A local Sobolev inequality on Ricci flow and its applications

TL;DR

This paper establishes a local Sobolev inequality for Ricci flows, linking the local $ u$-functional to Nash entropy and volume, and explores applications to understanding local geometric evolution and monotonicity formulas.

Contribution

The paper introduces a novel local Sobolev inequality for Ricci flows that relates the $ u$-functional to Nash entropy and volume, with applications to geometric analysis.

Findings

The local $ u$-functional depends only on Nash entropy at the disk's center.

The local Sobolev inequality helps analyze local geometric evolution.

Classical results related to Perelman's monotonicity are derived from these inequalities.

Abstract

In this article, we prove a local Sobolev inequality for complete Ricci flows. Our main result is that the local $\nu$-functional of a disk on a Ricci flow depends only on the Nash entropy based at the center of the disk, and consequently depends only on the volume of the disk. Furthermore, we introduce some applications of this local Sobolev inequality. These applications reveal the way in which the local geometry evolves along Ricci flow. In particular, we show that several classical theorems related to Perelman's monotonicity formula can be derived from our results.