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 √-entropy. In Euclidean space, this problem reduces to the characterization of the minimizers of the family of Gagliardo-Nirenberg inequalities studied by Del Pino and Dolbeault. We show that minimizers always exist on a compact manifold provided the 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 also give an example which shows that minimizers of the weighted Yamabe constant do not always exist.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology