# 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