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Elliptic PDEs on Compact Ricci Limit Spaces and Applications


About this Title

Shouhei Honda

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 253, Number 1211
ISBNs: 978-1-4704-2854-9 (print); 978-1-4704-4417-4 (online)
DOI: https://doi.org/10.1090/memo/1211
Published electronically: March 7, 2018
Keywords:Gromov-Hausdorff convergence, Ricci curvature, Elliptic PDEs.

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. $L^p$-convergence revisited
  • Chapter 4. Poisson’s equations
  • Chapter 5. Schrödinger operators and generalized Yamabe constants
  • Chapter 6. Rellich type compactness for tensor fields
  • Chapter 7. Differential forms

Abstract


In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrödinger operators, generalized Yamabe constants and eigenvalues of the Hodge Laplacian, with respect to the Gromov-Hausdorff topology. We apply these to the study of second-order differential calculus on such limit spaces.

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