Bôcher-type theorem on n-dimensional manifolds with conical metric

Li, Jiayu; Wan, Fangshu

doi:10.1090/proc/14554

Bôcher-type theorem on n-dimensional manifolds with conical metric
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by Jiayu Li and Fangshu Wan

Proc. Amer. Math. Soc. 147 (2019), 4527-4538

DOI: https://doi.org/10.1090/proc/14554

Published electronically: June 27, 2019

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Abstract:

We generalize the Bôcher-type theorem and give a sharp characterization of the behavior at the isolated singularities of a solution bounded on one side for the equation $\Delta _g u =0$ on singular manifolds with conical metrics. Furthermore, we also obtain a Liouville-type result which demonstrates that the fundamental solution is the unique nontrivial solution of $\operatorname {div}(|x|^\theta \nabla u)=0$ in $\mathbb {R}^n\setminus \{0\}$ that is bounded on one side in both a neighborhood of the origin as well as at infinity.

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