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Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1133–1144

# Volume bounds of the Ricci flow on closed manifolds

• Chih-Wei Chen
• Zhenlei Zhang
Article
• 49 Downloads

## Abstract

Let $$\{g(t)\}_{t\in [0,T)}$$ be the solution of the Ricci flow on a closed Riemannian manifold $$M^n$$ with $$n\ge 3$$. Without any assumption, we derive lower volume bounds of the form $$\mathrm{Vol}_{g(t)}\ge C (T-t)^{\frac{n}{2}}$$, where C depends only on n, T and g(0). In particular, we show that
\begin{aligned} \mathrm{Vol}_{g(t)} \ge e^{ T\lambda -\frac{n}{2}} \left( \frac{4}{(A(\lambda -r)+4B)T}\right) ^{\frac{n}{2}}\left( T-t\right) ^{\frac{n}{2}}, \end{aligned}
where $$r:=\inf _{\Vert \phi \Vert _2^2=1} \int _M R\phi ^2 \ d\mathrm{vol}_{g(0)}$$, $$\lambda :=\inf _{\Vert \phi \Vert _2^2=1} \int _M 4|\nabla \phi |^2+R\phi ^2\ d\mathrm{vol}_{g(0)}$$ and AB are Sobolev constants of (Mg(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 $$A=\frac{4}{n(n-2)}\omega _n^{-\frac{2}{n}}$$, $$B=\frac{n-1}{n-2}\omega _n^{-\frac{2}{n}}$$. On the other hand, if the diameter satisfies $$\mathrm{diam}_{g(t)}\le c_1\sqrt{T-t}$$ and there exists a point $$x_0\in M$$ such that $$R(x_0,t)\le c_2(T-t)^{-1}$$, then we have $$\mathrm{Vol}_{g(t)}\le C (T-t)^{\frac{n}{2}}$$ for all $$t>\frac{T}{2}$$, where C depends only on $$c_1,c_2,n,T$$ and g(0).

## Keywords

Ricci flow Volume estimate $$\mu$$-entropy

## Mathematics Subject Classification

Primary 53C44 Secondary 35A23

## Notes

### Acknowledgements

The first author appreciates Mao-Pei Tsui for suggesting him to compare the volume of sphere and other manifolds. He is always indebted to Shu-Cheng Chang and Huai-Dong Cao for their constant supports and discussions. He is supported by grant from MOST.

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## Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

• Chih-Wei Chen
• 1
• Zhenlei Zhang
• 2
1. 1.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan
2. 2.Department of MathematicsCapital Normal UniversityBeijingChina