Volume bounds of the Ricci flow on closed manifolds

Chih-Wei Chen; Zhenlei Zhang

Paper · Mar 26, 2018

Volume bounds of the Ricci flow on closed manifolds

Chih-Wei Chen, Zhenlei Zhang

math.DGarXiv:1803.09591

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Abstract

Let ${g (t)}_{t \in [0, T)}$ be the solution of the Ricci flow on a closed Riemannian manifold $M^{n}$ with $n \geq 3$ . Without any assumption, we derive lower volume bounds of the form $Vol_{g (t)} \geq C (T - t)^{\frac{n}{2}}$ , where $C$ depends only on $n$ , $T$ and $g (0)$ . In particular, we show that $Vol_{g (t)} \geq e^{T λ - \frac{n}{2}} (\frac{4}{( A ( λ - r ) + 4 B ) T})^{\frac{n}{2}} (T - t)^{\frac{n}{2}},$ where $r := in f_{∥ φ ∥_{2}^{2} = 1} \int_{M} R φ^{2} d vol_{g (0)}$ , $λ := in f_{∥ φ ∥_{2}^{2} = 1} \int_{M} 4∣\nabla φ ∣^{2} + R φ^{2} d vol_{g (0)}$ and $A, B$ are Sobolev constants of $(M, g (0))$ . This estimate is sharp in the sense that it is achieved by the unit sphere with scalar curvature $R_{g (0)} = n (n - 1)$ and $A = \frac{4}{n ( n - 2 )} ω_{n}^{- \frac{2}{n}}$ , $B = \frac{n - 1}{n - 2} ω_{n}^{- \frac{2}{n}}$ . On the other hand, if the diameter satisfies $diam_{g (t)} \leq c_{1} T - t$ and there exist a point $x_{0} \in M$ such that $R (x_{0}, t) \leq c_{2} (T - t)^{- 1}$ , then we have $Vol_{g (t)} \leq C (T - t)^{\frac{n}{2}}$ for all $t > \frac{T}{2}$ , where $C$ depends only on $c_{1}, c_{2}, n, T$ and $g (0)$ .

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Submitted: Mar 26, 2018
Subjects: math.DG

Ricci flow Scalar curvature

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Background reading

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.
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.

[1] 6C.4
The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

[2] 9B.6
Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds
Positive scalar curvature, its obstructions, and the positive mass theorem.

[3] 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.

Volume bounds of the Ricci flow on closed manifolds

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