SPHERE THEOREM VIA RICCI FLOW
Several geometric rigidity results are based on the invariance of certain cones of curvad
ture operators R along the ODE dt
R = R2 + R# . This strategy has been used to prove
the convergence of the Ricci flow in several di↵erent contexts. An important example
is the proof of the di↵erentiable sphere theorem for 1/4-pinched manifolds by Brendle
and Schoen ([BS], [B]).
Theorem: Let M be a compact simply-connected manifold admitting a Riemannian
metric whose sectional curvatures  satisfies 1/4 <   1. Then M is di↵eomorphic to
the sphere Sn .
The goal of this seminar is to understand the proof of the sphere theorem via Ricci
flow. Our main source will be the papers [H] by Richard Hamilton, [BW] by Christoph
Böhm and Burkhard Wilking, and [W] by Burkhard Wilking.
Prerequisites: good knowledge of di↵erential geometry and basics of Ricci flow, e.g.,
Lecture Advanced Di↵erential Geometry II SS2016 or [T]. There is a vast literature on
Ricci flow, any introduction to this topic which covers the the material of the lecture
”Advanced Di↵erential Geometry II” of the summer term 2016 is suitable.
1. Manifolds with positive curvature operator are space forms
Hamilton conjectured that compact Riemannian manifolds with positive curvature
operators are space forms. In the first part of the seminar we study a paper by Böhm
and Wilking [BW] in which this conjecture is proven. More generally, the authors show
that the conjecture holds for compact Riemannian manifolds with 2-positive curvature
operator.
The strategy and the main ideas of the proof are the following: Let gt be a solution
@
of the Ricci flow, i.e., @t
gt = 2Ric(gt ). Then the curvature operators Rt associated to
the metrics gt satisfy the evolution equation
@
@t R
=
R + 2(R2 + R# ),
where Rt : ⇤2 Tp M ! ⇤2 Tp M , ⇤2 Tp M is identified with so(Tp M ) and
R# = ad (R ^ R) ad⇤ .
Here ad denotes the adjoint representation.
2 (so(n)) denote the vector space of curvature operators, that is, the vector space
Let SB
of self-adjoint endomorphisms of so(n) satisfying the Bianchi identity. By Hamilton’s
2 (so(n)) which is
maximum principle, a closed, convex, O(n)-invariant subset C of SB
invariant under the ODE
d
dt R
= R2 + R#
is also invariant under the Ricci flow.
1
2
A key ingredient in the proof is the introduction of an equivariant linear map
2
2
`a,b : SB
(so(n)) ! SB
(so(n)),
where a, b 2 R. Böhm and Wilking prove that
Da,b = `a,b1 ((`a,b R)2 + (`a,b R)# )
R2
R#
is independent of the Weyl curvature and can be computed explicitly.
Consider the image of a known invariant curvature condition C under the linear map
`a,b for suitable a, b. This new subset or curvature condition is invariant under the ODE
1
d
2
#
2
#
dt R = R + R if `a,b ((`a,b R) + (`a,b R) ) lies in the tangent cone TR C of the invariant
set C. By assumption R2 + R# lies in that tangent cone, so it is sufficient to show that
Da,b 2 TR C. Is is verified for a suitable range of a, b.
This technique is used to construct a continuous family of invariant cones joining the
invariant cone of 2-positive operators to the invariant cone of multiples of the identity.
The authors prove that for any such family, a so-called generalized pinching set can be
constructed. A convergence argument of Hamilton then implies the main result.
1.1. Introduction and algebraic preliminaries (1/2 talk). Present the introduction (this is partially already down above), i.e., explain the background and the main
strategy of this paper. Furthermore, present Section 1 in [BW], i.e., introduce the
algebraic preliminaries. [Very easy]
1.2. Construction of new curvature conditions (1 talk). Present Section 2 in
[BW]. In particular, prove Theorem 2 in [BW], that is, show that Da,b is independent
of the Weyl curvature and can be computed explicitly. [Easy]
1.3. Construction of new invariant sets (1 talk). Present Section 3 in [BW], in
particular prove Theorem 3.1 in [BW].
1.4. Construction of a generalized pinching set (1/2 talk). Present Section 4 in
[BW]. [It is possible to merge this talk and the previous one.]
1.5. Proof of the main result (1 talk). Present Sections 5 and 6 in [BW]. In
particular prove Theorem 1 in [BW], i.e., that compact Riemannian manifolds with
2-positive curvature operator are space forms.
2. A Lie algebraic approach to Ricci flow invariant curvature
conditions and Harnack inequalities
B. Wilking provides a general invariance result for cones of metrics which covers
many existing invariance results. Furthermore, he proves a similar result where the
Riemannian tensor is replaced by a Harnack operator.
2.1. Introduction and proof of general invariance result for cones of metrics
(1 talk). Let S be a subset of the complex Lie algebra so(n, C) and C(S) the cone of
curvature operators which are nonnegative on S. Show that C(S) defines a Ricci flow
invariant curvature condition if S is invariant under AdSO(n,C) . (Section 1 of [W].)
3
2.2. Maximum principle for Harnack operators (1-2 talks). Establish a maximum principle for Harnack operators. (Section 2 in [W].)
2.3. Theorem 3 (1 talk). Generalize the result of the first talk to obtain Harnack
inequalities. (Section 3 in [W].)
2.4. Some pinching results (1 talk). Present Section 4 in [W]. In particular, provide
the missing step for the proof of the sphere theorem.
2.5. Kähler manifolds with positive orthogonal bisectional curvature (1 talk).
Prove that a compact Kähler manifold with orthogonal bisectional curvature evolves to
a metric with positive bisectional curvature, i.e., present Section 5 in [W].
References
[B]
[BS]
[BW]
[H]
[T]
[W]
S. Brendle, Ricci flow and the sphere theorem, Graduate Studies in Mathematics, 111. American
Mathematical Society.
S. Brendle, R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math.
Soc. 22 (2009), no. 1, 287?307.
C. Böhm, B. Wilking, Manifolds with positive curvature operators are space forms, Annals of
Math. 167 (2008), 1079–1097
R. Hamilton, Three manifolds with positive Ricci-curvature, J. Di↵. Geom. 17 (1982), 255–306.
P. Topping, Lectures on Ricci flow.
B. Wilking, A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack
inequalities, J. Reine Angew. Mathematik 679 (2013), 223-247.