Riemannian starshape and capacitary problems

TL;DR

This paper extends classical Euclidean results on starshaped level sets of capacitary potentials to Riemannian warped product spaces, including cases with the $q$-Laplacian, broadening geometric analysis understanding.

Contribution

It proves the Riemannian analogue of Euclidean starshaped level set results for capacitary potentials, including the $q$-Laplacian case, in warped product manifolds.

Findings

Level sets of capacitary potentials are starshaped in Riemannian warped products.

Extension of Euclidean results to Riemannian settings with warped products.

Inclusion of $q$-Laplacian generalization in the analysis.

Abstract

We prove the Riemannian version of a classical Euclidean result: every level set of the capacitary potential of a starshaped ring is starshaped. In the Riemannian setting, we restrict ourselves to starshaped rings in a warped product of an open interval and the unit sphere. We also extend the result by replacing the Laplacian with the $q$-Laplacian.