Calabi Metrics Research Articles

Ricci flat Calabi's metric is not projectively induced

We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kähler--Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [10] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of $\mathbb{C}^2$ at the origin is not projectively induced.

Classification of Calabi hypersurfaces with parallel Fubini-Pick form

In this paper, we present the classification of 2 and 3-dimensional Calabi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric.

The [formula omitted] enhancement of super Chern–Simons theories in [formula omitted], Calabi HyperKähler metrics and M2-branes on the [formula omitted] conifold

Considering matter coupled supersymmetric Chern–Simons theories in three dimensions we extend the Gaiotto–Witten mechanism of supersymmetry enhancement N3=3→N3=4 from the case where the hypermultiplets span a flat HyperKähler manifold to that where they live on a curved one. We derive the precise conditions of this enhancement in terms of generalized Gaiotto–Witten identities to be satisfied by the tri-holomorphic moment maps. An infinite class of HyperKähler metrics compatible with the enhancement condition is provided by the Calabi metrics on T⋆Pn. In this list we find, for n=2 the resolution of the metric cone on N0,1,0 which is the unique homogeneous Sasaki–Einstein 7-manifold leading to an N4=3 compactification of M-theory. This leads to challenging perspectives for the discovery of new relations between the enhancement mechanism in D=3, the geometry of M2-brane solutions and also for the dual description of super Chern–Simons theories on curved HyperKähler manifolds in terms of gauged fixed supergroup Chern–Simons theories.

The Calabi metric and desingularization of Einstein orbifolds

Consider an Einstein orbifold (M_0, g_0) of real dimension 2n having a singularity with orbifold group the cyclic group of order n in SU (n) which is generated by an nth root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is a natural “approximate” Einstein metric on the desingularization of M0 obtained by replacing a small neighborhood of the singular point of the orbifold with a scaled and truncated Calabi metric. In this paper, we identify the first obstruction to perturbing this approximate Einstein metric to an actual Einstein metric. If M_0 is compact, we can use this to produce examples of Einstein orbifolds which do not admit any Einstein metric in a neighborhood of the natural approximate Einstein metric on the desingularization. In the case that (M_0, g_0) is asymptotically hyperbolic Einstein and non-degenerate, we show that if the first obstruction vanishes, then there does in fact exist an asymptotically hyperbolic Einstein metric on the desingularization. We also obtain a non-existence result in the asymptotically hyperbolic Einstein case, provided that the obstruction does not vanish. This work extends a construction of Biquard in the case n = 2 , in which case the Calabi metric is also known as the Eguchi–Hanson metric, but there are some key points for which the higher-dimensional case differs.

A Bernstein theorem for affine maximal-type hypersurfaces

We obtain, in any dimension N and for a large range of values of θ, a Bernstein theorem for the fourth-order partial differential equation of affine maximal typeuijDijw=0,w=[det⁡D2u]−θ assuming the completeness of Calabi's metric. This contains the results of Li–Jia [A.M. Li, F. Jia, Ann. Glob. Anal. Geom. 23 (2003)] for affine maximal equations and of Zhou [B. Zhou, Calc. Var. Partial Differ. Equ. 43 (2012)] for Abreu's equation. In particular, we extend the result of Zhou from 2≤N≤4 to 2≤N≤5.

On new explicit Riemannian SU(2(n + 1))-holonomy metrics

Generalizing the Calabi metrics in dimension 4(n + 1), we construct a family of complete noncompact Ricci-flat metrics with holonomy SU(2(n + 1)) in explicit algebraic form.

Hyper-Kähler Calabi metrics, L2 harmonic forms, resolved M2-branes, and AdS 4/CFT 3 correspondence

We obtain a simple explicit expression for the hyper-Kähler Calabi metric on the cotangent bundle of CP n+1 , for all n, in which it is constructed as a metric of cohomogeneity one with SU( n+2)/ U( n) principal orbits. These results enable us to obtain explicit expressions for an L 2-normalisable harmonic 4-form in D=8, and an L 2-normalisable harmonic 6-form in D=12. We use the former in order to obtain an explicit resolved M2-brane solution, and we show that this solution is invariant under all three of the supersymmetries associated with the covariantly-constant spinors in the 8-dimensional Calabi metric. We give some discussion of the corresponding dual N=3 three-dimensional field theory. Various other topics are also addressed, including superpotentials for the Calabi metrics and the metrics of exceptional G 2 and Spin(7) holonomy in D=7 and D=8. We also present complex and quaternionic conifold constructions, associated with the cone metrics whose resolutions are provided by the Stenzel T ∗S n+1 and Calabi T ∗ CP n+1 metrics. In the latter case we relate the construction to the hyper-Kähler quotient. We then use the hyper-Kähler quotient to give a quaternionic rederivation of the Calabi metrics.

Quaternionic Kähler and hyper-Kähler non-linear σ-models

We construct a large class of quaternionic Kähler σ-models that can be coupled to N = 2 supergravity. We use a method similar to the symplectic quotient for Kähler manifolds. We obtain all hyper-Kähler manifolds constructed by Lindström and Roček as the κ → 0 flat superspace limit of some quaternionic Kähler spaces. An example, the Calabi metrics and their generalization, is given. The geometrical structure of quaternionic Kähler σ-models with respect to their relation to global and local N = 2 supersymmetry is discussed.

Quaternionic Kähler and hyper-Kähler non-linear σ-models

We construct a large class of quaternionic Kähler σ-models that can be coupled to N = 2 supergravity. We use a method similar to the symplectic quotient for Kähler manifolds. We obtain all hyper-Kähler manifolds constructed by Lindström and Roček as the κ → 0 flat superspace limit of some quaternionic Kähler spaces. An example, the Calabi metrics and their generalization, is given. The geometrical structure of quaternionic Kähler and hyper-Kähler σ-models with respect to their relation to global and local N = 2 supersymmetry is discussed.