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

Davide Barilari, Equipe INRIA GECO Saclay-ˆ Ile-de-France, Ugo Boscain Equipe INRIA GECO Saclay-ˆ Ile-de-France, Robert W. Neel Lehigh University,

Differential Geometry mathscidoc:1609.10288

Journal of Differential Geometry, 93, (1), 373-416, 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 pt(x, y) ! −d2(x, y) for t ! 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 pt(x, y)  t−5/4 exp(−d2(x, y)/4t) where y is reached from a - Riemannian point x by a minimizing geodesic which is conjugate at y.
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  author={Davide Barilari,, Ugo Boscain, and Robert W. Neel},
  booktitle={Journal of Differential Geometry},
Davide Barilari,, Ugo Boscain, and Robert W. Neel. SMALL-TIME HEAT KERNEL ASYMPTOTICS AT THE SUB-RIEMANNIAN CUT LOCUS. 2012. Vol. 93. In Journal of Differential Geometry. pp.373-416.
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