The Equivariant First Eigenvalues For Toric K(?)hler Manifolds

In this thesis we study the equivariant eigenvalues of the Laplace-Beltrami operatorΔ on toric Kahler manifolds.More precisely,for any integral weight a of the torus Tn,we consider the restriction of Δ to the space of functions that transform under the torus action by the weight a.We will construct equivariant isospectral manifolds who are not isometric.And We will prove that for any toric Kahler metric and any a,the first eigenvalue λ1α is simple.We will also prove that each λ1α,when viewed as a functional on the space of all compatible toric Kahler metrics on a fixed symplectic toric manifold,admits no critical metric,and also admits no upper bound.Finally we will show that in the class of compatible toric Kahler metrics with uniformly bounded scalar curvature,these λ1α does admit upper bounds.