Methods and Applications of Analysis

Volume 18 (2011)

Number 3

Locally nearly spherical surfaces are almost-positively $c$-curved

Pages: 269 – 302

DOI: https://dx.doi.org/10.4310/MAA.2011.v18.n3.a2

Authors

Philippe Delanoë (Laboratoire J.–A. Dieudonné, Faculté des Sciences, Université de Nice Sophia Antipolis, Nice, France)

Yuxin Ge (Centre de Mathématiques, Faculté des Sciences et Technologie, Université Paris Est Créteil Val de Marne, Créteil, France)

Abstract

The $c$-curvature of a complete surface with Gauss curvature close to 1 in $C²$ norm is almost-positive (in the sense of Kim–McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant curvature case, keeping track of an estimate on the closeness curvature condition.

Keywords

Monge problem, quadratic cost, compact surfaces, nearly spherical, positive ccurvature, stability

2010 Mathematics Subject Classification

Primary 53C21. Secondary 49N60.

Published 5 April 2012