Rosa Sena-Dias

HEARING DELZANT POLYTOPES FROM THE EQUIVARIANT SPECTRUM

Let M 2n be a symplectic toric manifold with a xed T n -action and with a toric Kahler metric g. Abreu (2) asked whether the spectrum of the Laplace operator g on C 1 (M) determines the moment polytope of M, and hence by Delzants 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 MR 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.

Toric Aspects of the First Eigenvalue

In this paper we study the smallest non-zero eigenvalue

Recovering

\lambda _1

-invariant toric K\"ahler metrics on

λ1 of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for

from the torus equivariant spectrum

\lambda _1

Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds

λ1 in terms of moment polytope data. We show that this bound can only be attained for