Granada) [email protected] ABSTRACT The notions of uniformly accelerated, unchanged direction and circular motion in General Relativity are introduced in the realm of the Lorentzian Geometry [1], [2]. We analyse the completeness of the inextensible trajectories of observers which obey these motions, when the ambient spacetime has certain conformal symmetry. References [1] D. de la Fuente and A. Romero, Uniformly accelerated motion in General Relativity: completeness of inextensible trajectories, Gen. Relativ. Gravit., (2015) 47:33, DOI 10.1007/s10714-015-1879-3 (13 pp.). [2] D. de la Fuente, A. Romero and P.J. Torres, Unchanged direction motion in General Relativity: the problem of prescribing acceleration, J. Math. Phys., DOI:10.1063/1.4935854 (2015), 1450006. 1 74 VIII International Meeting on Lorentzian Geometry Málaga, Spain Surfaces with a common isophote curve in Minkowski 3-space ∗ O. Oğulcan TUNCERa) , İsmail GÖKb) , Yusuf YAYLIc) Ankara University, Faculty of Science, Department of Mathematics, 06100, Ankara, TURKEY a) [email protected], b) [email protected], c) [email protected] ABSTRACT Isophote curve on a surface consists of a locus of surface points that have same light intensity from a given light source. In this study, we investigate the problem of generating a family of surfaces through a given spacelike (or timelike) isophote curve in Minkowski 3-space. Moreover, we give some illustrated examples via Bézier curves and some simple curves. References [1] Farouki R.T., Pythagorean-Hodograph Curves, Algebra and Geometry Inseparable, Springer, Berlin 2008. [2] Dogan F., YaylıY., On isophote curves and their characterizations, Turk J Math., Vol. 39 (2015), 650-664. [3] Dogan F., Isophote curves on timelike surfaces in Minkowski 3-space, Annals of the Al. I. Cuza Iasi Math., (2014), DOI: 10.2478/aicu-2014-0020. [4] Wang GJ., Tank K., Tai CL., Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des., Vol. 36 (2004), 447459. [5] Kasap E., Akyildiz F. T., Surfaces with common geodesic in Minkowski 3space, Appl. Math. Comput., Vol. 117 (2008), 260-270. [6] Poeschl T., Detecting surface irregularities using isophotes, Comput. Aided Geom. Des., Vol. 1 (1984), 163-168. This work was financially supported by University of Ankara, Scientific Research Projects Office (BAP) under Project Number 15H0430008. ∗ 1 75 Málaga, Spain VIII International Meeting on Lorentzian Geometry A curve whose position vector lies on the orthogonal complement of its any Frenet vector in Minkowski n-space Osman ATEŞ(a) , İsmail GÖK(b) , Yusuf YAYLI(c) Ankara University, Faculty of Science, Department of Mathematics (a) [email protected], (b) [email protected], (c) [email protected] ABSTRACT In the Minkowski 3-space, it is well known that rectifying, normal and osculator curves are defined by the property that the position vector of the curve in all points always lie on the orthogonal complement of their principle normal, tangent and binormal vector fields, respectively. The aim of this paper is to give a definition of harmonic curvature functions associate with curve whose position vector lies on the orthogonal complement of its any Frenet vector in Minkowski n-space. References [1] S. Cambie, W. Goemans, I. Bussche Rectifying curves in the n-dimensional Euclidean space, Turk J. Math. 40(2016), 210–223 [2] B.Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 110(2003), 147–152. [3] İ. Gök, Ç. Camcı, H.H. Hacisalihoğlu, Vn −slant helices in Euclidean n−space En , Math. Commun. 14(2009), 317–329. [4] İ. Gök, Ç. Camcı, H.H. Hacisalihoğlu, Vn −slant helices in Minkowski n−space En1 , Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 58(2009), 29–38. [5] K. Ilarslan, E. Nesovic, M. Petrovic-Torgasev, Some characterizations of rectifying curves in the Minkowski 3-space, Novi Sad J Math. 33(2003), 23–32. [6] K. Ilarslan, E. Nesovic, On Rectifying curves as centrodes and extremal curves in the Minkowski 3-space, Novi Sad J Math. 37(2007), 53–64. [7] M. Külahcı, M. Bektaş, M. Ergüt, On harmonic curvatures of a Frenet curve in Lorentzian space, Chaos, Solitons & Fractals 41, 1668–1675. [8] M. Grboviç, E. Nesovic, Some relations between rectifying and normal curves in Minkowski 3-space, Math. Commun. 17(2012), 655–664. [9] R. Uribe-Vargas, On singularities, “perestroikas” and differential geometry of space curves, Enseign. Math. 50(2004), 69–101. 1 76 VIII International Meeting on Lorentzian Geometry Málaga, Spain NORMAL FERMI-WALKER DERIVATIVE IN MINKOWSKI 3-SPACE Özgür KESKİN**, Yusuf YAYLI* Ankara University, Faculty of Science, Department of Mathematics (∗∗) [email protected], (∗) [email protected] ABSTRACT First, in Minkowski 3-Space E13 , we defined normal Fermi-Walker derivative and applied for adapted frame. Normal Fermi-Walker parallelism, normal non-rotating frame and normal Fermi-Walker derivative Darboux vector expressions according to normal Fermi-Walker derivative are given for adapted frame. Being conditions of normal Fermi-Walker derivative and normal non-rotating frame are analyzed for frames along spacelike, timelike, lightlike curves. It is shown that vector field which take part in [4] is normal Fermi-Walker parallel according to the normal Fermi-Walker derivative along the spacelike, timelike and lightlike general helix. Also, we show that the Frenet frame is normal nonrotating frame according to the normal Fermi-Walker derivative. Then, we proved that the adapted frame is normal non-rotating frame along the spacelike, timelike and lightlike general helix. Our aim is to show that the Fermi-Walker definitions can be defined by the first vector of other frames. References [1] R.Balakrishnan, Space curves, anholonomy and nonlinearity. Prama J. Phys. 64(4) (2005) 607-615. [2] I.M. Benn and R. W. Tucker, Wave mechanics and inertial guidance, Phys. Rev. D 39(6) (1989) 1594-1601. [3] M.V. Berry, Proc. Roy. Soc. London A 392 (1984). [4] B.Uzunoglu, İ.Gök and Y.Yayli, A new approach on curves of constant precession, Applied Mathematics and Computation, 275 (2016), 317–323. [5] R. Dandolof, Berry’s phase and Fermi-Walker parallel transport, Phys. Lett. A 139 (1,2)(1989) 19-20. [6] F. Karakuş and Y. Yayli, On the Fermi-Walker derivative and non-rotating frame, Int. Journal of Geometric Methods in Modern Physics,(9,8) (2012) 1250066. [7] F. Karakuş and Y. Yayli, The Fermi- Walker derivative in Lie groups, Int. Journal of Geometric Methods in Modern Physics, 10(7),Article ID 1320011,10p(2013). 77 Málaga, Spain VIII International Meeting on Lorentzian Geometry [8] F. Karakuş and Y. Yayli, The Fermi derivative in the hypersurfaces, Int. Journal of Geometric Methods in Modern Physics,12(1),Article ID 1550002,12p(2015). [9] F. Karakuş and Y. Yayli, The Fermi-Walker derivative in Minkowski 3-Space E13 , 2nd International Eurasian Conference On Mathematics Sciences And Applications, Proceedings(2013). [10] F. Karakuş and Y. Yayli, On the Surface the Fermi- Walker derivative in Minkowski 3-Space E13 , Advances in Applied Clifford Algebras,Springer International Publishing, pp 1-12(2015). [11] F. Karakuş and Y. Yayli, The Fermi-Walker derivative on the Spherical Indicatrix of Spacelike curve in Minkowski 3-Space E13 , Adv. Appl. Clifford Algebras, Springer International Publishing, Article DOI 10.1007/s00006-0150635-9(2016). [12] E. Fermi, Atti Accad. Naz. Lincei Cl. Sci. Fiz. Mat. Nat. 31 (1922) 184-306. [13] S.W.Hawking and G.F.R. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, 1973). [14] M. Crasmareanu and C. Frigioiu, Unitary vector fields are Fermi-Walker transported along Rytov-Legendre curves, Int. Journal of Geometric Methods in Modern Physics,(12) (2015) 1550111. [15] C. Calin and M. Crasmareanu, Slant Curves and Particles in three- dimensional Warped Products and their Lancret invariants, Bulletin of the Australian Mathematical Society,(88)Issue 01,(2013),128-142. [16] P.D. Scofield, Curves of Constant Precession,The American Mathematical Monthly(102),6(1995),531-537. [17] R.Lopez, Differential Geometry of Curves and Surfaces in Loretz-Minkowski Space, arXiv: 0810.3351[math. DG],(2008). 78 VIII International Meeting on Lorentzian Geometry Málaga, Spain A Quaternionic Representation for Canal Surfaces in Minkowski 3-Space Erdem KOCAKUŞAKLIa) , O. Oğulcan TUNCERb) , İsmail GÖKc) , F. Nejat EKMEKCİd) Ankara University, Faculty of Science, Department of Mathematics, 06100, Ankara, TURKEY a) [email protected], b) [email protected], c) [email protected], d) [email protected] ABSTRACT In this paper we have defined the canal surfaces determined by spherical indicatrices of any spatial curve in Minkowski 3-space, via the timelike and spacelike split quaternions. Moreover, using the orthogonal matrices corresponding to these quaternions, these surfaces have been obtained as homotetic motions. Then, we have investigated the ralationship between the canal surfaces and unit quaternions. Finally, we give some related examples with their figures. References [1] Aslan S. and Yayli, Y.: Split Quaternions and Canal Surfaces in Minkowski 3-Space, Adv. Appl. Clifford Algebras, (2015), DOI 10. 1007/s00006-0150602-5. [2] Bekar, M. and Yayli, Y.: Involutions of Complexified Quaternions and Split Quaternions, Advances in Applied Clifford Algebras, 23 (2013), 283-299. [3] Hamilton, W.R.: On Quaternions; or on a new system of imagniaries in algebra. Lond. Edinb. Dublin. Philos. Mag. J. Sci. 25(3), 489-495 (1844). [4] Izumiya, S. and Takeuchi, N.: New special curves and developable surfaces,Turk.J.Math.28(2004)153-163. [5] Kula, L. and Yayli, Y., : Split quaternions and rotations in semi Euclidean space E42 , J. Korean Math. Soc. 44(6) (2007) 1313-1327. [6] Ozdemir, M. and Ergin, A,A., : Rotations with unit timelike quaternions in Minkowski 3-space, J. Geom. Phys. 56(2) (2006), 322-326. [7] Tosun, M., Kucuk, A. and Gungor, M.A.: The homothetic motions in the Lorentzian 3-space, Acta Math. Sci. 26B (4) (2006), 711-719. [8] Yilmaz, B. , Gok, I. and Yayli, Y.: Extended Rectifying curves in Minkowski 3-space, Adv. Appl. Clifford Algebras, (2016), DOI 10. 1007/s00006-015-06377. 1 79 Málaga, Spain VIII International Meeting on Lorentzian Geometry Lorentzian generalized quasi-Einstein metrics and static structures M. Brozos-Vázquez, E. Garcı́a-Rı́o, X. Valle-Regueiro MBV: Departmento de Matemáticas, Escola Politécnica Superior, Universidade da Coruña, Spain EGR-XVR: Faculty of Mathematics, Universidade de Santiago de Compostela, Santiago de Compostela, Spain [email protected] [email protected] [email protected] ABSTRACT A Lorentzian manifold (M, g) is a generalized quasi-Einstein space if there is a solution f ∈ C ∞ (M ) of the equation [2] Hesf +ρ − µdf ⊗ df = λg (1) for some µ ∈ R and λ ∈ C ∞ (M ). Generalized quasi-Einstein metrics contain Einstein metrics, gradient Ricci solitons and gradient Ricci almost solitons as special cases. There are, however, other important situations when µ 6= 0. In such a case, Equation (1) is equivalent to Hesu −µuρ = −µλug (2) 1 and corresponds to conformally Einstein metrics [1] (when µ = − n−2 and λ = τ n−2 ∆u + , τ being the scalar curvature and n = dim M ) and to the static n u n ∆f 1 structures (when µ = 1 and λ = n (τ − f )). Our purpose is to present some new results on the classification of locally conformally flat generalized quasi-Einstein Lorentzian manifolds. References [1] Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul; Conformal invariants associated to a measure. Proc. Natl. Acad. Sci. USA 103 (2006), 2535–2540. [2] Jauregui, Jeffrey L.; Wylie, William; Conformal diffeomorphisms of gradient Ricci solitons and generalized quasi-Einstein manifolds. J. Geom. Anal. 25 (2015), 668–708. [3] Lafontaine, Jacques; A remark about static space times. J. Geom. Phys. 59 (2009), 50–53. [4] Lafontaine, Jacques; Sur la géométrie d’une généralisation de l’équation différentielle d’Obata. J. Math. Pures Appl. (9) 62 (1983), 63–72. [5] Qing, Jie; Yuan, Wei; A note on static spaces and related problems. J. Geom. Phys. 74 (2013), 18–27. 1 80 VIII International Meeting on Lorentzian Geometry Notes 81 Málaga, Spain Málaga, Spain VIII International Meeting on Lorentzian Geometry Notes 82
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