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

Li Ma

Vol. 296 (2018), No. 2, 357–369

DOI: 10.2140/pjm.2018.296.357

Abstract

We study volume growth, a Liouville theorem for $f$-harmonic functions, and a volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers and gradient steady Ricci solitons. We also study integral properties of $f$-harmonic functions and harmonic functions on complete manifolds, such as the Ricci–Einstein solitons.

Keywords

Ricci shrinkers, $f$-harmonic functions, Liouville theorem, volume comparison

Mathematical Subject Classification 2010

Primary: 53C21, 53C44, 53C25

Milestones

Received: 3 January 2017

Revised: 5 December 2017

Accepted: 2 March 2018

Published: 16 July 2018