A generalization of Aubin's result for a Yamabe-type problem on smooth metric measure spaces

TL;DR

This paper extends Aubin's classical result on the Yamabe problem to smooth metric measure spaces, specifically addressing nonlocally conformally flat manifolds with dimension ≥6 and parameters near nonnegative integers.

Contribution

It generalizes Aubin's result to Yamabe-type problems on smooth metric measure spaces for a broader class of manifolds and parameters close to nonnegative integers.

Findings

Extended Aubin's result to smooth metric measure spaces

Addressed nonlocally conformally flat manifolds with dimension ≥6

Applicable for parameters near nonnegative integers

Abstract

The Yamabe problem in compact closed Riemannian manifolds is concerned with finding a metric with constant scalar curvature in the conformal class of a given metric. This problem was solved by the combined work of Yamabe, Trudinger, Aubin, and Schoen. In particular, Aubin solved the case when the Riemannian manifold is compact, is nonlocally conformally flat and has a dimension equal to or greater than $6$. In $2015$, Case considered a Yamabe-type problem in the setting of smooth measure space in manifolds and for a parameter $m$, which generalizes the original Yamabe problem when $m=0$. Additionally, Case solved this problem when the parameter $m$ is a natural number. In the context of the Yamabe-type problem, we generalize Aubin's result for nonlocally conformally flat manifolds, with dimension equal and greater than 6 and parameter $m$ close to nonnegative integers.