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Volume bounds of the Ricci flow on closed manifolds

  • Chih-Wei ChenEmail author
  • Zhenlei Zhang
Article
  • 24 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

  1. 1.Department of Applied MathematicsNational Sun Yat-sen UniversityKaohsiungTaiwan
  2. 2.Department of MathematicsCapital Normal UniversityBeijingChina

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