Volume bounds of the Ricci flow on closed manifolds

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 $g(0)$.