Hearing Delzant polytopes from the equivariant spectrum
Authors:
Emily B. Dryden, Victor Guillemin and Rosa Sena-Dias
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
Published electronically:
October 4, 2011
MathSciNet review:
2846357
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a symplectic toric manifold with a fixed
-action and with a toric Kähler metric
. Abreu (2003) asked whether the spectrum of the Laplace operator
on
determines the moment polytope of
, and hence by Delzant's theorem determines
up to symplectomorphism. We report on some progress made on an equivariant version of this conjecture. If the moment polygon of
is generic and does not have too many pairs of parallel sides, the so-called equivariant spectrum of
and the spectrum of its associated real manifold
determine its polygon, up to translation and a small number of choices. For
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
.
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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
DOI:
https://doi.org/10.1090/S0002-9947-2011-05412-7
Keywords:
Laplacian,
symplectic manifold,
toric,
Delzant polytope,
equivariant spectrum
Received by editor(s):
August 24, 2009
Received by editor(s) in revised form:
June 18, 2010
Published electronically:
October 4, 2011
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.