A Yamabe-type problem on smooth metric measure spaces

TL;DR

This paper introduces a Yamabe-type problem on smooth metric measure spaces, connecting classical Yamabe issues with Perelman's entropy minimization, and explores conditions for the existence of minimizers.

Contribution

It formulates a new Yamabe-type problem on smooth metric measure spaces and analyzes the existence of minimizers based on the weighted Yamabe constant.

Findings

Minimizers exist when the weighted Yamabe constant is less than its Euclidean value.

Strict inequality for the weighted Yamabe constant holds in many cases.

Counterexample shows minimizers do not always exist.

Abstract

We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's $\nu$-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family of Gagliardo--Nirenberg--Sobolev inequalities studied by Del Pino and Dolbeault. We show that minimizers always exist on a compact manifold provided the so-called weighted Yamabe constant is strictly less than its value on Euclidean space. We also show that strict inequality holds for a large class of smooth metric measure spaces, but we will also give an example which shows that minimizers of the weighted Yamabe constant do not always exist.