On the heat diffusion for generic Riemannian and sub-Riemannian structures

In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic...

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Published in:arXiv.org 2013-12
Main Authors: Barilari, Davide, Boscain, Ugo, Charlot, Grégoire, Neel, Robert W
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Language:English
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Asymptotic properties
Brownian motion
Riemann manifold
Riemann surfaces
Singularities
Theorems
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Neel, Robert W
description In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As a byproduct, we show that, for generic low-dimensional Riemannian manifolds, the only singularities of the exponential map, as a Lagragian map, that can arise along a minimizing geodesic are \(A_3\) and \(A_5\) (in the classification of Arnol'd's school). We show that in the non-generic case, a cornucopia of asymptotics can occur, even for Riemannian surfaces.
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subjects Asymptotic properties
Brownian motion
Riemann manifold
Riemann surfaces
Singularities
Theorems
title On the heat diffusion for generic Riemannian and sub-Riemannian structures
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