Liouville property for $f$-harmonic functions with polynomial growth

TL;DR

This paper proves a Liouville property for polynomial growth $f$-harmonic functions on certain noncompact metric measure spaces with nonnegative Bakry-Émery Ricci curvature and sublinear geodesic sphere diameter growth.

Contribution

It establishes a Liouville theorem for $f$-harmonic functions under specific geometric conditions, extending previous results to new curvature and growth scenarios.

Findings

Liouville property holds for $f$-harmonic functions with polynomial growth

Nonnegative Bakry-Émery Ricci curvature is sufficient

Sublinear growth of geodesic sphere diameter is a key condition

Abstract

We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-\'Emery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.