Associated families of pluriharmonic maps¶and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are K?hler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic K?hler symmetric spaces. Received: 8 July 1997     

A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.     

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.     

Surfaces with zero mean curvature vector in neutral 4-manifolds

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces.     

Partial energies monotonicity and holomorphicity of Hermitian pluriharmonic maps

In this paper,we introduce the notion of Hermitian pluriharmonic maps from Hermitian manifold into K¨ahler manifold.Assuming the domain manifolds possess some special exhaustion functions and the vecotor field V=JMδJM satisfies some decay conditions,we use stress-energy tensors to establish some monotonicity formulas of partial energies of Hermitian pluriharmonic maps.These monotonicity inequalities enable us to derive some holomorphicity for these Hermitian pluriharmonic maps.     

Deformations of twisted harmonic maps and variation of the energy

We study the deformations of twisted harmonic maps \(f\) with respect to the representation \(\rho \). After constructing a continuous “universal” twisted harmonic map, we give a construction of every first order deformation of \(f\) in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kähler group, to show that the critical points of the energy functional \(E\) coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is a potential for the Kähler form of the “Betti” moduli space; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of \(E\) at critical points.     

A uniqueness theorem in Kähler geometry

We consider compact Kähler manifolds with their Kähler Ricci tensor satisfying F(Ric) = constant. Under the nonnegative bisectional curvature assumption and certain conditions on F, we prove that such metrics are in fact Kähler–Einstein.     

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.     

Curved flats,pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.      

Holomorphic Lefschetz Fixed Point Formula for Non-compact K¨ahler Manifolds

The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact “hyperbolic” Kǎihler manifolds （e.g. Kǎihler hyperbolic manifolds, bounded domains of holomorphy） by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.     

On the structure of nearly pseudo-Kähler manifolds

Firstly we give a condition to split off the Kähler factor from a nearly pseudo-Kähler manifold and apply this to get a structure result in dimension 8. Secondly we extend the construction of nearly Kähler manifolds from twistor spaces to negatively curved quaternionic Kähler manifolds and para-quaternionic Kähler manifolds. The class of nearly pseudo-Kähler manifolds obtained from this construction is characterized by a holonomic condition. The combination of these results enables us to give a classification result in (real) dimension 10. Moreover, we show that a strict nearly pseudo-Kähler six-manifold is Einstein.     

Trajectories for Sasakian magnetic fields on real hypersurfaces of type (B) in a complex hyperbolic space

On a real hypersurface in a Kähler manifold we can consider a natural closed 2-form associated with the almost contact metric structure induced by Kähler structure. We treat trajectories under magnetic fields which are constant multiples of this 2-form. We consider a condition for them to be also curves of order 2 on tubes around totally geodesic real hyperbolic spaces in a complex hyperbolic space.     

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 \(S^2\).     

Harmonic maps from Kähler manifolds

Some Liouville type theorems for harmonic maps from Kähler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (exceptR _IV(2)) to any Riemannian manifold with finite energy has to be constant.