Gradient estimates for positive eigenfunctions of L -operator on conformal solitons and its applications

We prove a local gradient estimate for positive eigenfunctions of L -operator on conformal solitons given by a general conformal vector field. As an application, we obtain a Liouville type theorem for 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.