Gradient estimates and Liouville properties for the drifted Laplacian

TL;DR

This paper investigates conditions under which positive solutions to the drifted Laplacian are constant, establishing Liouville properties on manifolds with specific curvature bounds and decay conditions for the vector field.

Contribution

It provides new gradient estimates and Liouville theorems for the drifted Laplacian under curvature and decay assumptions, extending previous results to more general settings.

Findings

Liouville property holds when Ric_X is non-negative and X decays at infinity.

Gradient estimates for positive solutions to the semilinear equation are established.

Manifolds with Ric_X ≥ -(n-1)K satisfy the Liouville property under certain conditions.

Abstract

In this paper, we discuss the validity of the Liouville property for $X$-harmonic functions, i.e. positive solution to $\Delta_{X}u=0$, where $X$ is a vector field on a complete, non-compact Riemannian manifold and $\Delta_{X}$ is the drifted Laplacian. In particular, we show that if the $X$-Bakry-\'Emery-Ricci curvature $\mathrm{Ric}_{X}$ is non-negative and the norm of $X$ decays to zero at infinity, then the manifold has the Liouville property for the $X$-Laplacian. The proof exploits a local gradient estimate for positive solutions to the semilinear equation $\Delta_{X}u+F(u)=0$, which holds when $F$ satisfies the structural conditions $tF'(t)-F(t)\le\alpha$ and $\vert F(t)\vert\le\beta t$, and the manifold has $\mathrm{Ric}_{X}\ge-(n-1)K$.