Chih-Wei Chen

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 $r:={\inf }_{\Vert \phi {\Vert }_2^2=1}{\int }_{M}R{\phi }^{2}d{\mathrm{vol}}_{g(0)}$, $\lambda :={\inf }_{\Vert \phi {\Vert }_2^2=1}{\int }_{M}4|\nabla \phi {|}^2+R{\phi }^{2}d{\mathrm{vol}}_{g(0)}$ and $A,B$ 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 $g(0)$.

Let $(M^3,g,f)$ be a nontrivial 3-dimensional steady gradient Ricci soliton. If the scalar curvature $R$ satisfies $c_1{r}^{-b}\le R\le c_2{r}^{-a}$ for some $a\in (0,1],b\ge a$, and $c_1,c_2>0$, then the umbilical ratio of the level sets of $f$ satisfies $\frac{2|A{|}^2-H^2}{H^2}\in O(r^{6a-\frac{8a^2}{b}})\cap O(r^{2b-4a})$.

We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\le K$ must satisfy $|Rm|\le C{t}^{-1}$ for all $t\in (0,T]$, where $C$ depends only on the dimension $n$ and $T$ depends on $K$ and the injectivity radius $in{j}_{g(t)}$. In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly unknown. In particular, another Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.