Isoperimetric inequality under K\"ahler Ricci flow

TL;DR

This paper establishes a uniform isoperimetric inequality for K"ahler Ricci flows with positive first Chern class, along with related gradient bounds and Poincaré inequalities, advancing understanding of geometric analysis under Ricci flow.

Contribution

It proves a uniform isoperimetric inequality and new bounds for harmonic functions without assuming Ricci curvature lower bounds in K"ahler Ricci flow.

Findings

Uniform isoperimetric inequality holds for all time

Cheng-Yau type log gradient bound established

Poincaré inequality proven without Ricci curvature lower bound

Abstract

Let $({\M}, g(t))$ be a K\"ahler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$, and a Poincar\'e inequality without assuming the Ricci curvature is bounded from below.