**Memoirs of the American Mathematical Society**

2015;
91 pp;
Softcover

Print ISBN: 978-1-4704-1420-7

Product Code: MEMO/236/1113

List Price: $76.00

Individual Member Price: $45.60

**Electronic ISBN: 978-1-4704-2279-0
Product Code: MEMO/236/1113.E**

List Price: $76.00

Individual Member Price: $45.60

# On the Differential Structure of Metric Measure Spaces and Applications

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*Nicola Gigli*

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 \(\Delta g=\mu\), where \(g\) is a function and \(\mu\) 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.

#### Table of Contents

# Table of Contents

## On the Differential Structure of Metric Measure Spaces and Applications

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Chapter 2. Preliminaries 916
- Chapter 3. Differentials and gradients 1926
- Chapter 4. Laplacian 3542
- Chapter 5. Comparison estimates 5966
- Appendix A. On the duality between cotangent and tangent spaces 7784
- Appendix B. Remarks about the definition of the Sobolev classes 8592
- References 8996
- Back Cover Back Cover1104