A generalization of Escobar-Riemann mapping type problem on smooth metric measure spaces

TL;DR

This paper extends the Escobar-Riemann mapping problem to smooth metric measure spaces with boundary, introducing a new weighted constant and solving the problem in the negative case, along with establishing an Aubin type inequality.

Contribution

It generalizes the Escobar-Riemann mapping problem to smooth metric measure spaces and provides solutions when the weighted constant is negative, along with related inequalities.

Findings

Resolved the Escobar-Riemann mapping type problem for negative weighted constant.

Introduced the Escobar weighted constant and proved its properties.

Established an Aubin type inequality linking the weighted constant and trace inequality.

Abstract

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization of Sobolev Trace Inequality deduced by Bolley at. al. This trace inequality allows us to introduce an Escobar quotient and its infimum. This infimum we call the Escobar weighted constant. The Escobar-Riemann mapping type problem for smooth metric measure spaces in manifolds with boundary consists of finding a function which attains the Escobar weighted constant. Furthemore, we resolve the later problem when Escobar weighted constant is negative. Finally, we get an Aubin type inequality connecting the weighted Escobar constant for compact smooth metric measure space and the optimal constant for the trace inequality due to Bolley at. al.