Improved Pseudolocality on Large Hyperbolic Balls

TL;DR

This paper presents an improved pseudolocality theorem for Ricci flows on 2D surfaces, showing curvature remains close to hyperbolic value over exponentially long times on large hyperbolic balls, with implications for higher dimensions.

Contribution

It provides a sharper pseudolocality result for 2D Ricci flows on hyperbolic regions, extending understanding of curvature behavior over large scales.

Findings

Curvature remains close to hyperbolic value for exponentially long times

Results are specific to two-dimensional surfaces with hyperbolic initial conditions

Conjectures are made for higher-dimensional generalizations

Abstract

We obtain an improved pseudolocality result for Ricci flows on two-dimensional surfaces that are initially almost-hyperbolic on large hyperbolic balls. We prove that, at the central point of the hyperbolic ball, the Gauss curvature remains close to the hyperbolic value for a time that grows exponentially in the radius of the ball. This two-dimensional result allows us to precisely conjecture how the phenomenon should appear in the higher dimensional setting.