p-Harmonic Maps and Convex Functions

p-harmonic maps (p ' 2) between Riemannian manifolds are investigated. Some theorems of Liouville type are given for such maps when target manifolds have convex functions.     

On subtransversality

Summary Let (X, A) and (Y, B) be pairs of manifolds. By means of a «completed» bijet bundle over the blow-up of X along A we give simple geometric interpretations of the notion of subtransversality of a smooth map f: (X, A)(Y, B) along A.     

Ein beweis des Grauertschen bildgarbensatzes nach ideen von B. Malgrange

In 1960, H. Grauert proved the following coherence theorem [2]: Let X, Y be complex spaces and : X Y a proper holomorphic map. Then, for every coherent analytic sheaf J on X, all direct image sheaves Rⁿ*J are coherent. We give a new proof of this theorem, based on ideas of B. Malgrange. This proof does not use induction on the dimension of the base space Y and can be generalized to relative-analytic spaces X Y where Y belongs to a bigger category of ringed spaces, which contains in particular all complex spaces and differentiable manifolds.     

Estimates of the singular numbers of a class of integral operators

Let X,Y be smooth compact manifolds in ^m of codimension 1. A pseudodifferential operator of negative order in ^m is contracted to an operator from L₂(Y) to L₂(X). Under the assumption that mes_m–1) (XY)=0, one finds estimates for the singular numbers of the operator.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 95–99, 1981.In conclusion the author expresses his gratitude to M. Z. Solomyak for useful discussions.     

Branched coverings and nonzero degree maps between Seifert manifolds

In this note we give a necessary and sufficient condition for the existence of a fiber preserving branched covering between two closed, orientable Seifert manifolds (for sufficiency we need the additional assumption that the genus of the base orbifold of the target manifold ). Combining this with two theorems of Rong we get a necessary and sufficient condition for the existence of a nonzero degree map between two such manifolds.

     

Strong density results in trace spaces of maps between manifolds

We deal with strong density results of smooth maps between two manifolds and in the fractional spaces given by the traces of Sobolev maps in W ^1,p .     

Instanton Floer homology and contact structures

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or without boundary—defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by Baldwin and Sivek (arXiv:1403.1930, 2014).     

Complete intersections in Stein manifolds

The purpose of this note is to prove some theorems on set theoretic complete intersections in Stein manifolds (or Stein spaces) which are analogous to results in affine algebraic geometry. Due to the Oka principle in Stein theory one gets stronger results. For example any locally complete intersection Y of dimension 3 in a Stein space X with dim X>2 dim Y is a set theoretic₆ complete intersection. A 4-dimensional submanifold of⁶ is a set theoretic complete intersection if sc ₁ ² (Y)=0 for some integer s>0.Der erstgenannte Autor dankt der Alexander-von-Humboldt-Stiftung für ein Stipendium zu einem Gastaufenthalt an der Universität Münster     

On the existence of Hermitian-harmonic maps from complete Hermitian to complete Riemannian manifolds

On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is not in divergence form.The case of noncompact complete preimage and target manifolds is considered. We give conditions for existence and uniqueness of Hermitian-harmonic maps and solutions of the corresponding parabolic system, which observe the non-divergence form of the underlying equations. Numerous examples illustrate the theoretical results and the fundamental difference to harmonic maps.Support from the research focus Globale Methoden in der komplexen Geometrie under the auspices of Deutsche Forschungsgemeinschaft is gratefully acknowledged.Acknowledgement We are grateful to Wolf von Wahl (University of Bayreuth) for his suggestion to investigate Hermitian-harmonic maps on noncompact manifolds.Dedicated to Prof. E. Heinz on the occasion of his 80^th birthday     

On weakly harmonic maps from Finsler to Riemannian manifolds

We prove global C0,α$C^{0,\alpha }$-estimates for harmonic maps from Finsler manifolds into regular balls of Riemannian target manifolds generalizing results of Giaquinta, Hildebrandt, and Hildebrandt, Jost and Widman from Riemannian to Finsler domains. As consequences we obtain a Liouville theorem for entire harmonic maps on simple Finsler manifolds, and an existence theorem for harmonic maps from Finsler manifolds into regular balls of a Riemannian target.     

Comparison and vanishing theorems for Kähler manifolds

In this paper, we consider orthogonal Ricci curvature \(Ric^{\perp }\) for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kähler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kähler manifolds with \(Ric^{\perp }>0\) is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233).     

The Canonical Decompositions of Some Family of Compact Orientable Hyperbolic 3-Manifolds with Totally Geodesic Boundary

L. Paoluzzi and B. Zimmermann constructed a family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary, and classified them up to homeomorphism. Our main purpose is to determine the canonical decompositions of these manifolds. Using the result, we can obtain an alternative proof of the classification theorem of these manifolds and determine their isometry groups. We also determine their unknotting tunnels. Some of these manifolds are related to certain spatial graphs, so-called Suzukis Brunnian graphs. The properties of these manifolds enable us to obtain those of the graphs. Moreover, we give an affirmative answer to Kinoshitas problem concerning these graphs. In the Appendix, we calculate the volume of these manifolds.     

Lokalkonvexe Unterräume in Topologischen Vektorräumen und Das ε-Produkt

It is well-known that the algebraic tensor product E Y of a not necessarily locally convex topological vector space E and a locally convex space Y can be identified with a subspace of the so-called -product EY (a space of continuous linear mappings from Y into E). So, whenever EY is complete, even the completed tensor product is (isomorphic to) a subspace of EY. As this occurs in many important cases, it is interesting to remark that, for each continuous linear operator u from a locally convex space F into E, there exists a locally convex U with continuous embedding jUE and a continuous linear map ûFU such that u=j·û. As main applications of a combination of these ideas, we obtain a characterization of the functions in as continuous functions with values in locally convex spaces (this gives new aspects for the intergration theory of Gramsch [5]) and a result extending a theorem in [6] on holomorphic functions with values in non locally convex spaces to arbitrary complex manifolds.     

The -harmonic forms on rotationally symmetric Riemannian manifolds revisited

We use separation of variables for generalized Dirac operators on rotationally symmetric Riemannian manifolds to recover a theorem of Dodziuk regarding the spaces of -harmonic forms on such manifolds.

     

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function onM. AsM supports a contact form, there exists a characteristic vector field dual to the contact structure. If induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature or of nonpositive curvature . By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When 0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.Dedicated to Professor Sasao Seiya for his sixtieth birthday     

On hyper-Kähler manifolds of typeA ∞

In this paper we shall construct new families of 4m dimensional non-compact complete hyper-Kähler manifolds on whichm dimensional torus acts. In the 4 dimensional case our manifolds should be considered as hyper-Kähler manifolds which correspond to the extended Dynkin diagram of typeA .     

Rakić duality principle in the almost Hermitian geometry

Raki? duality principle turns out to be one of the crucial steps in proving Osserman conjecture. Basically, it claims that if ${\mathcal{R}}$ is an Osserman algebraic curvature tensor and X and Y are unit vectors, then Y is an eigenvector of the Jacobi operator ${\mathcal{R}(\cdot, X)X}$ if and only if X is an unit eigenvector of ${\mathcal{R}(\cdot, Y)Y}$ with the same eigenvalue. We prove necessary and sufficient conditions for certain almost Hermitian manifolds, the so called AH ₃-manifolds, to have pointwise constant holomorphic curvature and pointwise constant antiholomorphic sectional curvature. It turns out that for this class of almost Hermitian manifolds these conditions are directly connected to the duality principle.     

An approach to the regularity for stable-stationary harmonic maps

In this paper we investigate the regularity of stable-stationary harmonic maps. By assuming that the target manifolds do not carry any stable harmonic , we obtain some compactness results and regularity theorems. In particular, we prove that the Hausdorff dimension of the singular set of these maps cannot exceed , and the dimension estimate is optimal.

     

On cohomology of ACH Kähler manifolds

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is a sharp vanishing theorem for the second cohomology of such manifolds under certain assumptions. The borderline case characterizes a Kähler-Einstein manifold constructed by Calabi.