Hearing Delzant polytopes from the equivariant spectrum
HTML articles powered by AMS MathViewer
- by Emily B. Dryden, Victor Guillemin and Rosa Sena-Dias PDF
- Trans. Amer. Math. Soc. 364 (2012), 887-910 Request permission
Abstract:
Let $M^{2n}$ be a symplectic toric manifold with a fixed $\mathbb {T}^n$-action and with a toric Kähler metric $g$. Abreu (2003) asked whether the spectrum of the Laplace operator $\Delta _g$ on $\mathcal {C}^\infty (M)$ determines the moment polytope of $M$, and hence by Delzant’s theorem determines $M$ up to symplectomorphism. We report on some progress made on an equivariant version of this conjecture. If the moment polygon of $M^4$ is generic and does not have too many pairs of parallel sides, the so-called equivariant spectrum of $M$ and the spectrum of its associated real manifold $M_{\mathbb {R}}$ determine its polygon, up to translation and a small number of choices. For $M$ of arbitrary even dimension and with integer cohomology class, the equivariant spectrum of the Laplacian acting on sections of a naturally associated line bundle determines the moment polytope of $M$.References
- Miguel Abreu, Kähler geometry of toric varieties and extremal metrics, Internat. J. Math. 9 (1998), no. 6, 641–651. MR 1644291, DOI 10.1142/S0129167X98000282
- Miguel Abreu, Kähler geometry of toric manifolds in symplectic coordinates, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001) Fields Inst. Commun., vol. 35, Amer. Math. Soc., Providence, RI, 2003, pp. 1–24. MR 1969265
- Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
- Ana Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR 1853077, DOI 10.1007/978-3-540-45330-7
- 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
- Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339 (French, with English summary). MR 984900
- Harold Donnelly, Spectrum and the fixed point sets of isometries. I, Math. Ann. 224 (1976), no. 2, 161–170. MR 420743, DOI 10.1007/BF01436198
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- Carolyn S. Gordon, Survey of isospectral manifolds, Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, 2000, pp. 747–778. MR 1736857, DOI 10.1016/S1874-5741(00)80009-6
- Victor Guillemin, Kaehler structures on toric varieties, J. Differential Geom. 40 (1994), no. 2, 285–309. MR 1293656
- Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR 201237, DOI 10.2307/2313748
- Yael Karshon, Liat Kessler, and Martin Pinsonnault, A compact symplectic four-manifold admits only finitely many inequivalent toric actions, J. Symplectic Geom. 5 (2007), no. 2, 139–166. MR 2377250
- J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. MR 162204, DOI 10.1073/pnas.51.4.542
- Toshikazu Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), no. 1, 169–186. MR 782558, DOI 10.2307/1971195
- Shǔkichi Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tǒhoku Math. J. (2) 25 (1973), 391–403. MR 0334086, DOI 10.2748/tmj/1178241341
Additional Information
- Emily B. Dryden
- Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
- Email: ed012@bucknell.edu
- Victor Guillemin
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Email: vwg@math.mit.edu
- Rosa Sena-Dias
- Affiliation: Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
- Email: senadias@math.ist.utl.pt
- Received by editor(s): August 24, 2009
- Received by editor(s) in revised form: June 18, 2010
- Published electronically: October 4, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 887-910
- MSC (2010): Primary 58J50, 53D20
- DOI: https://doi.org/10.1090/S0002-9947-2011-05412-7
- MathSciNet review: 2846357