On the corner contributions to the heat coefficients of geodesic polygons
Annales de l'Institut Fourier to appear, , 29 p.

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 γ=π/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 γ in a geodesic polygon.

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 γ=π/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 γ arbitraire dans un polygone géodésique.

Classification:  58J50
Keywords: Laplace operator, heat kernel, heat coefficients, orbifolds, cone points, corner contribution, distance function expansion
     author = {Schueth, Dorothee},
     title = {On the corner contributions to the heat coefficients of geodesic polygons},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
Schueth, Dorothee. On the corner contributions to the heat coefficients of geodesic polygons. Annales de l'Institut Fourier, to appear, 29 p.

[1] van den Berg, Michiel; Srisatkunarajah, Sivakolundu Heat equation for a region in 2 , J. Lond. Math. Soc., Tome 37 (1988), pp. 119-127 | Zbl 1200.53040

[2] Berger, Marcel Sur le spectre d’une variété riemannienne, C. R. Math. Acad. Sci. Paris, Tome 263 (1966), p. A13-A16 | Zbl 0141.38203

[3] Berger, Marcel Le spectre des variétés riemanniennes, Rev. Roum. Math. Pures Appl., Tome 13 (1968), pp. 915-931 | Zbl 0181.49603

[4] Berger, Marcel Eigenvalues of the Laplacian, Global Analysis (Proceedings of Symposia in Pure Mathematics) Tome 16, American Mathematical Society, 1970, pp. 121-125 | Zbl 0205.40102

[5] Berger, Marcel; Gauduchon, Paul; Mazet, Edmond Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Tome 194, Springer, 1971 | Zbl 0223.53034

[6] Branson, Thomas P.; Gilkey, Peter B. The asymptotics of the Laplacian on a manifold with boundary, Commun. Partial Differ. Equations, Tome 15 (1990) no. 2, pp. 245-272 | Zbl 1221.53067

[7] Chu, Wenchang; Marini, Alberto Partial fractions and trigonometric identities, Adv. Appl. Math., Tome 23 (1999) no. 2, pp. 115-175 | Zbl 07050767

[8] Donnelly, Harold Spectrum and the fixed points set of isometries. I, Math. Ann., Tome 224 (1976), pp. 161-170 | Zbl 0633.53001

[9] Dryden, Emily B.; Gordon, Carolyn S.; Greenwald, Sarah J.; Webb, David L. Asymptotic expansion of the heat kernel for orbifolds, Mich. Math. J., Tome 56 (2008) no. 1, pp. 205-238 | Zbl 0417.53030

[10] Gilkey, Peter B. Invariance theory, the heat equation, and the Atiyah–Singer index theorem, Studies in Advanced Mathematics, CRC Press, 1995 | Zbl 0856.58001

[11] Hsu, Elton P. On the principle of not feeling the boundary for diffusion processes, J. Lond. Math. Soc., Tome 51 (1995) no. 2, pp. 373-382 | Zbl 0822.58053

[12] Kac, Mark Can one hear the shape of a drum?, Am. Math. Mon., Tome 73 (1966) no. 4, pp. 1-23 | Zbl 0139.05603

[13] Mazzeo, Rafe; Rowlett, Julie A heat trace anomaly on polygons, Math. Proc. Camb. Philos. Soc., Tome 159 (2015) no. 2, pp. 303-319 | Zbl 1371.58017

[14] McKean, Henry P.; Singer, Isadore M. Curvature and the eigenvalues of the Laplacian, J. Differ. Geom., Tome 1 (1967) no. 1, pp. 43-69 | Zbl 0198.44301

[15] Minakshisundaram, Subbaramiah; Pleijel, Åke Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Can. J. Math., Tome 1 (1949), pp. 242-256 | Zbl 0041.42701

[16] Nicolaescu, Liviu I. Random Morse functions and spectral geometry (2012) (https://arxiv.org/abs/1209.0639)

[17] Sakai, Takashi On eigen-values of Laplacian and curvature of Riemannian manifold, Tôhoku Math. J., Tome 23 (1971), pp. 589-603 | Zbl 0237.53040

[18] Uçar, Eren Spectral invariants for polygons and orbisurfaces (2017) (Ph. D. Thesis)

[19] Watson, Simon The trace function expansion for spherical polygons, N. Z. J. Math., Tome 34 (2005) no. 1, pp. 81-95 | Zbl 1076.35042