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- W2110309295 endingPage "rnw141" @default.
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- W2110309295 abstract "In this article, 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 by-product, 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 nongeneric case, a cornucopia of asymptotics can occur, even for Riemannian surfaces. Let M be a complete Riemannian or sub-Riemannian manifold. Endow this structure with a smooth volume μ and consider the associated Laplace operator ∆ defined as the divergence of the (horizontal) gradient. Under the assumption that the Hörmander condition is satisfied (which is obvious in the Riemannian case), and thanks to the completeness of M, the operator ∆ is hypoelliptic and admits a smooth symmetric heat kernel pt(q1,q2)[25, 41]. (In this article, by sub-Riemannian manifold, we mean a constant rank sub-Riemannian manifold which is not Riemannian. By a (sub)-Riemannian manifold, we mean a constant-rank sub-Riemannian manifold, which is possibly Riemannian. However, several results of the article hold for more general rank-varying structures in the sense of [12, Appendix A], for instance, Grushin-like structures. This will be specified in the article.)" @default.
- W2110309295 created "2016-06-24" @default.
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- W2110309295 date "2016-07-13" @default.
- W2110309295 modified "2023-10-12" @default.
- W2110309295 title "On the Heat Diffusion for Generic Riemannian and Sub-Riemannian Structures" @default.
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- W2110309295 doi "https://doi.org/10.1093/imrn/rnw141" @default.
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