Soit une orbisurface riemannienne compacte. Nous calculons des formules pour la contribution des singularités coniques de au coefficient de du développement asymptotique de la trace du noyau de la chaleur de , les contributions de et étant connues. Comme application, nous calculons le coefficient de de la contribution d’un angle intérieur de la forme 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 . Nous terminons par une conjecture concernant la contribution analogue d’un angle arbitraire dans un polygone géodésique.
Let be a compact Riemannian orbisurface. We compute formulas for the contribution of cone points of to the coefficient at of the asymptotic expansion of the heat trace of , the contributions at and being known from the literature. As an application, we compute the coefficient at of the contribution of interior angles of the form 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 . We finish with a conjecture concerning the analogous contribution of an arbitrary angle in a geodesic polygon.
Classification : 58J50
Mots clés : Laplacien, noyau de la chaleur, coefficients de la chaleur, orbifolds, points coniques, contributions des coins, développement de la fonction distance
@article{AIF_2019__69_7_2827_0, author = {Schueth, Dorothee}, title = {On the corner contributions to the heat coefficients of geodesic polygons}, journal = {Annales de l'Institut Fourier}, pages = {2827--2855}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {69}, number = {7}, year = {2019}, doi = {10.5802/aif.3338}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3338/} }
TY - JOUR AU - Schueth, Dorothee TI - On the corner contributions to the heat coefficients of geodesic polygons JO - Annales de l'Institut Fourier PY - 2019 DA - 2019/// SP - 2827 EP - 2855 VL - 69 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3338/ UR - https://doi.org/10.5802/aif.3338 DO - 10.5802/aif.3338 LA - en ID - AIF_2019__69_7_2827_0 ER -
Schueth, Dorothee. On the corner contributions to the heat coefficients of geodesic polygons. Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 2827-2855. doi : 10.5802/aif.3338. http://archive.numdam.org/articles/10.5802/aif.3338/
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