Proposition
Let \(g_t\) be a Ricci flow on a complex Kahler manifold M. The volume of \(M\)is
\begin{align*} \operatorname{Vol}_{g_t}(M) = \int_M[\omega_t]^2. \end{align*}There are three possibilities.
(1)
\begin{align*} \lim_{t\to T} \operatorname{Vol}_{g_t}(M) > 0 \end{align*}(2)
\begin{align*} \operatorname{Vol}_{g_t}(M)\sim_{\mathbb P^1} T - t \end{align*}(3) This is called “extinction”:
\begin{align*} \operatorname{Vol}_{g_t}(M) \sim (T-t)^2 \end{align*}