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# Communications in Analysis and Geometry

## Volume 29 (2021)

### Number 8

### A generalization of the Escobar–Riemann mapping-type problem to smooth metric measure spaces

Pages: 1813 – 1846

DOI: https://dx.doi.org/10.4310/CAG.2021.v29.n8.a4

#### Author

#### Abstract

In this article, we introduce a problem analogous to the Yamabe-type problem considered by Case in [**4**], which generalizes the Escobar–Riemann mapping problem for smooth metric measure spaces with boundary. For this purpose, we consider the generalization of the Sobolev trace inequality deduced by Bolley *et al.* in [**3**]. This trace inequality allows us to introduce an Escobar quotient and its infimum. We call this infimum the weighted Escobar constant. The Escobar–Riemann mapping-type problem for smooth metric measure spaces in manifolds with boundary consists of finding a function that attains the weighted Escobar constant. Furthermore, we resolve this problem when the weighted Escobar constant is negative. Finally, we obtain an Aubin-type inequality, connecting the weighted Escobar constant for compact smooth metric measure spaces and the optimal constant for the trace inequality in [**3**].

Received 17 May 2018

Accepted 16 April 2019

Published 24 May 2022