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

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].

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Received 17 May 2018

Accepted 16 April 2019

Published 24 May 2022