arXiv research

A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. T…

The paper constructs Einstein Sasaki metrics on solvable Lie groups.

New Sasaki-Einstein 7-manifolds found, including rational homology 7-spheres and connected sums.

In this note, stimulated by the existence result of Futaki-Ono-Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on S^1-bundles associated to canonical line bundles of P^1-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here the …

Study finds infinite families of Sasaki-Einstein metrics on spheres.

This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.

Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.

In [11] it was proved that, given a compact toric Sasaki manifold of positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on com…

The study examines the stability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.

This article is a summary of some of the author's work on Sasaki-Einstein geometry. A rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki-Einstein geometry, in low dimensions, to superconformal field theory; properties of the latter are therefore reflected in the former, and vi…

We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to k#(S^2\times S^3), where k=2b_2(S)+1, and are given by a pencil of Sasaki embeddings of M in S and are given concretely by the zero set of a component of the 3-Sasaki…

New Sasaki-Einstein 7-spheres found via Berglund-Hübsch transpose.

The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.

In this paper we give a diameter bound for Sasaki manifolds with positive transverse Ricci curvature. As an application, we obtain the uniqueness of Sasaki-Einstein metrics on compact Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure.

We show that a connection with skew-symmetric torsion satisfying the Einstein metricity condition exists on an almost contact metric manifold exactly when it is D-homothetic to a cosymplectic manifold. In dimension five, we get that the existence of a connection with skew torsion satisfying the Einstein metricity condi…

By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from Kähler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we e…

We find a Sasaki-Einstein metric from a CFT state in AdS5.

New Einstein metrics found from para-Sasaki-like Riemannian manifolds.

We show that there are no irregular Sasaki-Einstein structures on rational homology 5-spheres. On the other hand, using K-stability we prove the existence of continuous families of non-toric irregular Sasaki-Einstein structures on odd connected sums of $S^2 \times S^3$.

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström and coupled Kähler-Ricci solitons by Hultgren, we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields…

We prove the existence of Sasaki-Einstein metrics on certain simply connected 5-manifolds where until now existence was unknown. All of these manifolds have non-trivial torsion classes. On several of these we show that there are a countable infinity of deformation classes of Sasaki-Einstein structures.

We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.

We decompose the de Rham Laplacian on Sasaki-Einstein manifolds as a sum over mostly positive definite terms. An immediate consequence are lower bounds on its spectrum. These bounds constitute a supergravity equivalent of the unitarity bounds in dual superconformal field theories. The proof uses a generalization of Kah…

The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.

New 5-manifolds allow Sasaki-Einstein structures.

Study finds critical points of volume functionals on Sasaki manifolds.

We consider the stability of Sasaki-extremal metrics under deformations of the complex structure on the Reeb foliation. Given such a deformation preserving the action of a compact subgroup of the automorphism group of a Sasaki-extremal structure, a sufficient condition is given involving the nondegeneracy of the relati…

We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use…

Smooth SE structures on Sasaki-joins and Bott orbifolds constructed.

Study local moduli of Sasaki-Einstein metrics on specific polynomial links.

New cohomology $η$ for deformed Sasaki-Einstein manifolds derived from Dolbeault cohomology.

Sasakian immersions prove Sasaki-Ricci solitons are η-Einstein with rational constants.

We show that the Einstein-Hilbert functional, as a functional on the space of Reeb vector fields, detects the vanishing Sasaki-Futaki invariant. In particular, this provides an obstruction to the existence of a constant scalar curvature Sasakian metric. As an application we prove that K-semistable polarized Sasaki mani…

We obtain infinite classes of new Einstein-Sasaki metrics on complete and non-singular manifolds. They arise, after Euclideanisation, from BPS limits of the rotating Kerr-de Sitter black hole metrics. The new Einstein-Sasaki spaces L^{p,q,r} in five dimensions have cohomogeneity 2, and U(1) x U(1) x U(1) isometry group…

Study on stability of minimal submanifolds in specific Einstein manifolds.

We prove that the identity component of the holomorphic isometry group of a Sasaki-Einstein metric is the identity component of a maximal compact subgroup of its automorphism group.

In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant $f_1$ for the first…

We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental compon…

Using the Sasakian join construction with homology 3-spheres, we give a countably infinite number of examples of Sasakian manifolds with perfect fundamental group in all odd dimensions greater than 1. These have extremal Sasaki metrics with constant scalar curvature. Moreover, we present further examples of both Sasaki…

Geometries and dual field theories linked by AdS/CFT.

Study on deformations of Einstein and nearly G2 structures in 3-Sasaki manifolds.

Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.

Given a Sasaki manifold S, we prove the Sasaki-Ricci flow converges exponentially fast to a Sasaki-Einstein metric if one exists, provided the automorphism group of the transverse holomorphic structure is trivial.

The paper shows convergence of Sasaki-Ricci flow on Sasakian 5-manifolds.

New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.

The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.

We study null Sasakian structures in dimension five. First, based on a result due to Kollár [Ko], we improve a result by Boyer, Galicki and Matzeu in [BGM] and prove that simply connected manifolds diffeomorphic to $# k(S^2\times S^3)$ admit null Sasaki $η$-Einstein structures if and only if $k\in \{3,..., 21\}$. After…

We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bound…