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GRADIENT ESTIMATES FOR POSITIVE EIGENFUNCTIONS OF THE
$\mathcal {L}$-OPERATOR ON CONFORMAL SOLITONS AND THEIR APPLICATIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 18 November 2024, pp. 1-12
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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.
