GRADIENT ESTIMATES FOR POSITIVE EIGENFUNCTIONS OF THE L -OPERATOR ON CONFORMAL SOLITONS AND THEIR APPLICATIONS

We prove a local gradient estimate for positive eigenfunctions of the L-operator on conformal solitons given by a general conformal vector field. As an application, we obtain a Liouville type theorem for Lu = 0, which improves the one of Li and Sun ['Gradient estimate for the positive solutions of Lu = 0 and Lu = partial derivative u/partial derivative t on conformal solitons', Acta Math. Sin. (Engl. Ser.) 37 (11) (2021), 1768-1782]. We also consider applications where manifolds are special conformal solitons and obtain a better Liouville type theorem in the case of self-shrinkers.