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 an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Codimension two Kähler submanifolds of space forms

In this article we study isometric immersions from Kähler manifolds whose (1, 1) part of the second fundamental form is parallel, theppmc isometric immersions. When the domain is a Riemann surface these immersions are precisely those with parallel mean curvature. P. J. Ryan has classified the Kähler manifolds that admit isometric immersions, as real hypersurfaces, in space forms. We classify the codimension twoppmc isometric immersions into space forms.     

An intrinsic characterization of isometric pluriharmonic immersions with codimension one

We give an intrinsic characterization of isometric pluriharmonic immersions of Kähler manifolds into semi-Euclidean spaces with real codimension one, which is a generalization of the Ricci-Curbastro theorem.Research partly supported by the Grants-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science and Culture, Japan.     

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.     

Affine Kähler hypersurfaces satisfying certain conditions on the curvature tensor

The notion of affine Kähler immersions has been recently introduced by Nomizu-Pinkall-Podestà ([N-Pi-Po]). This work is aimed at giving some results towards the classification of non degenerate affine Kähler hypersurfaces with symmetric and parallel Ricci tensor; this problem generalizes the classical results due to Nomizu-Smyth ([N-S]) in the theory of Kählerian hypersurfaces. In a second section we deal with the case of “semisymmetric” affine Kähler immersions, when the curvature tensor R satisfies R · R = 0 and the Ricci tensor is symmetric, providing a complete classification; for affine Kähler curves we prove that the conditions above are actually equivalent to saying that the immersion is isometric for a suitable Kähler metric in C².     

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.     

Splitting Theorems for Submanifolds of Nonnegative Isotropic Curvature

We study isometric immersions of complete manifolds of nonnegative isotropic curvature that have spaces of relative nullity. These manifolds decompose into a Riemannian product ${\bar{M}\times \mathbb R^k}$ . In the case that k??? 2, we use recent results of Brendle-Schoen to completely classify these manifolds. We then apply this classification to study complete non-compact K?hler submanifolds with relatively low codimension.     

Analysis of constrained Willmore surfaces

This paper investigates the regularity of constrained Willmore immersions into ?^m≥3 locally around both “regular” points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion and its first and second derivatives, given in terms of residues computed as circulation integrals. We deduce explicit “point removability” conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension.     

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.     

The Geometry of Positive Locally Quaternion Kähler Manifolds

A classification of locally quaternion Kähler manifolds M ⁴ⁿ with positive scalar curvature is obtained as a consequence of J. Wolf's work on space forms of irreducible symmetric spaces. We determine the Betti numbers of such manifolds M ⁴ⁿ as well as of the projective 3-Sasakian manifolds fibering over them. We study the geometry of the quaternion Kähler and locally quaternion Kähler submanifolds for each M ⁴ⁿ, which is particularly significant for 4n = 16 due to its relation with four quaternionic structures on the Grassmannian (R ⁸).     

Kähler–Einstein submanifolds of the infinite dimensional projective space

This paper consists of two main results. In the first one we describe all Kähler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kähler–Einstein metric with negative scalar curvature which admits a Kähler immersion into the infinite dimensional complex projective space.     

Extremal Kähler metrics and complex deformation theory

Let (M, J, g) be a compact Kähler manifold of constant scalar curvature. Then the Kähler class [] has an open neighborhood inH ^1,1 (M, ) consisting of classes which are represented by Kähler forms of extremal Kähler metrics; a class in this neighborhood is represented by the Kähler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [] is nondegenerate, every small deformation of the complex manifold (M, J) also carries Kähler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kähler metrics on certain compact complex surfaces.The first author is supported in part by NSF grant DMS 92-04093.     

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.     

Minimal surfaces in symmetric spaces with parallel second fundamental form

In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel second fundamental form. In particular, when the surface is \(S^2\), we discuss the special case and obtain a necessary and sufficient condition such that its second fundamental form is parallel. We also consider isometric minimal two-spheres immersed in complex two-dimensional Kähler symmetric spaces with parallel second fundamental form, and prove that the immersion is totally geodesic with constant Kähler angle if it is neither holomorphic nor anti-holomorphic with Kähler angle \(\alpha \ne 0\) (resp. \(\alpha \ne \pi \)) everywhere on \(S^2\).     

An extension of Calabi's rigidity theorem to complex submanifolds of indefinite complex space forms

The rigidity for full holomorphic isometric immersions of an indefinite Kähler manifold into an indefinite complex space form is proved. All such immersions between indefinite complex projective (and hyperbolic) spaces are founded and examples of non-congruents holomorphic isometric immersions are exposed.     

On the geometry of moduli spaces

We construct a Kähler metric on the moduli spaces of compact complex manifolds with c₁,<0 and of polarized compact Kähler manifolds with c₁=0, which is a generalization of the Petersson-Well metric. It is induced by the variation of the Kähler-Einstein metrics on the fibers that exist according to the Calabi-Yau theorem. We compute the above metric on the moduli spaces of polarized tori and symplectic manifolds. It turns out to be the Maaß metric on the Siegel upper half space and the Bergmann metric on a symmetric space of type III resp. In particular it is Kähler-Einstein with negative curvature.Dedicated to Karl SteinHeisenberg-Stipendiat der Deutschen Forschungsgemeinschaft     

A characterization of constant isotropic immersions by circles

Motivated by the well-known result of Nomizu and Yano [4], we provide a characterization of constant isotropic immersions into an arbitrary Riemannian manifold by circles on the submanifolds. As an immediate consequence of this result, we characterize Veronese imbeddings of complex projective spaces into complex projective spaces which are typical examples of Kähler immersions. Received: 11 January 2002     

Nonsymmetric Osserman indefinite Kähler manifolds

The authors prove the existence of Osserman manifolds with indefinite Kähler metric of nonnegative or nonpositive holomorphic sectional curvature which are not locally symmetric.

     

Complex contact structures and the first eigenvalue of the dirac operator on Kähler manifolds

In this paper Kählerian Killing spinors on manifolds of complex dimensionm=4l+3 are constructed. The construction is based on a theorem which states that a closed Kähler Einstein manifold of complex dimension 4l+3 and positive scalar curvature admits a Kählerian Killing spinor if and only if there is a complex (2l+1)-contact structure. In particular, any complex contact structure in the usual sense gives rise to such a generalized contact structure. Using this, new examples of Kählerian Killing spinors are obtained.