Gradient estimates for positive eigenfunctions of $\mathcal{L}$-operator on conformal solitons and its applications

Abstract

We prove a local gradient estimate for positive eigenfunctions of $\mathcal{L}$-operator on conformal solitons given by a general conformal vector field. As an application, we obtain a Liouville type theorem for $\mathcal{L}u=0$, which improves the one of Li--Sun (Acta Math. Sin. (Engl. Ser.), 37(11): 1768--1782, 2021.). We also consider applications where manifolds are special conformal solitons. Especially in the case of self-shrinkers, a better Liouville type theorem is obtained.

This version revises the statements of Corollary 1.2 and Corollary 1.6 in the version dated April 16, 2024, making them clearer, and also corrects some typographical errors