On the corner contributions to the heat coefficients of geodesic polygons

Soit $𝒪$ une orbisurface riemannienne compacte. Nous calculons des formules pour la contribution des singularités coniques de $𝒪$ au coefficient de $t^2$ du développement asymptotique de la trace du noyau de la chaleur de $𝒪$, les contributions de $t^0$ et $t^1$ étant connues. Comme application, nous calculons le coefficient de $t^2$ de la contribution d’un angle intérieur de la forme $\gamma =\pi /k$ dans un polygone géodésique sur une surface au développement asymptotique du noyau de la chaleur de Dirichlet du polygone, sous une hypothèse locale de symétrie près du sommet correspondant. La principale nouveauté ici est la détermination de la façon dont le Laplacien de la courbure de Gauss au sommet en question entre dans le coefficient de $t^2$. Nous terminons par une conjecture concernant la contribution analogue d’un angle $\gamma$ arbitraire dans un polygone géodésique.

Let $𝒪$ be a compact Riemannian orbisurface. We compute formulas for the contribution of cone points of $𝒪$ to the coefficient at $t^2$ of the asymptotic expansion of the heat trace of $𝒪$, the contributions at $t^0$ and $t^1$ being known from the literature. As an application, we compute the coefficient at $t^2$ of the contribution of interior angles of the form $\gamma =\pi /k$ in geodesic polygons in surfaces to the asymptotic expansion of the Dirichlet heat kernel of the polygon, under a certain symmetry assumption locally near the corresponding corner. The main novelty here is the determination of the way in which the Laplacian of the Gauss curvature at the corner point enters into the coefficient at $t^2$. We finish with a conjecture concerning the analogous contribution of an arbitrary angle $\gamma$ in a geodesic polygon.

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