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2011 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

The Ricci Flow in Riemannian Geometry

A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem

verfasst von: Ben Andrews, Christopher Hopper

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The relationship between curvature and topology has traditionally been one of the most popular and highly developed topics in Riemannian geometry. In this area, a central issue of concern is that of determining global topological structures from local metric properties. Of particular interest to us the so- called pinching problem and related sphere theorems in geometry. We begin with a brief overview of this problem, from Hopf’s inspiration to the latest developments in Hamilton’s Ricci flow.
Ben Andrews, Christopher Hopper
Chapter 2. Background Material
Abstract
This chapter establishes the notational conventions used throughout while also providing results and computations needed for later analyses. Readers familiar with differential geometry may wish to skip this chapter and refer back when necessary.
Ben Andrews, Christopher Hopper
Chapter 3. Harmonic Mappings
Abstract
When considering maps between Riemannian manifolds it is possible to associate a variety of invariantly defined ‘energy’ functionals that are of geometrical and physical interest. The core problem is that of finding maps which are ‘optimal’ in the sense of minimising the energy functional in some class; one of the techniques for finding minimisers (or more generally critical points) is to use a gradient descent flow to deform a given map to an extremal of the energy.
The first major study of Harmonic mappings between Riemannian mani– folds was made by Eells and Sampson [ES64]. They showed, under suitable metric and curvature assumptions on the target manifold, gradient lines do indeed lead to extremals.
We motivate the study of Harmonic maps by considering a simple problem related to geodesics. Following this we discuss the convergence result of Eells and Sampson. The techniques and ideas used for Harmonic maps provide some motivation for those use later for Ricci flow, and will appear again explicitly when we discuss the short-time existence for Ricci flow.
Ben Andrews, Christopher Hopper
Chapter 4. Evolution of the Curvature
Abstract
The Ricci flow is introduced in this chapter as a geometric heat-type equation for the metric. In Sect. 4.4 we derive evolution equations for the curvature, and its various contractions, whenever the metric evolves by Ricci flow. These equations, particularly that of Theorem 4.14, are pivotal to our analysis throughout the coming chapters. In Sect. 4.5.3 we discuss a historical re- sult concerning the convergence theory for the Ricci flow in n-dimensions. This will motivational much of the Böhm and Wilking analysis discussed in Chap. 11.
Ben Andrews, Christopher Hopper
Chapter 5. Short-Time Existence
Abstract
An important foundational step in the study of any system of evolutionary partial differential equations is to show short-time existence and uniqueness. For the Ricci flow, unfortunately, short-time existence does not follow from standard parabolic theory, since the flow is only weakly parabolic. To overcome this, Hamilton's seminal paper [Ham82b] employed the deep Nash –Moser implicit function theorem to prove short-time existence and uni- queness. A detailed exposition of this result and its applications can be found in Hamilton's survey [Ham82a]. DeTurck [DeT83]later found a more direct proof by modifying the flow by a time-dependent change of variables to make it parabolic. It is this method that we will follow.
Ben Andrews, Christopher Hopper
Chapter 6. Uhlenbeck’s Trick
Abstract
In Theorem 4.14 we derived an evolution equation for the curvature R under the Ricci flow, which took the form
Ben Andrews, Christopher Hopper
Chapter 7. The Weak Maximum Principle
Abstract
The maximum principle is the main tool we will use to understand the behaviourof solutions to the Ricci flow. While other problems arising in geo- metric analysis and calculus of variations make strong use of techniques from functional analysis, here – due to the fact that the metric is changing – most of these techniques are not available; although methods in this direction are developed in the work of Perelman [Per02]. The maximum principle, though very simple, is also a very powerful tool which can be used to show that pointwise inequalities on the initial data of parabolic pde are preserved by the evolution. As we have already seen, when the metric evolves by Ricci flow the various curvature tensors R, Ric, and Scal do indeed satisfy systems of parabolic pde. Our main applications of the maximum principle will be to prove that certain inequalities on these tensors are preserved by the Ricci flow, so that the geometry of the evolving metrics is controlled.
Ben Andrews, Christopher Hopper
Chapter 8. Regularity and Long-Time Existence
Abstract
In Chaps. 4 and 6 we saw that the curvature under Ricci flow obeys a parabolic equation with quadratic nonlinearity. By appealing to this view, we would expect the same kind of regularity that is seen in parabolic equa- tions to apply to the curvature. In particular we want to show that bounds on curvature automatically induce a priori bounds on all derivatives of the curvature for positive times. In the literature these are known as Bernstein– Bando–Shi derivative estimates as they follow the strategy and techniques introduced by Bernstein (done in the early twentieth century) for proving gradient bounds via the maximum principle and were derived for the Ricci flow in [Ban87] and comprehensively by Shi in [Shi89]. Here we will only need the global derivative of curvature estimates (for various local estimates see [CCG+08, Chap. 14]). In the second section we use these bounds to prove long-time existence.
Ben Andrews, Christopher Hopper
Chapter 9. The Compactness Theorem for Riemannian Manifolds
Abstract
The compactness theorem for the Ricci flow tells us that any sequence of complete solutions to the Ricci flow, having uniformly bounded curvature and injectivity radii uniformly bounded from below, contains a convergent subsequence. This result has its roots in the convergence theory developed by Cheeger and Gromov. In many contexts where the latter theory is applied, the regularity is a crucial issue. By contrast, the proof of the compactness theorem for the Ricci flow is greatly aided by the fact that a sequence of solutions to the Ricci flow enjoy excellent regularity properties (which were discussed in the previous chapter). Indeed, it is precisely because bounds on the curvature of a solution to the Ricci flow imply bounds on all derivatives of the curvature that the compactness theorem produces C8-convergence on compact sets.
The compactness result has natural applications in the analysis of singularities of the Ricci flow by ‘blow-up’, discussed here in Sect. 9.5: The idea is to consider shorter and shorter time intervals leading up to a singularity of the Ricci flow, and to rescale the solution on each of these time intervals to obtain solutions on long time intervals with uniformly bounded curvature. The limiting solution obtained from these gives information about the structure of the singularity.
As a remark concerning notation in this chapter, quantities depending on the metric gk or gk(t) will have a subscript k. For instance ?k and Rk denote the Riemannian connection and Riemannian curvature tensor of gk. Quantities without a subscript will depend on the background metric g.
Ben Andrews, Christopher Hopper
Chapter 10. The $$\mathcal{F}$$ -Functional and Gradient Flows
Abstract
After Ricci flow was first introduced, it appeared for many years that there was no variational characterisation of the flow as the gradient flow of a geometric quantity. In particular, Bryant and Hamilton established that the Ricci flow is not the gradient flow of any functional on Met – the space of smooth Riemannian metrics – with respect to the natural L2 inner product (with the exception of the two-dimensional case, where there is indeed such an ‘energy’). Considering the prominent role variational methods have played in geometric analysis, pde’s and mathematical physics, it seemed surprising that such a natural equation as Ricci flow should be an exception. One of the many important contributions Perel’man made was to elucidate a gradient flow structure for the Ricci flow, not on Met but on a larger augmented space. Part of this structure was already implicit in the physics literature [Fri85]. In this chapter we discuss this structure, at the centre of which is Perel’man’s F-functional [Per02]. The analysis will provide the ground work for the proof of a lower bound on injectivity radius at the end of Chap. 11.
Ben Andrews, Christopher Hopper
Chapter 11. The $$\mathcal{W}$$ -Functional and Local Noncollapsing
Abstract
The F-functional provides a gradient flow formalism for the Ricci flow, discussed in Chapt. 10. We hope to be able to use this to understand the singularities of Ricci flow, but the F-functional is not yet enough to do this, because it does not behave well under the scaling transformations needed in the blow-up analysis.
Ben Andrews, Christopher Hopper
Chapter 12. An Algebraic Identity for Curvature Operators
Abstract
In this chapter and the next we look at one of the most important recent developments in the theory of Ricci flow: The work of Böhm and Wilking [BW08] which gives a method for producing whole families of preserved convex sets for the Ricci flow from a given one. This remarkable new method has broken through what was an enormous barrier to further applications of Ricci flow: In particular the proof of the differentiable sphere theorem relies heavily on this work.
Ben Andrews, Christopher Hopper
Chapter 13. The Cone Construction of Böhm and Wilking
Abstract
In this section the remarkable formulas derived in the previous section, particularly the identities (12.13) and (12.14), will be applied to construct a family of cones preserved by the Ricci flow. We follow the argument presen- ted by Böhm and Wilking who applied it to produce a family of preserved cones interpolating between the cone of positive curvature operators and the line of constant positive curvature operators. The construction applies much more generally, so that given any preserved cone satisfying a few conditions, there is a family of cones linking that one to the ray of constant positive curvature operators. As we will see, this is a crucial step in proving that solutions the Ricci flow converge to spherical space forms.
Ben Andrews, Christopher Hopper
Chapter 14. Preserving Positive Isotropic Curvature
Abstract
The condition of positive curvature on totally isotropic 2-planes was first introduced by Micallef and Moore [MM88]. They were able to prove the following sphere theorem
Ben Andrews, Christopher Hopper
Chapter 15. The Final Argument
Abstract
We now have all the ingredients in place to prove the following: Theorem 15.1 (Differentiable 1/4-Pinched Sphere Theorem). A compact, pointwise 1/4-pinched Riemannian manifold of dimension n≥4 is diffeomorphic to a spherical space form.
Ben Andrews, Christopher Hopper
Backmatter
Metadaten
Titel
The Ricci Flow in Riemannian Geometry
verfasst von
Ben Andrews
Christopher Hopper
Copyright-Jahr
2011
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-16286-2
Print ISBN
978-3-642-16285-5
DOI
https://doi.org/10.1007/978-3-642-16286-2