Some results on the weighted Yamabe problem with or without boundary

TL;DR

This paper investigates Yamabe-type problems on smooth metric measure spaces with boundary, focusing on uniqueness, characterization of solitons, and existence of minimizers, extending classical results to weighted settings.

Contribution

It introduces new results on the weighted Yamabe problem with boundary, including uniqueness, soliton characterization, and existence of minimizers, in the context of smooth metric measure spaces.

Findings

Proved uniqueness of solutions to the weighted Yamabe problem with boundary.

Characterized weighted Yamabe solitons with boundary.

Established existence of positive minimizers in the weighted Escobar quotient.

Abstract

Let $(M^n,g,e^{-\phi}dV_g,e^{-\phi}dA_g,m)$ be a compact smooth metric measure space with boundary with $n\geqslant 3$. In this article, we consider several Yamabe-type problems on a compact smooth metric measure space with or without boundary: uniqueness problem on the weighted Yamabe problem with boundary, characterization of the weighted Yamabe solitons with boundary and the existence of positive minimizers in the weighted Escobar quotient.