Abstract. In a recent paper Donaldson [D1] explains how to use an older construction of Joyce [J] to obtain four dimensional local models for scalarflat Kähler metrics with a 2-torus symmetry. In [D2], using the same idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kähler metrics on R 4 . In this paper we generalize Donaldson's method and construct complete scalar-flat toric Kähler metrics on any symplectic toric 4-manifold with "strictly unbounded" moment polygon. These include the asymptotically locally Euclidean scalar-flat Kähler metrics previously constructed by Calderbank and Singer [CS], as well as new examples of complete scalar-flat toric Kähler metrics which are asymptotic to Donaldson's generalized Taub-NUT metrics. Our construction is in symplectic action-angle coordinates and determines all these metrics via their symplectic potentials. When the first Chern class is zero we obtain a new description of known Ricci-flat Kähler metrics.
We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show that the asymptotic equivariant spectrum of the Laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold; we also show that the asymptotic equivariant spectrum of a T n -invariant Schrödinger operator on R n determines its potential in some suitably convex cases. In addition, we prove that the asymptotic equivariant spectrum of an S 1 -invariant metric on S 2 determines the metric itself in many cases. Finally, we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces. As a crucial ingredient in these inverse results, we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi-classical differential operators invariant under a torus action.
Let M 2n be a symplectic toric manifold with a fixed T n -action and with a toric Kähler metric g. Abreu [2] asked whether the spectrum of the Laplace operator ∆g on C ∞ (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 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 .2000 Mathematics Subject Classification. 58J50, 53D20.
Abstract. In this paper we study the smallest non-zero eigenvalue λ 1 of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for λ 1 in terms of moment polytope data. We show that this bound can only be attained for CP n endowed with the Fubini-Study metric and therefore CP n endowed with the Fubini-Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of λ 1 which we denote by λ T 1 . It is the the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that λ T 1 is not bounded among toric Kähler metrics thus generalizing a result of Abreu-Freitas on S 2 . In particular, λ T 1 and λ 1 do not coincide in general.
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