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Mathematics > Differential Geometry

arXiv:1805.03694 (math)
[Submitted on 9 May 2018 (v1), last revised 6 Jul 2018 (this version, v2)]

Title:A generalization of Escobar-Riemann mapping type problem on smooth metric measure spaces

Authors:Jhovanny Muñoz Posso
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Abstract:In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization of Sobolev Trace Inequality deduced by Bolley at. al. This trace inequality allows us to introduce an Escobar quotient and its infimum. This infimum we call the Escobar weighted constant. The Escobar-Riemann mapping type problem for smooth metric measure spaces in manifolds with boundary consists of finding a function which attains the Escobar weighted constant. Furthemore, we resolve the later problem when Escobar weighted constant is negative. Finally, we get an Aubin type inequality connecting the weighted Escobar constant for compact smooth metric measure space and the optimal constant for the trace inequality due to Bolley at. al.
Comments: We have been informed by Nguyen Van Hoang that the Trace inequality, even in a general version, has been proven already by Bolley, F. et al. in 2018. However, the proof we did do in this particular case is different. In this version, we have changed the title and we have made some arrangements to give the credits. The other results work for $m\geq0$ instead of $m\in\mathbf{N}\cup\{0\}$
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
Cite as: arXiv:1805.03694 [math.DG]
  (or arXiv:1805.03694v2 [math.DG] for this version)
  https://doi.org/10.48550/arXiv.1805.03694

Submission history

From: Jhovanny Muñoz Posso [view email]
[v1] Wed, 9 May 2018 18:56:37 UTC (20 KB)
[v2] Fri, 6 Jul 2018 20:03:05 UTC (19 KB)
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