A Sharp Sobolev Interpolation Inequality on Finsler Manifolds

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Abstract

In this paper we study a sharp Sobolev interpolation inequality on Finsler manifolds. We show that Minkowski spaces represent the optimal framework for the Sobolev interpolation inequality on a large class of Finsler manifolds: (1) Minkowski spaces support the sharp Sobolev interpolation inequality; (2) any complete Berwald space with non-negative Ricci curvature which supports the sharp Sobolev interpolation inequality is isometric to a Minkowski space. The proofs are based on properties of the Finsler–Laplace operator and on the Finslerian Bishop–Gromov volume comparison theorem.

Original languageEnglish
Pages (from-to)2226-2240
Number of pages15
JournalJournal of Geometric Analysis
Volume25
Issue number4
DOIs
Publication statusPublished - Oct 1 2015

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Keywords

  • Finsler manifold
  • Minkowski space
  • Ricci curvature
  • Rigidity
  • Sharp constant
  • Sobolev interpolation inequality

ASJC Scopus subject areas

  • Geometry and Topology

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