Volume bounds of the Ricci flow on closed manifolds

TL;DR

This paper establishes sharp lower volume bounds for Ricci flow solutions on closed manifolds, depending on initial geometric data, and provides conditions under which volume estimates are also upper bounds.

Contribution

It derives explicit, sharp lower volume bounds for Ricci flow on closed manifolds without assumptions, and links geometric quantities to volume behavior near singularity.

Findings

Lower volume bounds depend only on initial data and time remaining.

Sharpness of bounds demonstrated on the unit sphere.

Upper volume bounds established under diameter and scalar curvature conditions.

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 ${\rm Vol}_{g(t)}\geq C (T-t)^{\frac{n}{2}}$, where $C$ depends only on $n$, $T$ and $g(0)$. In particular, we show that $${\rm Vol}_{g(t)} \geq e^{ T\lambda-\frac{n}{2}} \left(\frac{4}{(A(\lambda-r)+4B)T}\right)^{\frac{n}{2}}\left(T-t\right)^{\frac{n}{2}},$$ where $r:=\inf_{\|\phi\|_2^2=1} \int_M R\phi^2 \ d{\rm vol}_{g(0)}$, $\lambda:=\inf_{\|\phi\|_2^2=1} \int_M 4|\nabla\phi|^2+R\phi^2\ d{\rm vol}_{g(0)}$ 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 ${\rm diam}_{g(t)}\leq c_1\sqrt{T-t}$ and there exist a point $x_0\in M$ such that $R(x_0,t)\leq c_2(T-t)^{-1}$, then we have ${\rm Vol}_{g(t)}\leq C (T-t)^{\frac{n}{2}}$ for all $t>\frac{T}{2}$, where $C$ depends only on $c_1,c_2,n,T$ and $g(0)$.