THE MATHEMATICS OF
F. J. ALMGREN, JR.
Brian White
February 24, 1998 (revised March 11, then January 16, 1999)
Frederick Justin Almgren, Jr, one of the world’s leading geometric analysts and
a pioneer in the geometric calculus of variations, died on February 5, 1997 at the
age of 63 as a result of myelodysplasia. Throughout his career, Almgren brought
great geometric insight, technical power, and relentless determination to bear on a
series of the most important and difficult problems in his field. He solved many of
them, and, in the process, discovered ideas which turned out to be useful for many
other problems. This article is a more-or-less chronological survey of Almgren’s
mathematical research. (Excerpts from this article appeared in the December 1997
issue of the Notices of the American Mathematical Society.) Almgren was
also an outstanding educator, and he supervised the thesis work of nineteen PhD
students; the 1997 volume 6 issue of the journal Experimental Mathematics is
dedicated to Almgren and contains reminiscences by two of his PhD students and by
various colleagues. A general article about Almgren’s life appeared in the October
1997 Notices of the American Mathematical Society [MD]. See [T3] for a brief
biography.
Early Career
Almgren began his graduate work at Brown in 1958. It was a very exciting
place and time for geometric measure theory. Wendell Fleming had just arrived
and begun his collaboration with Herbert Federer, leading to their seminal paper
Normal and Integral Currents in 1960 [FF]. Among the major results in Normal and Integral Currents was a compactness theorem that implied existence of
k-dimensional rectifiable area minimizing varieties with prescribed boundaries in
Rn . (Similar existence theorems were proved independently by Reifenberg and, in
case n = k + 1, by De Giorgi.) Shortly afterward, Fleming (using earlier work of
Reifenberg) proved that if k = 2 and n = 3, then the varieties are in fact smooth
surfaces. Meanwhile De Giorgi (and subsequently, by a different argument, Reifenberg) did work implying that the varieties were smooth almost everywhere when
n = k + 1, the case of hypersurfaces.
I would like to thank W. Allard, H. Federer, W. Fleming, E. Lieb, F. Morgan, and J. E.
Taylor for helpful conversations and comments. This article was prepared with funding from NSF
grant DMS-95-04456. The article is printed here with permission of the editors of the Journal of
Geometric Analysis, where it will appear in a special issue devoted to the work of F. J. Almgren,
Jr.
Typeset by AMS-TEX
1
Fred came to Brown with an unusual background. He had just spent three
years as a Navy pilot, and before that his undergraduate degree at Princeton had
been not in mathematics but in engineering. In fact, as an undergraduate he took
only three math courses: two semesters of honors calculus and one semester of
differential equations and infinite series. Later he would jokingly accuse various
mathematicians at Brown of calling him the most ignorant person that they had
ever met.
It was clear that he had great raw talent and good intuition, says Federer,
but indeed he knew very little mathematics then. There were even basic things in
group theory, for example, that he had never heard of. When he asked me to be
his advisor, I suggested the problem I did because he didn’t know enough analysis
for most problems in geometric measure theory.
Federer suggested a problem that was as much topological as measure theoretic.
Four years earlier, Dold and Thom had showed that there was a natural isomorphism between the homology groups of a compact manifold M and the homotopy
groups of their associated symmetric product spaces. Federer realized that this
could be interpreted as a statement about the homotopy groups of the space of 0
dimensional integral cycles of M . He conjectured a generalization to k-dimensional
cycles, namely that the mth homotopy group of the space of integral k-cycles in M
is naturally isomorphic to the (m + k)th homology group of M . Almgren’s thesis,
published in the first volume of the journal Topology [1], proved that this was
indeed the case.
Oddly enough, its publication caused trouble. Almgren was supposed to sign the
copyright of his thesis over to Brown University, but could not since he had already
signed it over to Topology. To the dean, that meant Almgren could not graduate.
Federer actually had to go before the graduate council to have the dean overruled.
After his PhD, Almgren took a position at Princeton University. There he remained for the rest of his life, even taking most of his leaves of absence at the
Princeton Institute for Advanced Study, though he attended conferences and made
summer visits to institutes throughout the world.
After Almgren finished his thesis, Federer suggested that he develop a Morse theory for minimal varieties, analogous to the well-developed theory of closed geodesics.
In particular, although the Federer-Fleming paper proved existence of area minimizing varieties (integral currents) in homology classes of compact riemannian
manifolds, there were no theorems asserting the existence of unstable minimal varieties.
Almgren’s Topology result was a step in that direction, but it was not sufficient.
For although the integral currents introduced by Federer and Fleming had proved
to be ideally suited to the problem of minimizing area, they were not so suitable for
problems in which the solutions are unstable critical points of the area functional.
This is, in part, because the area functional is not continuous, but merely lowersemicontinuous, on the space of integral currents. To prove the existence of a
minimum, lower-semicontinuity suffices. But if one wishes to prove existence of a
critical point by a mountain pass lemma, having only lower semicontinuity poses
difficulties.
To handle these difficulties, Almgren turned to the class of surfaces that he
2
called integral varifolds1 . The area functional is continuous (with respect to weak
convergence) on the space of integral varifolds, and he was able to carry out a
Morse theory with them. In particular, he proved that every compact m-manifold
contains stationary integral varifolds of each dimension less than m. He also proved
a compactness theorem for integral varifolds (The Banach-Alaoglu theorem trivially
implies a compactness theorem for all varifolds, but the compactness for integral
varifolds is quite subtle.) He also proved a striking isoperimetric inequality for
stationary varifolds, about which I will say more later.
All this was done in Almgren’s 1965 Theory of Varifolds, mimeographed lecture
notes that, amazingly, were never published. Why not? Jean Taylor (Almgren’s
first PhD student and later wife) thinks it was submitted, perhaps to Acta Mathematica, and rejected as too long. Presumably, Almgren could have then published
it elsewhere. However, over the next few years, simpler and better definitions of
both varifold and first variation of a varifold were discovered, so that perhaps Almgren would have wanted to revise his Theory considerably before publishing it.
After several years, Allard (inspired in part by Almgren’s 1968 ellipticity paper,
described below) proved his celebrated regularity theorem, which implies that if V
is a stationary integral varifold, then there is an open dense subset of V that is
a smooth submanifold. Allard also streamlined and extended much of Almgren’s
work, and, with Almgren’s encouragement, published a full account of most aspects
of varifolds in his 1972 paper [Al1], which has become the standard reference on
the subject.
One aspect of the theory Allard did not describe was Almgren’s Morse-theoretic
theorem that every riemannian manifold contains stationary varifolds. Two years
later, Almgren’s second PhD student, Jon Pitts, wrote a thesis showing that, for the
case of two dimensional varifolds in 3-manifolds, the varifolds are actually smooth
surfaces. Later Pitts extended the result to m-dimensional surfaces for m ≤ 5 in
his book [P], the first chapter of which is a very clear, non-technical discussion of
Almgren’s morse-theoretic work together with Pitts’ extensions of it.
Early Regularity Results
Though Almgren did not in the 60s prove a regularity theorem for stationary
varifolds, he was making important advances in regularity theory. To describe them
accurately, it is necessary to distinguish between two possible interpretations of the
area of a surface. If two portions of a surface overlap, should the region of overlap
be counted once or twice? If we count it once, we find the size of the surface, that
is the area ignoring multiplicities. If we count it twice, we get the mass, the area
weighted by the multiplicity. Both mass-minimizing surfaces and size-minimizing
surfaces may be regarded as solutions to the Plateau problem, and although many
of their properties are essentially the same, mass-minimizing surfaces tend to have
smaller singular sets than do size-minimizers. Almgren’s first regularity paper [7]
1 Essentially
the same class of surfaces had been introduced in 1951 by L. C. Young under the
name ”generalized surfaces” [Y2]. That term is perhaps confusing since there are many other
generalizations of the familiar notion of surface: integral currents, integral flat chains, normal
currents, real flat chains, flat chains mod p, and so on. Indeed, Young himself had first used the
term generalized surface in a somewhat different context [Y1].
3
proved that 3-dimensional mass-minimizing surfaces (integral currents) in R4 have
no singularities away from the boundary. As mentioned earlier, the corresponding
result for 2-dimensional surfaces in R3 had been proved by Fleming. It was also
known that if there were any examples of singular mass-minimizing hypersurfaces
in R4 , then there would have to exist an example that was a cone regular except at
the origin. In particular, the intersection of the cone with the unit sphere S2 would
be a regular embedded oriented minimal surface M in S3 . Almgren ruled out such
minimizing cones as follows:
(1) First, he showed that there are no minimal immersed 2-spheres in S3 other
than the standard equatorial sphere (the cone over which is a plane and
therefore not singular.) He proved this in the same way that Hopf had
proved that the image of a constant mean curvature immersions of S2 in
R3 must be a standard round sphere.
(2) Then he showed that if the minimal surface M ⊂ S3 is not a sphere, then
the corresponding cone must be unstable. By separation of variables, the
cone is unstable if and only if the first eigenvalue λ1 of the jacobi (or second
variation) operator on M is below a certain critical value. He proved that
λ1 is below the critical value by applying the jacobi operator to the constant
unit normal vectorfield to M ⊂ S3 . One term of the resulting expression
is the integral of the scalar curvature; by Gauss-Bonnet, this term must be
≤ 0 (since M is not a sphere), which gives the necessary bound on λ1 .
Almgren’s theorem was superseded two years later by J. Simons, who proved
that there are non-planar stable minimal hypercones in Rn if and only if n ≥ 8,
which implies that mass-minimizing hypersurfaces of dimension ≤ 6 are smooth
away from their boundaries [SJ]. (In 1969, Bombieri, De Giorgi, and Giusti [BDG]
showed that Simons’ regularity result is sharp: there are singular mass-minimizing
hypersurfaces of dimension 7 and higher). However, Almgren’s result on minimal 2spheres remains quite interesting and does not follow from Simons’ theorem. Shortly
afterwards Calabi (apparently unaware of Almgren’s work) reproved the theorem
about minimal 2-spheres in S3 and also proved interesting results about minimal
2-spheres in Sn [Ca].
In the same paper, Almgren also discussed regularity of 2-dimensional area minimizing flat chains mod 2 in Rn . These are surfaces that minimize area among
all surfaces, with no restriction on genus or orientability. (Integral currents come
equipped with orientations.) He proved that such surfaces are smoothly immersed
with only isolated self-intersections. The key observation is that any tangent cone
must consist of a collection of multiplicity one planes that intersect only at the
origin.
So far the tremendous advances that had been made in the higher-dimensional
calculus of variations had been limited to the area functional. In his 1968 paper [9],
Almgren introduced the class of parametric elliptic functionals, and (extending the
techniques pioneered by De Giorgi) proved the fundamental regularity theorem: if a
minimizing surface is weakly close to a multiplicity one disk, then it is smooth near
the center of the disk. This implies that minimizing surfaces are smooth almost
everywhere (if multiplicity is not counted) or on an open dense set (if multiplicity
is counted).
4
Robert Hardt proved the analog of Almgren’s theorem for boundary points [Har].
Many years later, Bombieri [Bo] and Schoen and Simon [SS] gave different proofs of
Almgren’s theorem, replacing some of Almgren’s barehanded geometric constructions by more standard PDE techniques. However, the later proofs are limited to
oriented surfaces, whereas Almgren’s is not.
Almgren’s regularity theorem is used in almost all subsequent work on parametric
elliptic functionals.
Morrey [MCB] and much later Evans [E] applied ideas from Almgren’s paper to
systems of partial differential equations.
Soap-Film-Like Surfaces
Almgren’s next major work on regularity was his 1975 monograph, Existence
and regularity almost everywhere of solutions to elliptic variational problems
with constraints [19]. There were two major new features.
The first was that it developed a regularity theory for sets rather than currents.
The theory of integral currents, as well as the classical theory of Douglas and Rado,
can be regarded as providing solutions to Plateau’s problem. But neither theory
very accurately models soap films and soap bubbles, which were what Plateau
actually studied. For example, the classical Douglas-Rado solutions are sometimes
immersed, which real soap films never are. Conversely, soap films display types of
singularities that are present neither in the Douglas-Rado minimal surfaces nor in
mass-minimizing integral currents. (Reifenberg’s theory modeled soap films rather
well, but Almgren wanted to handle soap bubbles as well as films, and arbitrary
elliptic functionals, not just the area functional.) The difficulty with currents, as
Almgren saw it, was that they had structure that the the physical surfaces did not.
In this paper, he wrote, we do not wish to assume the existence of a boundary
operator . . . because in many of the geometric, physical, and biological phenomena
to which the results and methods [of this paper] are applicable there seems to be
no natural notion of such a boundary operator.
The lack of a boundary operator meant that comparison surfaces could no longer
be constructed by cut-and-paste operations. Instead, all comparison surfaces had
to be obtained by (not necessarily one-to-one) lipschitz deformations of the original
surface. This limitation meant that proofs were rather cumbersome compared to
those in Almgren’s earlier work on elliptic regularity.
Recent models [Br3, AmB, W6] of some of the physical surfaces Almgren was
interested in (e.g. soap films and soap bubble clusters, and immiscible fluid interfaces) do incorporate more structure, including a boundary operator, leading in
some cases to simpler existence and regularity proofs. To what extent these models
can replace Almgren’s models remains to be seen.
The second major new feature of the paper is the extremely useful idea of surfaces
that almost minimize a functional. Given an elliptic functional F , in addition to
the basic problem of finding a surface that minimizes F , there are many closely related problems of interest: minimize F among surfaces enclosing a specified volume,
or among surfaces in a riemannian manifold (rather than in RN ), or among surfaces
that avoid a specified smooth obstacle, etc. Almgren realized that a solution surface S to any one of these problems must be what he called (F, , δ)-minimizing.
5
Roughly speaking, the portion of S in a small ball has F -energy close to the smallest possible F -energy (ignoring volume constraints, obstacles, etc.); the smaller the
ball, the closer the F -energy to the minimum possible.
Consider, for example, a circle and the length functional. A small arc of the
circle comes close to minimizing length in that the ratio of its arclength to the chord
is only slightly greater than one, and indeed the ratio tends to one as the length
tends to 0.
It turns out that the fundamental regularity theorem (a minimizing surface
weakly near a multiplicity one disk is regular near the center of the disk) for minimizing surfaces is also true and only slightly more difficult to prove for (F, , δ)minimizing surfaces. (Here regular means C 1,α for a suitable α > 0.) Thus Almgren
simultaneously handled the various problems mentioned above.
(Of course there are more delicate questions whose answers depend on the particular problem. For example, for the obstacle problem, one can ask about the
smoothness of the set along which the surface meets the obstacle.)
Almgren also introduced a second, weaker notion of almost minimality. A set S
is called (γ, )-restricted if for every open set W (such as a ball) of diameter < ,
the area of S ∩ W is not more than γ times the minimum area (of any surface
obtained by deforming S ∩ W , leaving S ∩ ∂W fixed.) Almgren proved the beautiful
result that any such set S must be rectifiable and can in fact be approximated in a
strong sense by a polyhedral complex. This is analogous to (but much harder than)
the PDE theorem (proved 8 years later) that a function that quasi-minimizes the
dirichlet energy is holder continuous [GiG]. The notion of (F, , δ)-minimality was
also later applied in PDE theory [Anz].
A special case of the results in Almgren’s monograph is the existence and almost
everywhere regularity of surfaces in R3 that accurately model physical soap-films
and soap bubble clusters. Jean Taylor wrote a thesis [T1] leading to her celebrated
1976 theorem [T2] that Almgren’s soap-bubble-like surfaces do indeed have exactly
the structure described by Plateau: they consist of smooth constant mean curvature
surfaces, which meet in threes along smooth curves, which in turn meet in fours at
isolated points. Almgren and Taylor described their work in a beautiful Scientific
American article [21] on the physics and mathematics of soap films.
Further work on Elliptic Regularity
In 1977, Almgren, together with Rick Schoen and Leon Simon, wrote one more
important paper on elliptic regularity [ASS]. At that time, there was a big gap
between what was known about minimizing hypersurfaces for general elliptic functionals and what was known for the special case of the area functional. Whereas
m-dimensional hypersurfaces minimizing mass could have singular sets of dimension
at most m − 7, those minimizing other elliptic functionals (counting multiplicities)
were only known to be regular almost everywhere. That is, the singular sets had zero
m-dimensional measure, but conceivably could be m-dimensional. The AlmgrenSchoen-Simon paper closed the gap somewhat: they proved that the singular set
must have (m − 2)-dimensional measure 0.
Incidentally, a little later Almgren realized that from knowing that (for a given
elliptic functional) the singular sets all have (m − 2)-dimensional measure 0, one
6
can actually deduce by a general principle that their dimensions are bounded by
(m − 2 − ) for some > 0. Almgren did not attach much importance to this
principle, as it does not give a bound on , and he never wrote it down. When I
needed to use it later, he let me publish it as the work-raccoon lemma in [W3].
The principle was later used by Hardt, Kinderlehrer, and Lin [HKL] in some work
on liquid crystals.
We still do not know the maximum possible dimension of the singular set, but
14 years later Frank Morgan (Almgren’s 6th doctoral student) showed that the
(m − 2 − ) result is close to optimal: he proved that there are examples for which
the singular set has dimension (m − 3) [MF2].
Uniqueness of tangent cone
Once one knows that singularities occur in minimal varieties, one naturally wonders what the singularities are like. The first answer, already known from the 1960
Federer-Fleming paper, is that they weakly resemble cones. More precisely, if the
origin (for example) is an interior point of a minimal variety M , then any sequence
of dilates of M will have a subsequence that weakly converges to a minimal cone
(that is, a minimal variety invariant under dilations). Unfortunately, the simple
proof leaves open the possibility that a different subsequence might converge to a
different cone.
In other words, a minimal variety looked at under a microscope will resemble a
cone, but under higher magnification, it might (as far as anyone knows) resemble
a completely different cone. Whether this ever happens remains perhaps the most
fundamental open question about singularities of minimal varieties.
Almgren and Allard proved the first major result about uniqueness of tangent
cone. They considered the case of a strongly isolated singularity, that is, a point at
which at least one of the tangent cones (call it C) is a multiplicity one cone whose
only singularity is at the vertex. This is certainly an interesting class of singularities;
it essentially includes, for example, all singularities in mass-minimizing hypersurfaces in R8 (the lowest dimensional space in which mass-minimizing hypersurfaces
can have singularities.)
Of course the intersection of C with the unit sphere around the vertex is a
smooth minimal submanifold M of the sphere. Almgren and Allard proved that if
M satisfies a certain condition, namely integrability of jacobi fields, then C is the
only tangent cone at the point in question.
They knew that the integrability condition held for a number of minimal cones,
including what is still the only known example of an area minimizing hypercone in
R8 . However, Dan Burns came up with example in which the integrability does
not hold. A little later Allard calculated that when M is a product of k spheres (of
various dimensions), the integrability holds if and only if k ≤ 2.
There are no hypercones for which the integrability is known to fail, so it is
conceivable that the integrability may hold for all hypercones. Unless a cone is
homogeneous (that is, unless its intersection with the sphere has a transitive group
of isometries), it is quite difficult to determine whether or not the integrability
condition holds. However, Bruce Solomon (Almgren’s 9th doctoral student) has
proved that it does hold for an interesting class of nonhomogeneous hypercones
[So2].
7
In 1983, Leon Simon proved that one could drop the integrability hypothesis
from the Allard-Almgren theorem [SL1, SL2]. However, the Allard-Almgren proof
remains important for two reasons. First, when the integrability hypothesis is
satisfied, their proof shows that the dilated surfaces converge to the cone much more
quickly than is asserted by Simon’s theorem. Indeed, the integrability condition is
actually necessary and sufficient for such rapid convergence. (Precisely, if a cone C
does not satisfy the integrability condition, then there is a minimal variety whose
dilates about a point converge slowly to C [AdS].)
Both Simon’s theorem and the Allard-Almgren theorem also apply to singularities of harmonic maps (and some other geometric problems) [SL2]. However
(and this is the second reason the Allard-Almgren theorem remains important),
Simon’s proof requires that the target manifold be real-analytic, whereas the AllardAlmgren proof does not. (Indeed, there is an example [W6] of a harmonic map into
a C ∞ manifold with a continuum of tangent-cone maps at a strongly isolated singularity.) For maps of 3-dimensional domains into 2-dimensional C ∞ manifolds (the
lowest dimensions for which singularities occur), the Allard-Almgren hypotheses
hold at every singular point [GW].
For a streamlined treatment of both the Allard-Almgren and Simon theorems,
see [SL2].
Surfaces of higher codimension
In the early 1970s, very little was known about regularity of mass-minimizing
surfaces of codimension greater than one. The singular set of such an m-dimensional
surface was known to be closed and nowhere dense, but it was not known to have
m-dimensional measure 0. Around 1974, Almgren started on what would become
his most massive project, culminating ten years later in a three-volume, 1700 page
proof that the singular set not only has m-dimensional measure 0, but in fact has
dimension at most (m−2) [34]. This dimension is optimal, since by an earlier result
of Federer, there are many examples in which the dimension of the singular set is
exactly (m − 2). In 1986, Sheldon Chang, Almgren’s eleventh PhD student, proved
in his 1986 thesis that for the case m = 2, the singular set is not only 0-dimensional
(which some cantor sets are), but is locally finite away from the boundary [Cha1].
The reader may wonder why Almgren’s theorem is so much more complicated
than all the regularity results that preceded it. Perhaps the most fundamental
difference is the following. For the various surfaces (e.g. mass-minimizing hypersurfaces, size minimizing surfaces, mass-minimizing surfaces mod 2) in which almost
everywhere regularity was already known, one can determine whether a point is a
regular point just by examining its tangent cone: the point is regular if and only if
it has a plane as a tangent cone.
For mass-minimizing surfaces of higher codimension, this is no longer true. Suppose for example that a two-dimensional mass-minimizing surface has as a tanget
cone (at a certain point) a plane with multiplicity 2. The point could be a regular
point, namely part of a smooth multiplicity 2 surface, or it could be a singular
branch point (such as the origin in the surface {(z 2 , z 3 ) : z ∈ C} ⊂ C2 ∼
= R4 .)
Thus the problem is multiplicity. If a tangent cone is a plane with multiplicity
one, the standard regularity theory applies. In codimension 1, multiplicity is not a
8
problem: an integral current weakly close to a multiplicity k disk can be decomposed
into layers (from top to bottom), each of which is close to the same disk with
multiplicity 1. But in codimension greater than 1, such decomposition is impossible
because there is no appropriate analog of top and bottom.
To get an idea of how subtle Almgren’s theorem is, imagine an m-dimensional
mass-minimizing surface consisting of two distinct smooth pieces, which intersect
and are tangent to each other along a set S. Almgren’s theorem asserts in this
case that the set S has dimension ≤ (m − 2), and thus it can be regarded as a
delicate unique continuation theorem. Now imagine that the two pieces are actually
joined by infinitely many little handles near S. The surface can no longer be locally
represented by graphs of functions solving a PDE, but the same conclusion about
S has to be reached.
Since Sheldon Chang is writing an article about Almgren’s paper, I will not try
to describe the proof. The enormity of the paper has deterred all but a few people
from reading very much of it. Nevertheless, some ideas from it have begun to have
an impact.
One important ingredient is a theorem that stratifies minimal varieties by tangent
cone type. Roughly speaking, it says that if a cone is very complicated, then it
can be the tangent cone at only a small set of points2 . This has the following
extremely useful consequence. To prove that some property holds except on a set
of small dimension, it is not necessary first to classify all tangent cones. Rather, it
is sufficient to classify the simplest tangent cones and analyze the behavior at the
points where those tangent cones occur. In the case of Almgren’s (m − 2) result,
it meant that he only needed to worry about those points having a plane (with
multiplicity) as a tangent cone.
The stratification theorem has been generalized to harmonic maps and mean
curvature flows, and has been used to prove results about partial regularity and
about structure of singular sets [SL36, W7].
Another important ingredient is the monotonicity of frequency formula. A
special case is the following. Consider a harmonic function on R2 , and express it in
terms of polar coordinates: u(r, θ). If we fix r, we can expand the resulting function
of θ as a Fourier series. Now as r decreases, the high frequency terms in the Fourier
series die off faster than the low frequency terms.
N. Garafolo and Fang-Hua Lin have used the monotonicity of frequency to prove
unique continuation for solutions of elliptic partial differential equations even when
the coefficients are only lipschitz [GaL]. Later Lin used it bound the size of nodal
sets of eigenfunctions of laplacian on compact manifolds [L]. Gromov and Schoen
also used it in their analysis of harmonic maps into polyhedral complexes [GS].
A third important ingredient involves multivalued functions. In earlier regularity
proofs (starting with De Giorgi), a minimal variety near a multiplicity one disk
would be approximated by the graphs of a harmonic function. Almgren realized
that a minimal variety near a multiplicity k disk could be well approximated by
the graph of a multivalued function minimizing a suitable analog of the ordinary
dirichlet integral. That idea was later used in proving that, for two-dimensional
2 More precisely, if S (M ) is the set of points in minimal variety V whose tangent cones have
k
at most k independent directions of translational invariance, then Sk has dimension at most k.
9
mass-minimizing integral currents, each point has only one tangent cone [W1], and
also in analyzing structure of branch points [MW] and the boundary behavior [W9]
of classical minimal surfaces and pseudoholomorphic curves.
The first of the three volumes of Almgren’s paper is largely devoted to properties
of such multivalued functions minimizing the dirichlet integral. These functions
are exactly equivalent (by an explicit isomorphism) to energy minimizing maps
into a certain polyhedral cone. Much of Almgren’s analysis does not depend on
the particular cone. Thus when Hardt and Lin realized that certain liquid crystal
models were equivalent to energy minimizing maps into round cones, they were able
to use some of Almgren’s ideas in their analysis of the structure of those crystals
[HL].
The great length of Almgren’s paper made publication problematic. He was
exploring the possibility of making it available on the web. Not long after his death,
Vladimir Scheffer (Almgren’s third doctoral student) began converting the typed
manuscript into TeX files, which he plans to make widely available.
On a personal note, I vividly remember my first encounter with Almgren’s (m−2)
paper. I had lost interest in the first thesis topic he had suggested to me, and,
after letting way too much time go by, I asked him for another topic. He described
a problem that I quite liked. Then he added, at this point, since you need to
make up for lost time, it will be good to have a problem you can start working
on immediately. For this problem, you need only read the first two sections of
my new paper. Like Fred, I enjoy working on problems much more than doing
relevant background reading, so that sounded great to me. Thus I was quite taken
aback when he handed me the two sections, which resembled two large telephone
directories.
Multivalued Functions
For some time, Almgren believed that multivalued functions would play a very
large role in the future of geometric analysis. To this end, he wrote a long paper
[38] recasting the the theory of integral currents in terms of multivalued functions.
(Later he also handled flat chains mod p in the same way [52].) The idea was to
approximate arbitrary integral currents by graphs of lipschitz multivalued functions
(allowing points with positive and negative multiplicities), and then to prove the
fundamental compactness property of integral currents by an Arzela-Ascoli theorem
for such functions. Though very ingenious and in a sense more elementary than
the Federer-Fleming proof, it is actually much longer and rather complicated. It
does bypass the notoriously difficult Federer structure theorem, but there are now
several other, more direct proofs that also have that feature. [So1,W3,W10].
The main new theorem in [38] is a compactness theorem for real rectifiable
currents (assuming bounds on both size and mass). Afterward, Federer showed how
to prove this without Almgren’s multivalued function machinery [F]. Other proofs
of a more general result are given in [W10] (using the Federer structure theorem)
and in [W11] (not using the structure theorem).
Almgren continued to discuss multivalued functions in expository articles and
he continued to believe that his paper [38] was the right way to develop integral
current theory. However, they play a role in only one [52] of his subsequent research
10
papers. On the other hand, multivalued functions have been useful in several papers
by other authors [So1, Cha1, Hut, Br1, MW, W1, W8].
The isoperimetric inequality
Almgren discovered several beautiful theorems relating to the classical isoperimetric inequality. The classical inequality states that if S is a closed k-dimensional
surface in Rk+1 and if M is the region bounded by S, then
(1)
1
Hk+1 (M ) ≤ ck Hk (S)1+ k
with the constant ck chosen so that equality holds if S is a sphere. (Here Hk denotes
k-dimensional Hausdorff measure: length if k = 1, area if k = 2, and so on.)
In their 1960 paper Normal and Integral Currents [FF], Federer and Fleming
proved a beautiful generalization to k-dimensional surfaces in Rn . Of course if
n > k + 1, there will be many (k + 1)-dimensional surfaces bounded by S. Federer
and Fleming proved that for any S, there exists a surface M bounded by S for
which essentially the same inequality still holds, though their proof did not give the
best constant ck . Of course one may as well take M to be the least area surface
with boundary S. Thus their isoperimetric inequality can be stated as follows: if
M is a (k + 1)-dimensional mass-minimizing integral current in Rn with boundary
S, then
(2)
1
mass(M ) ≤ ck,n (mass(S))1+ k
Almgren’s first discovery along these lines was his beautiful 1965 proof [5] that
(2) is true (possibly with a worse constant) not only for minimizing surfaces M ,
but also for surfaces that are merely stationary for the area functional. (To say
that M is stationary means that it satisfies the first-derivative test for minima: if
we deform M , then the initial derivative of area is 0.) In particular, it applies to
any smooth surface M whose mean curvature is 0 at every point. See [Al1, §7].
In the mid 80’s, Almgren made an important observation [38] that sharpened the
the Federer-Fleming inequality and also clarified the role of multiplicities. Namely,
he proved that if M is a (k + 1)-dimensional mass-minimizing integral current with
boundary S, then
(3)
mass(M ) ≤ ck,n (mass(S))(size(S))1/k
Notice this is sharper than (2) because the size of an integral current is always less
than or equal to its mass. Almgren’s inequality is also important because it applies
to real currents, i.e. surfaces in which the multiplicities (or densities) are allowed
to be any real numbers, not just integers. (More generally, the multiplicities can
be elements of any normed abelian group.) The Federer-Fleming inequality was
known to be false for real currents. Suppose for example that M is a unit square
with density . Then the mass of M is · 12 = and the mass of ∂M is 4, so that
the right side of (2) goes to 0 faster than the left side as → 0. In contrast, both
sides of Almgren’s inequality (3) are proportional to .
11
The value of the best constants in these various inequalities remained open for
16 years after the Federer-Fleming paper was published. Then, in 1986, Almgren
published a beautiful proof [37] that the best constant for the Federer-Fleming
inequality (2) (and for his improved version (3) incorporating size) is the same as
for the classical isoperimetric inequality. Thus among all (k + 1)-dimensional massminimizing integral currents of a given mass, a ball in Rk+1 of the appropriate
radius has the smallest possible boundary mass.
Almgren proved this optimal isoperimetric inequality as follows. First, he proves
(using standard compactness theorems) that among all k + 1-dimensional surfaces
(integral currents) of a given mass (say mass equal to the volume of the unit ball B
in Rk+1 ) there exists a surface M of least possible boundary mass. The theorem
then reduces to showing that ∂M has area greater or equal to the area of ∂B.
For simplicity, let us assume (as Almgren could not!) that the boundary ∂M is a
smooth surface.
Almgren next used the defining property of M to show that the mean-curvature
of ∂M is everywhere less than or equal to the k, the mean curvature of ∂Bk+1 .
After this, the surface M no longer plays a role in the proof. Instead, Almgren
proves the fact (quite interesting in its own right, even for smooth surfaces) that
if S is any closed k-dimensional surface with mean curvature everywhere bounded
above by k, then the area of S is greater than or equal to the area of the unit
k-sphere ∂Bk+1 , with equality if and only if S is isometric to ∂Bk+1 .
Almgren’s proof of this fact is typical of his geometric ingenuity and ability to
look at problems from an unusual point of view. First he takes the convex hull C
of S. Then he thickens it a little to get a convex body C 0 with a well-defined unit
normal at every boundary point. Let R be the set of points x in ∂C 0 such that the
point in C nearest to x belongs to S. Of course the image of ∂C 0 under the gauss
map is the unit sphere ∂Bn . Now Almgren proves that the image of ∂C 0 \ R has
area 0, and he bounds the area of the image of R (by a straightforward calculation
of the the jacobian determinants) in terms the area of S and the given bound on
mean curvature of S. Putting these together gives the inequality.
(See [W5] for a more detailed expository account of Almgren’s proof.)
Incidentally, it has long been conjectured that if M is a complete simply connected m-dimensional riemannian manifold with sectional curvatures bounded above
by k ≤ 0 and if Ω is a region in M of volume v, then the area of ∂M is bounded
below by the surface area of a ball of volume v in a simply-connect m-manifold of
constant sectional curvature k. Almgren’s use of the convex hull was a key ingredient in Bruce Kleiner’s elegant proof that the conjecture is true for 3 manifolds
[K]. (The conjecture had been proved for surfaces by many authors starting in the
40’s: see [O, p. 1206] for discussion and references. Earlier, in 1926, A. Weil [We]
proved the k = 0 case for surfaces. In 1984, C. Croke [Cr] proved the k = 0 case
for 4-dimensional manifolds. Very recently, J. Cao and J. F. Escobar [CE] proved
a PL-version of the k = 0 in all dimensions. But the full conjecture remains open.)
Almgren’s optimal isoperimetric inequalities paper applies only to minimizing
surfaces. Whether his inequality for stationary (but not necessarily minimizing)
surfaces is true with the same constant is still not known. The only general result
is due to Michael and Simon [MS], who proved that (2) is true for stationary
surfaces with a constant depending only on k and not on n. The optimal constant
12
(corresponding to a disk) is known to hold for some classes of classical 2-dimensional
surfaces. For instance it is true if the boundary has only one component, or (for
disconnected boundaries) it the boundary components are not too far apart [Cho1],
[LSY]. (See also [Cho2].)
Rearrangements
In 1989, Almgren together with Elliott Lieb wrote a paper [47] on spherically
symmetric rearrangements, which are useful in proving optimal inequalities. Let C
be the class of nonnegative measurable functions on Rn such that for every positive
a, the set {x : f (x) > a} has finite lebesgue measure. For any such f , let g = T (f )
be the function in C such that for every a > 0, {x : g(x) > a} is a ball centered at
the origin and having the same measure as {x : f (x) > a}.
Of course g has the same Lp norm as f . For many norms k·k involving derivatives
(such as the Sobolev W 1,p norms), one can show that kgk must be less than or
equal to kf k. This lets one find optimal constants for many inequalities (such as
the Sobolev and Hardy-Littlewood-Sobolev inequalities) since it reduces them to
the simpler spherically symmetric case.
It is fairly straightforward to show that T is continuous with respect to the Lp
norms (in fact, it is a contraction), but it was not known to be continuous with
respect to W 1,p norms, except (by a paper of Coron) for n = 1. Such continuity
is not important for proving optimal inequalities, but it would imply existence of
solutions to some PDE problems.
Because of Coron’s n = 1 result, Almgren and Lieb (and just about everyone
else) thought that only a technical leap was needed to prove continuity in the n > 1
case. They tried to use the coarea formula to construct a proof, and only slowly
began to realize, to their great surprise, that the coarea formula is insufficient. The
trouble comes from the critical set
{x : Df (x) = 0}
concerning which the coarea formula is silent.
Almgren and Lieb showed that T is continuous (with respect to the W 1,p norm)
at a function f if and only if f has a property they call coarea regularity, which
reflects the behavior of f on its critical set. They prove that if n = 1, all functions
are coarea regular (thus giving a new proof of Coron’s theorem), but that if n > 1,
the coarea regular functions and the coarea irregular functions are both dense in
W 1,p ∩ C.
Incidentally, only rather bizarre functions fail to be coarea regular. For example,
if f is a function whose critical values
{f (x) : Df (x) = 0}
form a Lebesgue nullset, then f must be coarea regular. Thus by the Morse-SardFederer theorem, any sufficiently smooth (namely C n−1,1 ) function on Rn must
be coarea regular. However, for each α < 1, Almgren and Lieb construct coarea
irregular C n−1,α functions on Rn .
13
Liquid crystals and harmonic maps
Several of Almgren’s later papers involve harmonic maps. (Actually, as mentioned earlier, the first volume of his magnum opus can be viewed as an analysis
of harmonic maps into certain cones.) Although harmonic maps had been studied
since 1964, until 1982 the work was limited to smooth harmonic maps. Schoen and
Uhlenbeck initiated a new era by proving existence of energy minimizing harmonic
maps into arbitrary compact riemannian manifolds. Such maps often are forced to
have discontinuities (also called singularities) for topological reasons, but Schoen
and Uhlenbeck cleverly adapted many techniques from measure-theoretic minimal
surface theory to prove that the maps must be smooth except for small sets of
discontinuities.
Energy minimizing maps from a domain Ω in R3 to to the 2-sphere are an
important special case because they model a certain kind of liquid crystal. The
Schoen-Uhlenbeck theory implies that the discontinuities occur at isolated points in
the domain. A simple example of such a discontinuous map is the radial projection
from the unit ball to its boundary, which of course is discontinuous at the origin.
Brezis, Coron, and Lieb proved that locally the discontinuities all have this form
up to an isometry. More precisely, if x is a point of discontinuity, then there is a
isometry T : S2 → S2 such that
lim f (x + ry) = T
r→0+
y
|y|
Let us call the discontinuity positive or negative according to whether T is orientation preserving or reversing.
Note that if f is continuous at the boundary of Ω, then the mapping degree of
f : ∂Ω → S2 is equal to the number of positive discontinuities minus the number
of negative discontinuities:
(*)
deg(f |∂Ω) = P − N
This gives a lower bound on the number of singularities, but there was no upper
bound.
Almgren and Lieb gave an upper bound, proving that the total number of singularities is bounded by a constant times the energy of f |∂Ω:
P + N ≤ CΩ
Z
|D(f |∂Ω)|2
∂Ω
where the constant CΩ depends on the smoothness of the domain’s boundary. Examples of Hardt and Lin showed that this is sharp, except for the constant (which
the Almgren-Lieb proof, being based on compactness arguments, does not estimate.)
One might think that a single map f should not have both a positive and a
negative discontinuity, since topologically such a pair can come together and cancel
each other out. If this were were true, then (*) would imply that the number of
discontinuities is equal to the absolute value of the mapping degree of f |∂Ω. But
14
Almgren and Lieb give ingenious examples showing that it is emphatically not true.
For instance, they give examples in which Ω = B3 and in which the image f (∂B3 )
of the boundary is a simple closed curve (a perturbation of great circle), but such
that the map has many discontinuities.
They also give an examples in which positive and negative discontinuities are
close together (though necessarily far apart relative to their distances from the
boundary.) For example, given a finite set of points S in ∂Ω and an integer N , they
show that there is an energy minimizing map
f : Ω → S2
such that for each p ∈ S, there is a sequence of N discontinuities near p approximately along the normal line to ∂Ω at p. Furthermore, the signs of the discontinuities of the pi alternate as they approach p!
Algmren and Lieb also give interesting examples of symmetry breaking. That
is, they show that an energy minimizing map need not have symmetries present in
the boundary data.
Ordinarily in the study of energy minimizing maps f : Ω ⊂ R3 → S2 , the
boundary values are prescribed, but the locations of the discontinuities are not;
they occur at the places that permit f to acheive the smallest energy. If one tries
to minimize energy among maps with discontinuities at prescribed locations, the
infimum will typically not be attained. A key result in the Brezis-Coron-Lieb paper
was a formula for this infimum. Furthermore, they showed that, although the
infimum is not attained, one can in a certain sense take a limit of a minimizing
sequence, the result of which is map that is discontinuous along entire line segments.
Almgren, W. Browder and Lieb proved a beautiful generalization of this result
[43]. Let M be an oriented riemannian m-manifold, which we will assume compact
(though compactness is not required in their paper.) Let C be a closed orientable
submanifold of dimension k < m. Now let
f : M \ C → Sm−k−1
be a smooth map from the complement of C to a unit (m − k − 1)-sphere. By
Sard’s theorem, almost every level set of f is a smooth oriented k + 1-dimensional
submanifold, with boundary in C. In particular, such a level set determines an
element s[f ] of the relative homology group
Hk+1 (M, C; Z)
(This element does not depend on the particular level set chosen.)
Now fix an element α of Hk+1 (M, C; Z) and consider the following two problems:
(1) Minimize the area (mass) of T among all surfaces (i.e.integral currents) in
the homology class α.
(2) Minimize
Z
|Df |m−k−1
M \C
among all smooth f : M \ C → Sm−k−1 such that s[f ] = α.
15
Almgren, Browder, and Lieb prove that these seemingly very different problems
are in a sense equivalent. In particular, the minimum area in (1) is equal to a
constant (depending only on dimensions) times the infimal energy in (2). Also,
although the infimum in problem (2) is not attained, it is approached arbitrarily
closely by maps f which are constant except in an -neighborhood of T , where T is
a least-area surface for problem (1). Of course in the limit one gets a map f which
is constant except along T , where it is discontinuous.
In other words, roughly speaking, one can find a least area solution surface for
problem (1) by taking the set of discontinuities of the limit of a minimizing sequence
for problem (2).
Incidentally, De Giorgi’s formulation of the Plateau problem in terms of BV functions has always been thought valid only for hypersurfaces. However, the AlmgrenBrowder-Lieb theorem can be regarded as a generalization of De Giorgi’s approach
to higher codimension. Indeed, the authors raise the intriguing possibility of using
problem (2) to analyze regularity of mass-minimizing surfaces.
Their paper is described as an announcement, but the theorems are stated precisely, and the proofs are outlined in some detail.
Curvature Flows
In recent years, Almgren was very interested in dynamic problems in which surfaces move through space with velocities related to their curvatures. His first major
contribution to this field was not a paper, but a graduate student: in 1975, Ken
Brakke wrote a remarkable thesis on mean-curvature flow for varifolds [Br1]. However, the thesis was ahead of its time, and had little impact for many years. After
Hamilton’s stunning success with the Ricci curvature flow in 1982 [Ham], Gerhardt
Huisken, Mike Gage, Matt Grayson, and Hamilton himself began obtaining striking
theorems about mean curvature flow, using methods of classical PDE and differential geometry rather than geometric measure theory. The field became a large and
active one, as it continues to be. After a few years, researchers in the field became
aware of Brakke’s thesis and were amazed by the wealth of insights it contained.
Meanwhile, Brakke had also become interested in numerical work, and today he is
even more widely known for his extremely powerful Surface Evolver software for
simulating geometric flows [Br2].
Smooth surfaces evolving under curvature flows typically become singular at a
finite time, after which it is not clear even how to define the flow with classical PDE
and differential geometry. Thus people have proposed various notions of weak or
generalized solutions to geometric evolution equations.
These notions allow one to prove existence of solutions for all time. Typically it
is relatively easy to prove that the generalized solutions agree with the classical ones
as long as the surfaces are smooth. It is rather difficult to prove partial regularity
results for solutions after singularities form. (Indeed, I believe that Brakke’s regularity theorem is the only such result known. That is, the various partial regularity
theorems all use Brakke’s.)
Almgren and his collaborators Jean Taylor and Lihe Wang wanted a model that
would apply to many kinds of geometric evolutions occurring in nature. Brakke’s
mean curvature flow models motion of grain interfaces in annealing metals, but
16
mean-curvature flows in general are relevant only when the physical surface energies
are isotropic. Almgren, Taylor, and Wang also wanted to model crystal growth, in
which the surfaces energies are highly non-isotropic; indeed, for crystals the surface
energy density typically depends in a non-differentiable way on the unit normal.
They proposed such a model in [56]. The idea was to replace the parabolic
problem by a series of minimization problems at discrete time steps. The flow
is then the limit (as the time step size goes to 0) of the minimization problem.
More precisely, at each time, one chooses a surface that minimizes the sum of two
quantities: the surface energy and a term that measures how far the surface is from
where it was at the previous time step.
In another paper [61], Almgren lists three advantages of this approach:
(1) Powerful existence and regularity theorems for the minimization problems
already exist.
(2) By formulating problems in a variational setting, one typically needs only
one approximate derivative or tangential information with which to make
sense out of the problem rather than requiring second derivative or curvature
information. This means the analysis can remain valid even if curvatures,
say, become unbounded in spots. One can also formulate and solve problems
for the nondifferentiable surface energies which lead to faceted crytals in
nature.
(3) The geometry and topology of the solution is an output of the problem
rather than an input. This is in contrast to other formulations for sharp interface motion, which involve moving an interface according to information
about its normal directions.
The main theorem of [57] asserts that the region bounded by the evolving
surface is a holder-continuous function of time. The authors also prove that for
smooth initial surfaces and smooth elliptic surface energy density functions, their
flow agrees with the classical one until singularities appear.
Jean Taylor had previously modelled crystal growth for crystals with finitely
many faces by systems of ODE’s, using one dependent variable for the position
of each face. (This model is particularly well-suited to numerical simulations.)
Almgren and Taylor showed that, in the absence of certain conceivable pathologies,
the Almgren-Taylor-Wang flow agrees with Taylor’s ODE model for such finitelyfaceted crystals in R2 [59].
In [62], Almgren and Wang incorporated temperature into the model so that
one could, for example, model melting or freezing ice. (Somewhat earlier, a similar,
but less general model, was discovered independently by S. Luckhaus [Lu].) Thus
in addition to a moving region (the ice), there is a temperature function. Both
in the region and in its complement, the temperature should satisfy a heat equation. Whereas mathematicians studying this problem routinely use the same heat
equation in both regions, Almgren, characteristically, does not, since (as he points
out elsewhere [61]) the heat capacity of water is twice that of ice. Furthermore,
within the crystal the model allows heat to flow more easily in some directions in
others. Naively, one might define the ice region to be the set of points at which
the temperature is below freezing, but this neglects (among the other things) the
Gibbs-Thomson effect, that is, the dependence of the local freezing temperature
17
(along an interface) on curvature (of that interface). As Almgren explains, small
crystals with high curvatures melt even thought the temperature of the surrounding
liquid is slightly below the freezing temperature given in handbooks[61].
Almgren and Wang [62] define a kind of evolution that includes these many
physical effects, they prove existence for all time, and they prove (among other
results) that temperature depends Holder-continuously on time.
Miscellaneous Research Papers
Almgren wrote a number of other research articles that do not fit into the main
groups I have described above. I will only mention two here. His fourth paper [4]
is perhaps the only paper in geometric measure theory that assumes the continuum
hypothesis. It proves that the dual space of the Banach space of real rectifiable
k-chains (with the mass norm) consists of differential forms. Almgren mentions
the following striking consequence: there is a differential one-form on R2 whose
integral on the line segment from (x, y) to (X, Y ) is X − x or 0 according to
whether the slope of the segment is rational or not. The 1-form is necessarily nonmeasurable, but its pull-back to every C 1 curve is measurable! Almgren regarded
his use of the continuum hypothesis as a cheap trick, and he once told me that
if he had to withdraw one of his theorems, this would be the one. Cheap or not,
it turns out that the trick is unavoidable: T. De Pauw recently demonstrated [DP]
that Almgren’s theorem cannot be proved without using the continuum hypothesis.
(More precisely, it cannot be proved in ZFC.) Had Almgren known this, perhaps
he would have regarded his paper more favorably.
In 1977, Almgren and Thurston published a paper about the convex hull genus
of a curve C in R3 ; this is the smallest possible genus of an embedded surface
bounded by C and contained in the convex hull of C. Since any mass-minimizing
current T bounded by C is such a surface, the convex hull genus of C is a lower
bound on the genus of T . Almgren and Thurston gave examples of unknotted curves
for which the convex hull genus is large. Their work inspired Hubbard to write a
beautiful and elementary paper [Hub] giving an exact formula for the convex hull
genus for a large class of interesting curves.
Expository Papers
Almgren’s research papers are often exceedingly difficult to read, even allowing
for the inherent complexity of the mathematics. However, he also wrote a great
many expository papers, and those are a delight to read. They are characterized by
vivid descriptions and by well-chosen examples and pictures. He was particularly
good at explaining physical or chemical phenomena related to the mathematics
under discussion. He also did a superb job of showing how concrete geometric and
scientific problems lead naturally to seemingly abstract notions such as integral
currents and varifolds. Some of my favorites among his expository works are [10],
[21] (jointly written with Jean Taylor), [58], [61], and [45] (jointly written with
Elliott Lieb). The first of these, though written in 1968, still provides an excellent
introduction to the geometric calculus of variations. An undergraduate classmate,
not a mathematician, describes the Almgren-Taylor Scientific American article [21]
as gloriously inviting. In Mathematical Reviews, A. J. Tromba describes the
18
Almgren-Lieb article [45] as beautifully written, and adds, the reviewer rarely
has come across a paper like this one that he could highly recommend as pleasurable
reading for people with a general interest in the calculus of variations.
Unfinished Work
Almgren was busy with a number of projects when illness struck in the summer of
1996. He wanted to prove regularity theorems for the Almgren-Taylor and AlmgrenTaylor-Wang curvature flows. Brakke’s work gives some partial regularity, but only
for isotropic surface energy densities. For general elliptic densities, one of the major
difficulties is proving lower density bounds, i.e., ruling out long thin spikes or tubes
(or filigree as Almgren would say.) This is why, for example, it is not known
whether a surface stationary for an elliptic functional must be regular on an open
dense set [Al2]. Almgren recently was able to prove a lower density bound for a
class of elliptic functionals, and wrote a draft of a paper on that [66].
He was also working on several papers that he had announced previously. For
many years, there have been several closely-related open questions about the foundations of integral currents and flat chains, such as: if 2Ti is sequence of integral
flat chains that converge in the integral flat topology, must the sequence Ti also
converge? Is the integral flat norm of an integral surface bounded above by a multiple of the its real flat norm? (The latter question turns out to be important
in the Giaquinta-Modica-Souček approach to elasticity using Cartesian currents
[GMS1, GMS2].) Paradoxical examples by L. C. Young [Y3], Frank Morgan [MF1],
and myself [W2] indicate that the questions are quite subtle. Several years ago,
Almgren thought he had discovered the answers to a number of these questions
(in particular, an affirmative answer to the second one mentioned above), and he
was planning to write proofs. On several occasions, he described the proofs, which
seemed to use every weapon in his arsenal, including his enormous (m − 2) regularity paper. As Federer said, Almgren used to be a fighter pilot, and that’s the
way he did mathematics.
Almgren’s PhD Students
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
Jean Taylor (1973)
Jon Pitts (1974)
Harold Parks (1974)
Vladimir Scheffer (1974)
Kenneth Brakke (1975)
Frank Morgan (1977)
Robert Kohn (1979)
Brian White (1982)
Bruce Solomon (1982)
Dana Nance Mackenzie (1983)
Sheldon Chang (1986)
Kevin Perry (1986)
John Sullivan (1990)
John Steinke (1991)
Alice Underwood (1993)
19
(16) David Caraballo (1996)
(17) NungKwan (Aaron) Yip (1996)
(18) Kin Yan Chung (1997)
Students who wrote undergraduate theses under
Almgren’s Supervision
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Thomas J. Givnish (’73)
Joseph H.G. Fu (’78)
Peter E. Oppenheimer (’79)
Michael J. Barall (’80)
Steven H. Strogatz (’80)
Boaz J. Super (’83)
Bruce J. Esrig (’82)
Alexander J. Oss (’91)
Ivan A. Blank (’93)
Stephen P. Lardieri (’94)
Papers of F. J. Almgren, Jr
(1). The homotopy groups of the integral cycle groups, Topology 1 (1962), 257299.
(2). An isoperimetric inequality, Proc. Amer. Math. Soc. 15 (1964), 284285.
(3). Three theorems on manifolds with bounded mean curvature, Bull. Amer. Math. Soc. 71
(1965), 755756.
(4). Mass continuous cochains are di erential forms, Proc. Amer. Math. Soc. 16 (1965),
12911294.
(5). The theory of varifolds. A variational calculus in the large for the k-dimensional are
integrand, Multilithed notes (Princeton U. Library), 1965, 178 pages.
(6). Plateau’s Problem. An Invitation to Varifold Geometry, W. A. Benjamin, Inc., New York,
1966.
(7). Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s
theorem, Annals of Mathematics 84 (1966), 277292.
(8). Existence and regularity of solutions to elliptic calculus of variations problems among
surfaces of varying topological type and singularity structure, Bull. Amer. Math. Soc. 73
(1967), 576680.
(9). Existence and regularity almost everywhere of solutions to elliptic variational problems
among surfaces of varying topological type and singularity structure, Annals of Mathematics 87 (1968), 321391.
(10). Measure theoretic geometry and elliptic variational problems, Bull. Amer. Math. Soc. 75
(1969), 285304.
(11). A maximum principle for elliptic variational problems, Journal of Functional Analysis 4
(1969), 380389.
(12). Measure theoretic geometry and elliptic variational problems, Proceedings of the Symposium on Continuum Mechanics and Related Problems of Analysis (Tbilisi, 1971), (in
Russian), vol. II, Izdat. Mecniereba, Tbilisi, 1974, pp. 307324.
(13). Geometric measure theory and elliptic variational problems, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, pp. 813
819.
(14). with W. K. Allard, An introduction to regularity theory for parametric elliptic variational
problems, Partial differential equations, Proc. Symp. Pure Math., vol. XXIII, Amer. Math.
Soc., Providence, RI, 1973, pp. 231260.
(15). Geometric variational problems from a measure-theoretic point of view, Global analysis
and its applications (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys.,
Trieste, 1972), Vol. II, Internat. Atomic Energy Agency, Vienna, 1974, pp. 122.
20
(16). Geometric measure theory and elliptic variational problems, Geometric Measure Theory
and Minimal Surfaces, (C.I.M.E. Lectures, III Ciclo, Varenna, 1972), Ediziono Cremonese,
Rome, 1973, pp. 31117.
(17). The structure of limit varifolds associated with minimizing sequences of mappings, Symposia Mathematica, (Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali,
INDAM, Rome, 1973), vol. XIV, Academic Press, London, 1974.
(18). Existence and regularity almost everywhere of solutions to elliptic variational problems
with constraints, Bull. Amer. Math. Soc. 81 (1975), 151154.
(19). Existence and regularity almost everywhere of solutions to elliptic variational problems
with constraints, viii + 199 pages, Memoirs Amer. Math. Soc. 4 (1976), no. 165.
(20). with W. K. Allard, The structure of stationary one-dimensional varifolds with positive
density, Inventiones Mathematicae 34 (1976), 8397.
(21). with J. E. Taylor, The geometry of soap lms and soap bubbles, Scientific American (July
1976), 8293.
(22). with W. P. Thurston, Examples of unknotted curves which bound only surfaces of high
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21
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(55). with J. Sullivan, Visualization of soap bubble geometries, Leonardo 25 (1992), 267271.
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(57). with J. E. Taylor and L. Wang, Curvature driven ows: A variational approach, SIAM
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(58). Questions and answers about area minimizing surfaces and geometric measure theory,
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22
(59). with J. E. Taylor, Flat ow is motion by crystalline curvature for curves with crystalline
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Benoit Mandelbrot (Curaçao, 1995), Fractals 3 (1995), no. 4, 713723.
(61). Questions and answers about geometric evolution processes and crystal growth, The
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(62). with L. Wang, Mathematical existence of crystal growth with Gibbs-Thomson curvature
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(63). with I. Rivin, The mean curvature integral is invariant under bending, Geometry and
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(65). Global analysis, preprint (survey/expository).
(66). Isoperimetric inequalities for anisotropic surface energies, unfinished manuscript.
(67). A new look at at chains mod n, unfinished manuscript.
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