GRADIENT ESTIMATES FOR POSITIVE EIGENFUNCTIONS OF THE $\mathcal {L}$-OPERATOR ON CONFORMAL SOLITONS AND THEIR APPLICATIONS

Abstract We prove a local gradient estimate for positive eigenfunctions of the $\mathcal {L}$ -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 and Sun [‘Gradient estimate for the positive solutions of $\mathcal Lu = 0$ and $\mathcal Lu ={\partial u}/{\partial t}$ 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.