The Ricci Flow in Riemannian Geometry A Complete Proof of the Differentiable 1 4 Pinching Sphere Theorem 1st Edition by Ben Andrews, Christopher Hopper ISBN 9783642162855 by Ben Andrews, Christopher Hopper (auth.) 9783642162855, 3642162851 instant download after payment.

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ISBN 13: 9783642162855
Author: Ben Andrews, Christopher Hopper

The Ricci Flow in Riemannian Geometry A Complete Proof of the Differentiable 1 4 Pinching Sphere Theorem 1st Table of contents:

• 1. Introduction

• 2. Background Material

  • (This chapter would cover fundamental concepts in differential geometry necessary for understanding Ricci flow, such as Riemannian manifolds, metrics, connections, curvature tensors, etc.)

• 3. Harmonic Mappings

• 4. Evolution of the Curvature

  • (This is where the Ricci flow equation is introduced and its impact on the curvature of the manifold is analyzed.)

• 5. Short-Time Existence

  • (Discusses the local existence of solutions to the Ricci flow equation for a short period of time.)

• 6. Uhlenbeck's Trick

  • (A specific technical tool or method used in the analysis of geometric flows.)

• 7. The Weak Maximum Principle

  • (A crucial analytical tool used for proving various estimates and properties of solutions to partial differential equations, including the Ricci flow.)

• 8. Regularity and Long-Time Existence

  • (Explores the smoothness of solutions and conditions under which the Ricci flow exists for a longer duration, possibly leading to singularities.)

• 9. The Compactness Theorem for Riemannian Manifolds

  • (Covers important results that allow for sequences of Riemannian manifolds to converge, which is vital for understanding the behavior of the flow.)

• 10. The F-Functional and Gradient Flows

  • (Introduces Perelman's F-functional, a key tool in his work on the Ricci flow, and its connection to gradient flows.)

• 11. The W-Functional and Perelman's Monotonicity Formulas

  • (Delves into Perelman's powerful W-functional and associated monotonicity formulas, which were central to his proof of the Poincaré and Geometrization Conjectures.)

• 12. The Noncollapsing Result

  • (Covers Perelman's noncollapsing theorem, which describes how the volume of the manifold behaves under the Ricci flow and prevents the formation of certain types of singularities.)

• 13. The Canonical Neighborhood Theorem

• 14. The Differentiable 1/4-Pinching Sphere Theorem

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Tags: Ben Andrews, Christopher Hopper, Ricci, Riemannian