Vol. 13, No. 7, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 6, 1671–1976
Issue 5, 1333–1669
Issue 4, 985–1332
Issue 3, 667–984
Issue 2, 323–666
Issue 1, 1–322

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
New formulas for the Laplacian of distance functions and applications

Fabio Cavalletti and Andrea Mondino

Vol. 13 (2020), No. 7, 2091–2147

The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially nonbranching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds together with lower bounds and a precise description of the singular part. The exact representation formula for the Laplacian of a general 1-Lipschitz function holds also (and seems new) in a general complete Riemannian manifold.

We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic splitting theorem for infinitesimally Hilbertian, essentially nonbranching spaces satisfying MCP(0,N).

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Ricci curvature, optimal transport, Laplacian comparison, distance function
Mathematical Subject Classification 2010
Primary: 49J52, 53C23
Received: 20 November 2018
Revised: 21 July 2019
Accepted: 6 September 2019
Published: 10 November 2020
Fabio Cavalletti
Mathematics Area
Andrea Mondino
Mathematical Institute
University of Oxford
United Kingdom
Mathematics Institute
University of Warwick
United Kingdom