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Abstract
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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.
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Keywords
Laplace method, Kim–McCann metric, Ma–Trudinger–Wang
tensor, heat kernel asymptotics
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Mathematical Subject Classification
Primary: 41A60, 49Q22, 53B12, 53C55, 90B06
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Milestones
Received: 25 January 2023
Revised: 19 July 2023
Accepted: 4 September 2023
Published: 15 December 2023
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© 2023 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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