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

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.