Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers

TL;DR

This paper investigates geometric and analytic properties of Ricci shrinkers, including volume growth, Liouville theorems, and gradient estimates for harmonic functions, enhancing understanding of their structure and behavior.

Contribution

It provides new results on volume comparison, Liouville theorems, and integral properties of harmonic functions specifically for complete noncompact gradient Ricci shrinkers.

Findings

Established volume growth estimates for Ricci shrinkers

Proved Liouville theorems for f-harmonic functions on these manifolds

Derived local gradient estimates and volume comparison results

Abstract

In this paper, we study volume growth, Liouville theorem and the local gradient estimate for $f$-harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.