Calculus of Variations and Geometric Measure Theory

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

Small time heat kernel asymptotics at the sub-Riemannian cut locus

created by barilari on 12 Apr 2012
modified on 28 Nov 2012


Published Paper

Inserted: 12 apr 2012
Last Updated: 28 nov 2012

Journal: Journal of Differential Geometry
Volume: 92
Number: 3
Pages: 373-416
Year: 2012


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 endowed with the Euclidean volume 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 $y$.