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...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2013-12
Main Authors: Barilari, Davide, Boscain, Ugo, Charlot, Grégoire, Neel, Robert W
Format: Article
Language:English
Subjects:
Asymptotic properties
Brownian motion
Riemann manifold
Riemann surfaces
Singularities
Theorems
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422