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Biquard-Gauduchon have shown that conformally K\"ahler, Ricci-flat, ALF toric metrics on the complement of toric divisors are: the Taub-NUT metric with reversed orientation, in the Kerr-Taub-bolt family or in the Chen-Teo family. The same authors have also given a unified construction for the above families relying on an axi-symmetric harmonic function on $\mathbb{R}^3$. In this work, we reverse this construction and use methods from a paper of the second named author, "Uniqueness among scalar-flat K\"ahler metrics on non-compact toric 4-manifolds", to show that all conformally K\"ahler, Ricci-flat, toric metrics on the complement of toric divisors, under some mild assumptions on the associated moment polytope, are among the families above. In particular all such metrics are ALF., Comment: 23 pages, 1 figure

Using an ansatz due to LeBrun we construct complete scalar-flat K\"ahler metrics with a prescribed varying conical singularity along a divisor.

Drawing on results of Derdzi\'nski's from the 80's, we classify conformally K\"ahler, $U(2)$-invariant, Einstein metrics on the total space of $\mathcal{O}(-m)$, for all $m \in \mathbb{N}$. This yields infinitely many $1$-parameter families of metrics exhibiting several different behaviours including asymptotically hyperbolic metrics (more specifically of Poincar\'e type), ALF metrics, and metrics which compactify to a Hirzebruch surface $\mathbb{H}_m$ with a cone singularity along the "divisor at infinity". This allows us to investigate transitions between behaviours yielding interesting results. For instance, we show that a Ricci--flat ALF metric known as the Taub-bolt metric can be obtained as the limit of a family of cone angle Einstein metrics on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ when the cone angle converges to zero. We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat K\"ahler metric. Such metrics cannot be obtained by applying Derdzi\'nski's theorem., Comment: In theorem 1.2, the limiting Ricci-flat metric was incorrectly identified in a previous version. This is now corrected

We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric K\"ahler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric K\"ahler manifold. Amongst other questions we investigate and give partial answers to the following: (1) How many such minimal Lagrangian tori exist? (2) Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? (3) Given a toric symplectic manifold, for which sets of orbits $S$, is there a compatible toric K\"ahler metric whose set of minimal Lagrangian orbits is $S$?

Abreu-Sena-Dias have constructed two distinct families of scalar-flat K\"ahler non-compact toric metrics using Donaldson's rephrasing of Joyce's construction in action-angle coordinates. In this paper and using the same set-up, we show that these are the only J-complete scalar-flat K\"ahler metrics on any given strictly unbounded toric surface. We also show that the asymptotic behaviour of such a metric determines it uniquely., Comment: 23 pages

In [LS], it is shown shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric K\"ahler manifold (i.e. $\lambda_1^\mathbb{T}$, the invariant first eigenvalue) is an unbounded function of the toric K\"ahler metric. In this note we show that, seen as a function on the space of toric K\"ahler metrics on a fixed toric manifold, $\lambda_1^\mathbb{T}$ admits no analytic critical points. We also show that on $S^2$, the first eigenvalue of the Laplacian restricted to the space of $S^1$-equivariant functions of any given integer weight admits no critical points., Comment: 14 pages

In this note we prove that toric K\"ahler metrics on complex projective space which are also $U(n)$-invariant are determined by their equivariant spectrum i.e. the list of eigenvalues of the Laplacian together with weights of the torus representation on the eigenspaces., Comment: 11 pages

In this paper we study the smallest non-zero eigenvalue $\lambda_1$ of the Laplacian on toric K\"ahler manifolds. We find an explicit upper bound for $\lambda_1$ in terms of moment polytope data. We show that this bound can only be attained for $\mathbb{CP}^n$ endowed with the Fubini-Study metric and therefore $\mathbb{CP}^n$ endowed with the Fubini-Study metric is spectrally determined among all toric K\"ahler metrics. We also study the equivariant counterpart of $\lambda_1$ which we denote by $\lambda_1^T$. It is the the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that $\lambda_1^T$ is not bounded among toric K\"ahler metrics thus generalizing a result of Abreu-Freitas on $S^2$. In particular, $\lambda_1^T$ and $\lambda_1$ do not coincide in general., Comment: 24 pages. Added some more details and corrected an estimate. Fixed some typos

We prove an inverse spectral result for $S^1$-invariant metrics on $S^2$ based on the so-called asymptotic equivariant spectrum. This is roughly the spectrum together with large weights of the $S^1$ action on the eigenspaces. Our result generalizes an inverse spectral result of the first and last named authors, together with Victor Guillemin, concerning $S^1$-invariant metrics on $S^2$ which are invariant under the antipodal map. We use higher order terms in the asymptotic expansion of a natural spectral measure associated with the Laplacian and the $S^1$ action., Comment: 16 pages; minor revisions throughout following comments from referees

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 Schrodinger 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., Comment: 35 pages

In this paper, we show that the complete scalar-flat Kahler metrics constructed by Abreu and the author on strictly unbounded toric 4-dimensional orbifolds have finite $L^2$ norm of the full Riemannian tensor. In particular, this answers a question of Donaldon's on the corresponding Generalized Taub-NUT metric on $R^4$. This norm is explicitly determined when the underlying toric manifold is the minimal resolution of a cyclic singularity. In the Ricci-flat case corresponding to gravitational instantons, this recovers a recent result of Atiyah-Lebrun., Comment: 29 pages

Let O be a symplectic toric 2n-dimensional orbifold with a fixed T^n-action and with a toric Kahler metric g. We previously explored whether, when O is a manifold, the equivariant spectrum of the Laplace operator acting on smooth functions on (O,g) determines the moment polytope of O, and hence by Delzant's theorem determines O up to symplectomorphism. In the setting of toric orbifolds we significantly improve upon our previous results and show that the moment polytope of a generic toric orbifold is determined by its equivariant spectrum, up to two possibilities and up to translation. This involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry. We also show that the equivariant spectrum determines whether the toric Kahler metric has constant scalar curvature., Comment: 19 pages

In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In this paper we generalize Donaldson's method and construct complete scalar-flat toric Kahler metrics on any symplectic toric 4-manifold with "strictly unbounded" moment polygon. These include the asymptotic locally Euclidean scalar-flat Kahler metrics previously constructed by Calderbank and Singer, as well as new examples of complete scalar-flat toric Kahler 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 Kahler metrics., Comment: 30 pages, 3 figures. Revised version: minor corrections, added two references. Version 3: minor changes, corrected sign mystake, updated references

Let M^{2n} be a symplectic toric manifold with a fixed T^n-action and with a toric K\"ahler metric g. Abreu 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_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., Comment: 23 pages, 9 figures; v2 is published version

We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the scalar curvature of a toric metric on a toric variety in terms of polytope data., Comment: 21 pages