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A geometric Laplace method

Flavien Léger and François-Xavier Vialard

Vol. 5 (2023), No. 4, 1041–1080

A classical tool for approximating integrals is the Laplace method. The first- and higher-order Laplace formulas are most often written in coordinates without any geometrical interpretation. In this article, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation to the first-order term of the Laplace method. The central tool is a metric introduced by Kim and McCann in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two metrics arising naturally in the Kim–McCann framework. We give an explicitly quantified version of the Laplace formula, as well as examples of applications.

Laplace method, Kim–McCann metric, Ma–Trudinger–Wang tensor, heat kernel asymptotics
Mathematical Subject Classification
Primary: 41A60, 49Q22, 53B12, 53C55, 90B06
Received: 25 January 2023
Revised: 19 July 2023
Accepted: 4 September 2023
Published: 15 December 2023
Flavien Léger
François-Xavier Vialard
Université Gustave Eiffel