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. 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.
"A Yamabe-type problem on smooth metric measure spaces." J. Differential Geom. 101 (3) 467 - 505, November 2015. https://doi.org/10.4310/jdg/1445518921