Local solution and extension to the Calabi flow

TL;DR

This paper studies the local existence of solutions to the Calabi flow starting from initial metrics with limited regularity and establishes conditions under which the flow can be extended on compact Kähler surfaces by controlling metric bounds.

Contribution

It provides a new local existence result for the Calabi flow with C^α initial metrics and extends the flow on compact Kähler surfaces under bounded metric conditions.

Findings

Established local solutions for the Calabi flow with C^α initial metrics.

Proved extension of the flow on compact Kähler surfaces when metrics are bounded in L^ sense.

Derived higher order derivative estimates from second order derivatives for a fourth order nonlinear PDE.

Abstract

We consider the local solution to the Calabi flow for C^\alpha initial metric. We also prove that the Calabi flow on compact Kaehler surfaces can be extended once the metrics along the flow are bounded in L^\infty sense. This can be viewed as obtaining higher order derivative estimates from second order derivatives for a fourth order nonlinear equation.