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. S...

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Bibliographic Details
Published inarXiv.org
Main Author Zhao, Guangwen
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 07.05.2024
Subjects
Eigenvectors
Fields (mathematics)
Liouville theorem
Solitary waves
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Summary: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.
ISSN:2331-8422