Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T13:06:45.873Z Has data issue: false hasContentIssue false
  • English
  • Français

An Isospectral Deformation on an Infranil-Orbifold

Published online by Cambridge University Press:  20 November 2018

Emily Proctor  and
Elizabeth Stanhope
Emily Proctor
Affiliation:
Middlebury College, Department of Mathematics, Middlebury, VT, U.S.A. e-mail: eproctor@middlebury.edu
Elizabeth Stanhope
Affiliation:
Lewis & Clark College, Department of Mathematical Sciences, Portland, OR, U.S.A. e-mail: stanhope@lclark.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's theorem due to DeTurck and Gordon.

Keywords

58J5353C20spectral geometryglobal Riemannian geometryorbifoldnilmanifold
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 4 , 01 December 2010 , pp. 684 - 689
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Bérard, P. and Webb, D., On ne peut pas entendre l’orientabilité d’une surface. C. R. Acad. Sci. Paris Sér. I Math. 320(1995), no. 5, 533–536.Google Scholar
[2] Buser, P., Conway, J., Doyle, P., and Semmler, K.-D., Some planar isospectral domains. Internat. Math. Res. Notices 1994, no. 9.Google Scholar
[3] Chiang, Y.-J., Spectral geometry of V-manifolds and its application to harmonic maps. In: Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54, Part 1, American Mathematical Society, Providence, RI, 1993, pp. 93–99.Google Scholar
[4] DeTurck, D. M. and Gordon, C. S., Isospectral deformations II. Trace formulas, metrics and potentials. With an appendix by Lee, K. B., Comm. Pure Appl. Math. 42(1989), no. 8, 1067–1095. doi:10.1002/cpa.3160420803Google Scholar
[5] Donnelly, H., Asymptotic expansions for the compact quotients of properly discontinuous group actions. Illinois J. Math. 23(1979), no. 3, 485–496.Google Scholar
[6] Doyle, P. and Rossetti, J., Isospectral hyperbolic surfaces having matching geodesics. New York J. Math. 14(2008), 193–204.Google Scholar
[7] Gordon, C. S. and Rossetti, J., Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal. Ann. Inst. Fourier 53(2003), no. 7, 2297–2314.Google Scholar
[8] Gordon, C. S. and Wilson, E. N., Isospectral deformations of compact solvmanifolds. J. Differential Geom. 19(1984), no. 1, 241–256.Google Scholar
[9] Gordon, C. S. and Wilson, E. N., Isometry groups of Riemannian solvmanifolds. Trans. Amer. Math. Soc. 307(1988), no. 1, 245–269. doi:10.2307/2000761Google Scholar
[10] Rosetti, J. P., Schueth, D., and Weilandt, M., Isospectral orbifolds with different maximal isotropy orders. Ann. Glob. Anal. Geom. 34(2008), no. 4, 351–366. doi:10.1007/s10455-008-9110-3Google Scholar
[11] Satake, I., On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. U.S.A. 42(1956), 359–363. doi:10.1073/pnas.42.6.359Google Scholar
[12] Scott, P., The geometries of 3-manifolds. Bull. London Math. Soc. 15(1983), no. 5, 401–487. doi:10.1112/blms/15.5.401Google Scholar
[13] Shams, N., Stanhope, E., and Webb, D. L., One cannot hear orbifold isotropy type. Arch. Math. 87(2006), no. 4, 375–384.Google Scholar
[14] Stanhope, E., Spectral bounds on orbifold isotropy. Ann. Global Anal. Geom. 27(2005), no. 4, 355–375. doi:10.1007/s10455-005-1584-7Google Scholar
[15] Sutton, C. J., Equivariant isospectrality and isospectral deformations of metrics on spherical orbifolds. http://arxiv.org/abs/math/0608557.Google Scholar
[16] Thurston, W. P., The geometry and topology of three-manifolds. Lecture notes, Princeton University, Princeton, NJ, 1979.Google Scholar