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Heat kernel asymptotics at the cut locus

We give small-time asymptotic expansions for the gradient and Hessian of the logarithm of the heat kernel at the cut locus of a Riemannian manifold. We relate the leading terms of the expansions to the structure of the cut locus, especially to conjugacy, and we provide a probabilistic interpretation in terms of the Brownian bridge. In particular, we can characterize the cut locus in terms of the behavior of the log-Hessian. We also mention how the distributional asymptotics can be used to compute the distributional Hessian of the energy function.

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