Small time heat kernel asymptotics at the cut locus on surfaces of revolution

Barilari, Davide; Jendrej, Jacek

doi:10.1016/j.anihpc.2013.03.003

Cited by 7 publications

(6 citation statements)

References 23 publications

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“…Notice that this is consistent with the results obtained in [12] on surfaces of revolution. Figure 1: Cut locus and first conjugate locus in a generic 3D contact sub-Riemannian structure close to the starting point.…”

Section: The Riemannian Casesupporting

confidence: 93%

“…For the four-cusp case, two of the cusps are reached by cut-conjugate geodesics, while in the six-cusp case this happens for three of them. Notice that the first conjugate locus at a generic point looks like a suspension of the first conjugate locus that one gets on a Riemannian ellipsoid [12]. The precise statement of these facts can be found in [6,17] (see also [2]).…”

Section: D Contact Casementioning

confidence: 93%

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On the Heat Diffusion for Generic Riemannian and Sub-Riemannian Structures

Barilari

Boscain

Charlot

et al. 2016

Int Math Res Notices

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In this paper 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 byproduct, 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 A 3 and A 5 (in the classification of Arnol'd's school). We show that in the non-generic case, a cornucopia of asymptotics can occur, even for Riemannian surfaces.MSC classes: 53C17 · 57R45 · 58J35 1 In this paper, 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 paper hold for more general rank-varying structures in the sense of [11, Appendix A], for instance, Grushin-like structures. This will be specified in the paper.

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Section: The Riemannian Casesupporting

confidence: 93%

Section: D Contact Casementioning

confidence: 93%

On the Heat Diffusion for Generic Riemannian and Sub-Riemannian Structures

Barilari

Boscain

Charlot

et al. 2016

Int Math Res Notices

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“…and the remaining question is the size of the error term. On S 2 , this is rather well understood [4,19] and we have for x, y sufficiently close to have uniqueness of geodesics (the formula would be slightly different if x and y were antipodal points but because of the rapid decay of the heat kernel, this does not play a role),…”

mentioning

confidence: 87%

On the Logarithmic Energy of Points on S^2

Steinerberger

2020

Preprint

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We revisit a classical question: how large is the minimal logarithmic energy of n points on S 2Bétermin & Sandier (building on work of Sandier & Serfaty) showed thatwhere the constant c log is characterized by a certain renormalized minimization problem. Brauchart, Hardin & Saff conjectured a closed form expression for c log (∼ −0.05) assuming analytic continuation. We describe a simple renormalization approach that results in a purely local problem involving superpositions of Gaussians. In particular, if the hexagonal lattice minimizes Gaussians energy, this would prove that c log indeed coincides with the conjectured value.We also improve the lower bound from c log ≥ −0.223 to c log ≥ −0.095.

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“…La divergence d'un champ de vecteur est définie à partir du volume riemannien par div(X)vol = L X vol. [20,3,8,7]). Mais, de façon générale, pour pouvoir définir partout un laplacien en sousriemannien, il faut choisir un volume lisse.…”

Section: Résultats Connus En Riemannien Et Sous-riemannienunclassified

Asymptotiques en temps petit du noyau de la chaleur des métriques riemanniennes et sous-riemanniennes

Barilari¹,

Boscain²,

Charlot³

et al. 2014

Séminaire De Théorie Spectrale Et Géométrie

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Nous établissons l'asymptotique en temps petit du noyau de la chaleur au lieu de coupure dans les situations génériques, en géométrie riemannienne en dimension inférieure ou égale à 5, en géométrie sous-riemannienne de contact en dimension 3 ou de quasi-contact en dimension 4. La preuve nous permet de montrer qu'en dimension inférieure ou égale à 5 les seules singularités d'une application exponentielle riemannienne générique qui peuvent apparaître le long d'une géodésique minimisante sont A 3 et A 5 .

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Small time heat kernel asymptotics at the cut locus on surfaces of revolution

Cited by 7 publications

References 23 publications

On the Heat Diffusion for Generic Riemannian and Sub-Riemannian Structures

On the Heat Diffusion for Generic Riemannian and Sub-Riemannian Structures

On the Logarithmic Energy of Points on S^2

Asymptotiques en temps petit du noyau de la chaleur des métriques riemanniennes et sous-riemanniennes

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