Determinants of Laplacians on the space of conical metrics on the sphere
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- by Hala Khuri King PDF
- Trans. Amer. Math. Soc. 339 (1993), 525-536 Request permission
Abstract:
On a compact surface with smooth boundary, the determinant of the Laplacian associated to a smooth metric on the surface (with Dirichlet boundary conditions if the boundary is nonempty) is a well-defined isospectral invariant. As a function on the moduli space of such surfaces, it is a smooth function whose boundary behavior in certain cases is well understood; see [OPS and K]. In this paper, we restrict ourselves to a certain class of singular metrics on closed surfaces called conical metrics. We show that the determinant of the associated Laplacian is still well defined and that it is a real analytic function on a suitably restricted subset of the space of conical metrics on the sphere.References
- Jochen Brüning and Robert Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), no. 2, 369–429. MR 899656, DOI 10.1016/0022-1236(87)90073-5
- Constantine J. Callias, The heat equation with singular coefficients. I. Operators of the form $-d^{2}/dx^{2}+\kappa /x^{2}$ in dimension $1$, Comm. Math. Phys. 88 (1983), no. 3, 357–385. MR 701923
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Jeff Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106. MR 530173, DOI 10.1073/pnas.76.5.2103
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- Hala Halim Khuri, Heights on the moduli space of Riemann surfaces with circle boundaries, Duke Math. J. 64 (1991), no. 3, 555–570. MR 1141285, DOI 10.1215/S0012-7094-91-06427-6
- H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR 217739 R. Melrose, Analysis on manifolds with corners, Lecture notes, M.I.T., 1988.
- Masayoshi Nagase, The fundamental solutions of the heat equations on Riemannian spaces with cone-like singular points, Kodai Math. J. 7 (1984), no. 3, 382–455. MR 760044, DOI 10.2996/kmj/1138036957
- B. Osgood, R. Phillips, and P. Sarnak, Moduli space, heights and isospectral sets of plane domains, Ann. of Math. (2) 129 (1989), no. 2, 293–362. MR 986795, DOI 10.2307/1971449 M. Reed and B. Simon, Methods of mathematical physics, vol. 4, Academic Press, 1978.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 339 (1993), 525-536
- MSC: Primary 58G26
- DOI: https://doi.org/10.1090/S0002-9947-1993-1102890-7
- MathSciNet review: 1102890