Results 1  10
of
15
Second order families of special Lagrangian 3folds
, 7
"... Abstract. A second order family of special Lagrangian submanifolds of C m is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian submanifolds of C 3 is characterized by a single algebraic equatio ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
Abstract. A second order family of special Lagrangian submanifolds of C m is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian submanifolds of C 3 is characterized by a single algebraic equation on the second fundamental form. While the ‘generic ’ set of such conditions turns out to be incompatible, i.e., there are no special Lagrangian submanifolds that satisfy them, there are many interesting sets of conditions for which the corresponding family is unexpectedly large. In some cases, these geometrically defined families can be described explicitly, leading to new examples of special Lagrangian submanifolds. In other cases, these conditions characterize already known families in a new way. For example, the examples of LawlorHarvey constructed for the solution of the angle conjecture and recently generalized by Joyce turn out to
Selfdual Einstein Hermitian four manifolds, preprint
, 2000
"... Abstract. We provide a local classification of selfdual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, nonhyperkähler structure compatible with the negative orientation. We finally show that selfdual Eins ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
Abstract. We provide a local classification of selfdual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, nonhyperkähler structure compatible with the negative orientation. We finally show that selfdual Einstein 4manifolds obtained as quaternionic quotients of the
Canonical Sasakian metrics
"... Abstract. Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2norm of the scalar curvature ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Abstract. Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L 2norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a SasakiFutaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their SasakiFutaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that just the standard polarization can be represented by a SasakiEinstein metric. 1.
Homogeneous paraKáhler Einstein manifolds
 Russian Math. Surveys
"... Dedicated to E.B.Vinberg on the occasion of his 70th birthday Abstract. A paraKähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skewsymmetric paracomplex structures K, i.e. a parallel field of skewsymmetric endomorphisms with K 2 = Id or, equivalently, as a sym ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Dedicated to E.B.Vinberg on the occasion of his 70th birthday Abstract. A paraKähler manifold can be defined as a pseudoRiemannian manifold (M, g) with a parallel skewsymmetric paracomplex structures K, i.e. a parallel field of skewsymmetric endomorphisms with K 2 = Id or, equivalently, as a symplectic manifold (M, ω) with a biLagrangian structure L ± , i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold M = G/H of a semisimple Lie group G admits an invariant paraKähler structure (g, K) if and only if it is a covering of the adjoint orbit AdGh of a semisimple element h. We give a description of all invariant paraKähler structures (g,K) on a such homogeneous manifold. Using a paracomplex analogue of basic formulas of Kähler geometry, we prove that any invariant paracomplex structure K on M = G/H defines a unique paraKähler Einstein structure (g,K) with given nonzero scalar curvature. An explicit formula for the Einstein metric g is given. A survey of recent results on paracomplex geometry is included. Contents
Special symplectic connections
"... By a special symplectic connection we mean a torsion free connection which is either the LeviCivita connection of a BochnerKähler metric of arbitrary signature, a BochnerbiLagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold w ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
By a special symplectic connection we mean a torsion free connection which is either the LeviCivita connection of a BochnerKähler metric of arbitrary signature, a BochnerbiLagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case. Keywords: BochnerKähler metrics, Ricci type connections, Symplectic holonomy MSC: 53D35, 53D05, 53D10
Kähler metrics generated by functions of the timelike distance in the flat KählerLorentz space, ArXiv: math.DG/0510468
"... Abstract. We prove that every Kähler metric, whose potential is a function of the timelike distance in the flat KählerLorentz space, is of quasiconstant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above mention ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We prove that every Kähler metric, whose potential is a function of the timelike distance in the flat KählerLorentz space, is of quasiconstant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kähler structures. We prove that these rotational hypersurfaces carry Kähler metrics of quasiconstant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kähler metrics are BochnerKähler (especially of constant holomorphic sectional curvatures) are also described. 1.
BochnerKaehler metrics and connections of Riccitype, eprint arXiv: 0710.0164
, 2007
"... We apply the results from [CS] about special symplectic geometries to the case of BochnerKaehler metrics. We obtain a (local) classification of these based on the orbit types of the adjoint action in su(n, 1). The relation between Sasaki and BochnerKaehler metrics in cone and transveral metrics co ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We apply the results from [CS] about special symplectic geometries to the case of BochnerKaehler metrics. We obtain a (local) classification of these based on the orbit types of the adjoint action in su(n, 1). The relation between Sasaki and BochnerKaehler metrics in cone and transveral metrics constructions is discussed. The connection of the special symplectic and Weyl connections is outlined. The duality between the Riccitype and BochnerKaehler metrics is shown. Keywords: BochnerKähler metric, Sasaki metric, Ricci type connection, Weyl structure. 1 BochnerKaehler metrics The curvature tensor of the LeviCivita connection of a Kaehler metric g decomposes (under the action of u(n)) into its Ricci and Bochner part ([Bo]). The metric is said to be BochnerKaehler, iff the Bochner part of its curvature tensor vanishes. A remarkable relationship was revealed among following types of geometric structures in the article [CS]: manifolds with a connection of Ricci type, manifolds with a connection with the special symplectic holonomy, pseudoRiemannian BochnerKähler structures, manifolds with a BochnerbiLagrangian
TORIC GEOMETRY OF CONVEX QUADRILATERALS
, 2009
"... We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler– Einstein and toric Sasaki–Einstein metrics constructed in [6, 23, 14]. As a ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler– Einstein and toric Sasaki–Einstein metrics constructed in [6, 23, 14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbisurfaces, including Kähler–Einstein ones, and show that for a toric orbisurface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kähler metric with constant scalar curvature. Our results also provide explicit examples of relative K–unstable toric orbisurfaces that do not admit extremal metrics.
BY HEISENBERG, SPHERICAL CR GEOMETRY
"... Abstract. A Bochner flat Kähler manifold is a Kähler manifold with vanishing Bochner curvature tensor. We shall give a uniformization of Bochner flat Kähler manifolds. One of the aims of this paper is to give a correction to the proof of our previous paper [9] concerning uniformization of Bochner fl ..."
Abstract
 Add to MetaCart
Abstract. A Bochner flat Kähler manifold is a Kähler manifold with vanishing Bochner curvature tensor. We shall give a uniformization of Bochner flat Kähler manifolds. One of the aims of this paper is to give a correction to the proof of our previous paper [9] concerning uniformization of Bochner flat Kähler manifolds. A Bochner flat locally conformal Kähler manifold is a locally conformal Kähler manifold with vanishing Bochner curvature tensor. We shall apply our result to Bochner flat locally conformal Kähler manifolds. Contents
Special symplectic connections
, 2003
"... By a special symplectic connection we mean a torsion free connection which is either the LeviCivita connection of a BochnerKähler metric of arbitrary signature, a BochnerbiLagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold w ..."
Abstract
 Add to MetaCart
By a special symplectic connection we mean a torsion free connection which is either the LeviCivita connection of a BochnerKähler metric of arbitrary signature, a BochnerbiLagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case. Keywords: BochnerKähler metrics, Ricci type connections, Symplectic holonomy MSC: 53D35, 53D05, 53D10