An existence time estimate for K\"ahler-Ricci flow

TL;DR

This paper provides estimates for the existence time, curvature, and bounds of the Kähler-Ricci flow on noncompact manifolds, depending only on initial data and initial bounds, including cases with unbounded initial curvature.

Contribution

It introduces new estimates for the existence time and bounds of the Kähler-Ricci flow that depend solely on initial conditions, extending to unbounded curvature scenarios.

Findings

Derived bounds depend only on initial data and initial bounds.

Extended estimates to cases with unbounded initial curvature.

Provided existence time estimates for the Kähler-Ricci flow.

Abstract

Fix a complete noncompact \K manifold $(M^n,h_0)$ with bounded curvature. Let $g(t)$ be a bounded curvature solution to the \KR flow starting from some $g_0$ uniformly equivalent to $h_0$. We estimate the existence time of $g(t)$ together with $C^0$ bounds and curvature bounds, where the estimates depend only on $h_0$ and the $C^0$ distance between $g_0$ and $h_0$. We also generalize these results to cases when $g_0$ may have unbounded curvature.