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On the differential structure of metric measure spaces and applications


About this Title

Nicola Gigli

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 236, Number 1113
ISBNs: 978-1-4704-1420-7 (print); 978-1-4704-2279-0 (online)
DOI: http://dx.doi.org/10.1090/memo/1113
Published electronically: November 13, 2014
Keywords:Differential geometry, metric geometry

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. Differentials and gradients
  • Chapter 4. Laplacian
  • Chapter 5. Comparison estimates
  • Appendix A. On the duality between cotangent and tangent spaces
  • Appendix B. Remarks about the definition of the Sobolev classes

Abstract


The main goals of this paper are: (i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative. (ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like , where is a function and is a measure. (iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.

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