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November 2015 A Yamabe-type problem on smooth metric measure spaces
Jeffrey S. Case
J. Differential Geom. 101(3): 467-505 (November 2015). DOI: 10.4310/jdg/1445518921


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.


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Jeffrey S. Case. "A Yamabe-type problem on smooth metric measure spaces." J. Differential Geom. 101 (3) 467 - 505, November 2015.


Published: November 2015
First available in Project Euclid: 22 October 2015

zbMATH: 1334.53031
MathSciNet: MR3415769
Digital Object Identifier: 10.4310/jdg/1445518921

Rights: Copyright © 2015 Lehigh University


Vol.101 • No. 3 • November 2015
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