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GRADIENT ESTIMATES FOR POSITIVE EIGENFUNCTIONS OF THE 
$\mathcal {L}$-OPERATOR ON CONFORMAL SOLITONS AND THEIR APPLICATIONS
Published online by Cambridge University Press: 18 November 2024
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.
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- Research Article
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
This work is partially supported by the National Natural Science Foundation of China (Grant No. 12001410).
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