Paper · Sep 6, 2021

Ricci curvature integrals, local functionals, and the Ricci flow

As on arXiv

Abstract

Consider a Riemannian manifold $(M^{m}, g)$ whose volume is the same as the standard sphere $(S^{m}, g_{r o u n d})$ . If $p > \frac{m}{2}$ and $\int_{M} {R c - (m - 1) g}_{-}^{p} d v$ is sufficiently small, we show that the normalized Ricci flow initiated from $(M^{m}, g)$ will exist immortally and converge to the standard sphere. The choice of $p$ is optimal.

Ricci flow Long-time existence & convergence

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.
11A.1
The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles
A heat equation for the metric: spheres, cylinders and scaling.

Ricci curvature integrals, local functionals, and the Ricci flow

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