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

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 Δ g u = 0 \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 div ⁡ ( | x | θ ∇ u ) = 0 \operatorname {div}(|x|^\theta \nabla u)=0 in R n ∖ { 0 } \mathbb {R}^n\setminus \{0\} that is bounded on one side in both a neighborhood of the origin as well as at infinity.