Abstract
We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that allows us to reduce to the case of surfaces hyperbolic near infinity recently studied by Borthwick and Perry, or to the closed case by Osgood, Phillips, and Sarnak if there are only cusps.
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Notes
The function ξ is closely related to the ‘scattering phase’; see [10].
When acting on half-densities.
References
Albin, P., Aldana, C.L., Rochon, F.: Ricci flow and the determinant of the Laplacian on non-compact surfaces. Commun. Partial Differ. Equ. 38(4), 711–749 (2013)
Aldana, C.: Isoresonant conformal surfaces with cusps and boundedness of the relative determinant. Commun. Anal. Geom. 18(5), 1009–1048 (2010)
Anderson, M.T.: Remarks on the compactness of isospectral sets in low dimensions. Duke Math. J. 63(3), 699–711 (1991)
Brooks, R., Glezen, P.: An L p spectral bootstrap theorem. In: Geometry of the Spectrum, Seattle, WA, 1993. Contemp. Math., vol. 173, pp. 89–97. Am. Math. Soc., Providence (1994)
Borthwick, D., Judge, C., Perry, P.A.: Determinants of Laplacians and isopolar metrics on surfaces of infinite area. Duke Math. J. 118(1), 61–102 (2003)
Borthwick, D., Perry, P.A.: Inverse scattering results for metrics hyperbolic near infinity. J. Geom. Anal. 21(2), 305–333 (2011)
Brooks, R., Perry, P., Peter Petersen, V.: Compactness and finiteness theorems for isospectral manifolds. J. Reine Angew. Math. 426, 67–89 (1992)
Brooks, R., Perry, P., Yang, P.: Isospectral sets of conformally equivalent metrics. Duke Math. J. 58(1), 131–150 (1989)
Bunke, U.: Relative index theory. J. Funct. Anal. 105, 63–76 (1992)
Carron, G.: Déterminant relatif et la fonction Xi. Am. J. Math. 124(2), 307–352 (2002)
Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982)
Chang, S.-Y.A., Qing, J.: The zeta functional determinants on manifolds with boundary. II. Extremal metrics and compactness of isospectral set. J. Funct. Anal. 147(2), 363–399 (1997)
Chen, R., Xu, X.: Compactness of isospectral conformal metrics and isospectral potentials on a 4-manifold. Duke Math. J. 84(1), 131–154 (1996)
Chang, S.-Y.A., Yang, P.C.-P.: Isospectral conformal metrics on 3-manifolds. J. Am. Math. Soc. 3(1), 117–145 (1990)
Donnelly, H.: Essential spectrum and heat kernel. J. Funct. Anal. 75, 326–381 (1987)
Gilkey, P.B.: Leading terms in the asymptotics of the heat equation. In: Geometry of Random Motion, Ithaca, NY, 1987. Contemp. Math., vol. 73, pp. 79–85. Am. Math. Soc., Providence (1988)
Gordon, C., Perry, P., Schueth, D.: Isospectral and isoscattering manifolds: a survey of techniques and examples. In: Geometry, Spectral Theory, Groups, and Dynamics. Contemp. Math., vol. 387, pp. 157–179 (2004)
Guillarmou, C.: Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. Duke Math. J. 129(1), 1–37 (2005)
Guillopé, L., Zworski, M.: Upper bounds on the number of resonances for non-compact Riemann surfaces. J. Funct. Anal. 129(2), 364–389 (1995)
Hassell, A., Zelditch, S.: Determinants of Laplacians in exterior domains. Int. Math. Res. Not. 18, 971–1004 (1999)
Kim, Y.-H.: Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality. Duke Math. J. 144(1), 73–107 (2008)
Melrose, R.B.: Isospectral sets of drumheads are compact in \(\mathcal{C}^{\infty}\) (1983). Unpublished preprint from MSRI, 048-83. Available online at http://math.mit.edu/~rbm/paper.html
Melrose, R.B.: Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces. In: Spectral and Scattering Theory, Sanda, 1992. Lecture Notes in Pure and Appl. Math., vol. 161, pp. 85–130. Dekker, New York (1994)
Mazzeo, R., Melrose, R.B.: Meromorphic extension of the resolvent on complete spaces with asymptotically negative curvature. J. Funct. Anal. 75(2), 260–310 (1987)
McKean, H.P., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 1, 43–69 (1967)
Osgood, B., Phillips, R., Sarnak, P.: Compact isospectral sets of plane domains. Proc. Natl. Acad. Sci. USA 85(15), 5359–5361 (1988)
Osgood, B., Phillips, R., Sarnak, P.: Compact isospectral sets of surfaces. J. Funct. Anal. 80(1), 212–234 (1988)
Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal. 80(1), 148–211 (1988)
Osgood, B., Phillips, R., Sarnak, P.: Moduli space, heights and isospectral sets of plane domains. Ann. Math. 129, 293–362 (1989)
Zhou, G.: Compactness of isospectral compact manifolds with bounded curvatures. Pac. J. Math. 181(1), 187–200 (1997)
Acknowledgements
The authors are grateful to David Borthwick, Gilles Carron, Andrew Hassell, Rafe Mazzeo, and Richard Melrose for helpful conversations and to the anonymous referee for their comments.
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Communicated by Steven Zelditch.
P. Albin was partially funded by NSF grant DMS-1104533. C.L. Aldana was partially funded by FCT project ptdc/mat/101007/2008. Registered at MPG, AEI-2012-059. F. Rochon was supported by grant DP120102019.
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Albin, P., Aldana, C.L. & Rochon, F. Compactness of Relatively Isospectral Sets of Surfaces Via Conformal Surgeries. J Geom Anal 25, 1185–1210 (2015). https://doi.org/10.1007/s12220-013-9463-0
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DOI: https://doi.org/10.1007/s12220-013-9463-0