In general, the exact maximal time $T$ of a Riemannian Ricci flow may not be easy to find. However, fortunately, for Kähler-Ricci flows, the maximal time of existence $T$ is explicitly determined by the initial Kähler class $[ω_0]$ and the first Chern class.

Theorem (Tian-Zhang ). Let $(M,ω(t))$ be an Kahler-Ricci flow $∂ω_t/∂t=-Ric(ω_t) $ on a compact Kahler manifold $M$, with $dim_C=n$, then the maximal existence time $T$ is given by

$T=sup \{t:[ω_0 ]-tc_1 (M)>0\}$.

we have Hyperbolic Kahler Ricci flow

$\frac{∂^2}{∂t}g_{i\bar{j}}(t)=−Ri\bar{j}$ So how can we formulate maximal time $T$ in Hyperbolic Kahler Ricci flow ?

In general, the exact maximal time $T$ of a Riemannian Ricci flow may not be easy to find. However, fortunately, for Kähler-Ricci flows, the maximal time of existence $T$ is explicitly determined by the initial Kähler class $[ω_0]$ and the first Chern class.

Theorem (Tian-Zhang ). Let $(M,ω(t))$ be an Kahler-Ricci flow $∂ω_t/∂t=-Ric(ω_t) $ on a compact Kahler manifold $M$, with $dim_C=n$, then the maximal existence time $T$ is given by

$T=sup \{t:[ω_0 ]-tc_1 (M)>0\}$.

we have Hyperbolic Kahler Ricci flow

$\frac{∂^2}{∂t}g_{i\bar{j}}(t)=−Ri\bar{j}$ So how can we formulate the maximal time $T$ in Hyperbolic Kahler Ricci flow ?