15809 articles – 31799 references  [version française]
 HAL: hal-00687651, version 1
 arXiv: 1201.3023
 Small time heat kernel asymptotics at the sub-Riemannian cut locus
 Davide Barilari 1, Ugo Boscain 1
 (2012-03-23)
 For a sub-Riemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point $y$ of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e.\ all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre $4t\log p_t(x,y)\to -d^2(x,y)$ for $t\to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane we get the expansion $p_t(x,y)\sim t^{-5/4}\exp(-d^2(x,y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$.
 1: GECO (INRIA Saclay - Ile de France / CMAP Centre de Mathématiques Appliquées) INRIA : SACLAY - ÎLE-DE-FRANCE – CNRS : UMR7641 – Polytechnique - X 2: Department of Mathematics Lehigh University, Bethlehem, USA
 Subject : Mathematics/Analysis of PDEsMathematics/Differential Geometry