Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Let F be a Kähler spin foliation of codimension q=2n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75-90] that the eigenvalue λ of the basic Dirac operator Db satisfies the inequality , where σ∇ is the transversal scalar curvature of F. In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, F is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.     

Conformally Kähler base metrics for Einstein warped products

A Riemannian metric g with Ricci curvature r is called nontrivial quasi-Einstein, in a sense given by Case, Shu and Wei, if it satisfies (−a/f)∇df+r=λg, for a smooth nonconstant function f and constants λ and a>0. If a is a positive integer, it was noted by Besse that such a metric appears as the base metric for certain warped Einstein metrics. This equation also appears in the study of smooth metric measure spaces. We provide a local classification and an explicit construction of Kähler metrics conformal to nontrivial quasi-Einstein metrics, subject to the following conditions: local Kähler irreducibility, the conformal factor giving rise to a Killing potential, and the quasi-Einstein function f being a function of the Killing potential. Additionally, the classification holds in real dimension at least six. The metric, along with the Killing potential, form an SKR pair, a notion defined by Derdzinski and Maschler. It implies that the manifold is biholomorphic to an open set in the total space of a CP¹ bundle whose base manifold admits a Kähler-Einstein metric. If the manifold is additionally compact, it is a total space of such a bundle or complex projective space. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler quasi-Einstein metrics is reproduced in dimension at least six in a more explicit form.     

Minimal 2-spheres in ℂPn with constant Kähler angle

In 1988 the author and J. Bolton conjectured that a minimally immersed 2-sphere in ℂP ⁿ with constant K?hler angle θ≠ 0, π/2,π necessarily has constant curvature. In 1995 Li Zhen-qi showed that the simplest candidates for counterexamples must be linearly full in ℂP ¹⁰ with tan² (θ/2) = 3/4, and produced an explicit 3-parameter family of them. In the present paper it is shown that these counterexamples may be completely characterised using almost complex curves in the nearly K?hler S ⁶ and that the space of such counterexamples, modulo ambient isometries, is a 14-cell with a single point removed. Received: 7 April 1999     

Totally Real Minimal Surfaces with Non-Circular Ellipse of Curvature in the Nearly Kahler S6

In [2] we discussed almost complex curves in the nearly KählerS⁶. These are surfaces with constant Kähler angle 0 or and, as a consequence of this, are also minimal and have circularellipse of curvature. We also considered minimal immersionswith constant Kähler angle not equal to 0 or , but withellipse of curvature a circle. We showed that these are linearlyfull in a totally geodesic S⁵ in S⁶ and that (in the simplyconnected case) each belongs to the S¹-family of horizontallifts of a totally real (non-totally geodesic) minimal surfacein CP². Indeed, every element of such an S¹-family has constantKähler angle and in each family all constant Kählerangles occur. In particular, every minimal immersion with constantKähler angle and ellipse of curvature a circle is obtainedby rotating an almost complex curve which is linearly full ina totally geodesic S⁵.     

On real and complex Berwald connections associated to strongly convex weakly Kähler-Finsler metric

In this paper, we give a definition of weakly complex Berwald metric and prove that, (i) a strongly convex weakly Kähler-Finsler metric F on a complex manifold M is a weakly complex Berwald metric iff F is a real Berwald metric; (ii) assume that a strongly convex weakly Kähler-Finsler metric F is a weakly complex Berwald metric, then the associated real and complex Berwald connections coincide iff a suitable contraction of the curvature components of type (2,0) of the complex Berwald connection vanish; (iii) the complex Wrona metric in Cn is a fundamental example of weakly complex Berwald metric whose holomorphic curvature and Ricci scalar curvature vanish identically. Moreover, the real geodesic of the complex Wrona metric on the Euclidean sphere S2n−1⊂Cn is explicitly obtained.     

Nearly Kähler manifolds with vanishing Tricerri-Vanhecke Bochner curvature tensor

We study the local structures of nearly Kähler manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke (TV). We show that there does not exist a TV Bochner flat strict nearly Kähler manifold in 2n(?10) dimension and determine the local structures of the manifolds in 6 and 8 dimensions.     

Enveloppes inférieures de fonctions admissibles sur l'espace projectif complexe. Cas symétrique

We prove that admissible functions for Fubini-Study metric on the complex projective space PmC of complex dimension m, invariant by a convenient automorphisms group, are lower bounded by a function going to minus infinity on the boundary of usual charts of PmC. A similar lower bound holds on some projective manifolds. This gives an optimal constant in a Hörmander type inequality on these manifolds, which allows us, using Tian's invariant, to establish the existence of Einstein-Kähler metrics on them.     

Positive Quaternion Kähler Manifolds with fourth Betti number equal to one

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. In this article we are mainly concerned with Positive Quaternion Kähler Manifolds M satisfying b₄(M)=1. Generalising a result of Galicki and Salamon we prove that M4n in this case is homothetic to a quaternionic projective space if 2≠n?6.     

Constant curved minimal 2-spheres in G(2,4)

This paper gives a complete classification for minimal 2-spheres with constant Gaussian curvature immersed in the complex Grassmann manifold G(2,4). Received: 14 May 1998 / Revised version: 12 October 1998     

Isotropic ppmc immersions

Isometric immersions with parallel pluri-mean curvature (“ppmc”) in euclidean n-space generalize constant mean curvature (“cmc”) surfaces to higher dimensional Kähler submanifolds. Like cmc surfaces they allow a one-parameter family of isometric deformations rotating the second fundamental form at each point. If these deformations are trivial the ppmc immersions are called isotropic. Our main result drastically restricts the intrinsic geometry of such a submanifold: Locally, it must be a symmetric space or a Riemannian product unless the immersion is holomorphic or a superminimal surface in a sphere. We can give a precise classification if the codimension is less than 7. The main idea of the proof is to show that the tangent holonomy is restricted and to apply the Berger-Simons holonomy theorem.     

On the existence of complete Kähler metrics of negative Riemannian curvature bounded away from zero on ellipsoidal domains inC n

Summary The purpose of this paper is to prove that every ellipsoidal domain in Cⁿ admits a complete Kähler metric whose Riemannian sectional curvature is bounded from above by a negative constant (Theorem 1). We construct a Kähler metric, in a natural way, as potential of a suitable function defining the boundary (§2). Directly we compute the curvature tensor and we find upper and lower bounds for the holomorphic sectional curvature (§ 4, § 5). In order to prove the boundness of Riemannian sectional curvature we use finally a classical pinching argument (§ 6). We also obtain that for certain ellipsoidal domains the curvature tensor is very strongly negative in the sense of [15] (§ 3). Finally we prove that the metric constructed on ellipsoidal domains in Cⁿ is the Bergman metric if and only if the domain is biholomorphic to the ball (Theorem 2). In [8], [9] R. E. Greene and S. G. Krantz gave large families of examples of complete Kähler manifolds with Riemannian sectional curvature bounded from above by a negative constant; they are sufficiently small deformations of the ball in Cⁿ, with the Bergman metric. Before the only known example of complete simply-connected Kähler manifold with Riemannian sectional curvature upper bounded by a negative constant, not biholomorphic to the ball, was the surface constructed by G. D. Mostow and Y. T. Siu in [14], to the best of the author's knowledge, is not known at present if this example is biholomorphic to a domain in Cⁿ.     

Extremal Kähler metrics on projective bundles over a curve

Let M=P(E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E→Σ over a compact complex curve Σ of genus ?2. Building on ideas of Fujiki (1992) [27], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.     

Construction of homogeneous minimal 2-spheres in complex Grassmannians

In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU (2). Furthermore, we compute induced metrics, Gaussian curvatures, Khler angles and the square lengths of the second fundamental forms of these homogeneous minimal 2-spheres in G(2, n + 1) by making use of Veronese sequence.     

An application of Guillemin-Abreu theory to a non-abelian group action

This note is a step towards demonstrating the benefits of a symplectic approach to studying equivariant Kähler geometry. We apply a local differential geometric framework from Kähler toric geometry due to Guillemin and Abreu to the case of the standard linear SU(n) action on Cn?{0}. Using this framework we (re)construct certain Kähler metrics from data on moment polytopes.     

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.     

Complex Monge-Ampère equations and totally real submanifolds

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kähler metrics.     

Hyper-Hermitian Quaternionic Kähler Manifolds

We call a quaternionic Kähler manifold with nonzero scalar curvature, whosequaternionic structure is trivialized by a hypercomplex structure, ahyper-Hermitian quaternionic Kähler manifold. We prove that every locallysymmetric hyper-Hermitian quaternionic Kähler manifold is locally isometricto the quaternionic projective space or to the quaternionic hyperbolic space.We describe locally the hyper-Hermitian quaternionic Kähler manifolds withclosed Lee form and show that the only complete simply connected suchmanifold is the quaternionic hyperbolic space.     

Minimal S3 with Constant Curvature in CPn

Various kinds of minimal 3-spheres with constant sectional curvaturein the complex projective space CPⁿ are studied.     

Métriques autoduales sur la boule

A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishes), one may hope to find in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies the Positive Frequency Conjecture of LeBrun. The proof uses twistor theory, which enables to translate the problem in terms of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable by a complex domain. Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by quaternionic-K?hler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic contact structure on the (4m−1)-sphere is the boundary of a quaternionic-K?hler metric on the (4m)-ball. Oblatum 29-XI-2000 & 7-XI-2001?Published online: 1 February 2002