Calculus of Variations and Geometric Measure Theory

D. Barilari - U. Boscain - G. Charlot - R. W. Neel

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

created by barilari on 11 Oct 2013
modified on 16 Apr 2021


Published Paper

Inserted: 11 oct 2013
Last Updated: 16 apr 2021

Journal: Int. Math. Res. Not.
Year: 2017


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 A3 and A5 (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.