Volume bounds of the Ricci flow on closed manifolds

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 $${\mathrm{Vol}}_{g(t)}\ge e^{T\lambda -\frac{n}{2}}{\left(\frac{4}{(A(\lambda -r)+4B)T}\right)}^{\frac{n}{2}}{\left(T-t\right)}^{\frac{n}{2}},$$ where $r:={\inf }_{\Vert \varphi {\Vert }_2^2=1}{\int }_{M}R{\varphi }^{2}d{\mathrm{vol}}_{g(0)}$, $\lambda :={\inf }_{\Vert \varphi {\Vert }_2^2=1}{\int }_{M}4|\nabla \varphi {|}^2+R{\varphi }^{2}d{\mathrm{vol}}_{g(0)}$ 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 $A=\frac{4}{n(n-2)}{\omega }_n^{-\frac{2}{n}}$, $B=\frac{n-1}{n-2}{\omega }_n^{-\frac{2}{n}}$. On the other hand, if the diameter satisfies ${\mathrm{diam}}_{g(t)}\le c_1\sqrt{T-t}$ and there exist 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 $g(0)$.