Maximal complexifications of certain homogeneous Riemannian manifolds

Let be a homogeneous Riemannian manifold with , where denotes the universal complexification of . Under certain extensibility assumptions on the geodesic flow of , we give a characterization of the maximal domain of definition in for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and naturally reductive homogeneous Riemannian spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex.

     

Stein fillability and the realization of contact manifolds

There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though a border always exists, its germ is not unique; nevertheless the germ of the Dolbeault cohomology of any border is unique. We also point out that any Stein fillable compact contact -manifold has a geometric realization in via an embedding, or in via an immersion.

     

Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds

In this paper, a notion of generalized gradient on Riemannian manifolds is considered and a subdifferential calculus related to this subdifferential is presented. A characterization of the tangent cone to a nonempty subset S of a Riemannian manifold M at a point x is obtained. Then, these results are applied to characterize epi-Lipschitz subsets of complete Riemannian manifolds.     

Large discrete sets in Stein manifolds

We construct infinite discrete subsets of Stein manifolds with remarkable properties. These generalize results of Rosay and Rudin on discrete subsets of . Received August 21, 1998; in final form October 11, 1999 / Accepted Published online February 5, 2001     

Induced local actions on taut and Stein manifolds

Let act by biholomorphisms on a taut manifold . We show that can be regarded as a -invariant domain in a complex manifold on which the universal complexification of acts. If is also Stein, an analogous result holds for actions of a larger class of real Lie groups containing, e.g., abelian and certain nilpotent ones. In this case the question of Steinness of is discussed.

     

Generalized Arrow-Barankin-Blackwell theorems in locally convex spaces

This paper deals with generalizations of the Arrow-Barankin-Blackwell theorem in locally convex spaces, partially ordered by cones whose duals have nonempty quasi-interiors.This research has been partially supported by the World Laboratory, Lausanne, Switzerland, by the Department of Mathematics, University of Pisa, Pisa, Italy, and by the National Natural Sciences Foundation of China. Useful discussions with Professor F. Ferro are gratefully acknowledged.     

Proximal point method for minimizing quasiconvex locally Lipschitz functions on Hadamard manifolds

In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex locally Lipschitz objective functions on Hadamard manifolds. To reach this goal, we use the concept of Clarke subdifferential on Hadamard manifolds and assuming that the function is bounded from below, we prove the global convergence of the sequence generated by the method to a critical point of the function.     

Equivalence of domains arising from duality of orbits on flag manifolds

S. Gindikin and the author defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of non-holomorphic type by computing many examples. In this paper, we first prove this conjecture for the open -orbit on an ``arbitrary' flag manifold generalizing the result of Barchini. This conjecture for closed was solved by J. A. Wolf and R. Zierau for Hermitian cases and by G. Fels and A. Huckleberry for non-Hermitian cases. We also deduce an alternative proof of this result for non-Hermitian cases.

     

Equivalence of domains arising from duality of orbits on flag manifolds II

S. Gindikin and the author defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of nonholomorphic type. This conjecture was proved for closed in the works of J. A. Wolf, R. Zierau, G. Fels, A. Huckleberry and the author. It was also proved for open by the author. In this paper, we prove the conjecture for all the other orbits when is of non-Hermitian type.

     

Equivalence of domains arising from duality of orbits on flag manifolds III

In Gindikin and Matsuki 2003, we defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of nonholomorphic type. This conjecture was proved for closed in Wolf and Zierau 2000 and 2003, Fels and Huckleberry 2005, and Matsuki 2006 and for open in Matsuki 2006. It was proved for the other orbits in Matsuki 2006, when is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed -orbit when is of Hermitian type. Thus the conjecture is completely solved affirmatively.

     

Harmonic analysis on symmetric Stein manifolds from the point of view of complex analysis

The classical theory of finite dimensional representations of compact and complex semisimple Lie groups is discussed from the perspective of multidimensional complex geometry and analysis. The key tool is the complex horospherical transform which establishes a duality between spaces of holomorphic functions on symmetric Stein manifolds and dual horospherical manifolds. Communicated by: Toshiyuki Kobayashi     

Strict lower semicontinuity of the level sets and invexity of a locally Lipschitz function

The strict lower semicontinuity property (slsc property) of the level sets of a real-valued functionf defined on a subsetCR ⁿ was introduced by Zang, Choo, and Avriel (Ref. 1). They showed a class of functions for which the slsc property is equivalent to invexity, i.e., the statement that every stationary point off overC is a global minimum. In this paper, we study the relationship between the slsc property of the level sets and invexity for another class of functions. Namely, we consider the class formed by all locally Lipschitz real-valued functions defined on an open set containingC. For these functions, invexity implies the slsc property of the level sets, but not conversely.The authors would like to thank Dr. B. D. Craven and the referees for helpful comments and suggestions.     

Compact coverings for Baire locally convex spaces

Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space Cp(X) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff Cp(X) is Baire (i.e. of the second Baire category) and is covered by a family of compact sets such that Kα⊂Kβ if α?β. Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.     

On Riemannian coverings of manifolds admitting locally splitting actions

By studying the structure of certain covering manifolds, associated with a series of subgroups of ₁(M), we approximate the structure of the Riemannian manifoldM, provided that it admits a locally splitting action. The above series arises because of the local splitting of the action and the study is carried out via initial data. Also, we indicate how manifolds without conjugate points can be investigated using locally splitting actions of abelian Lie groups.     

Stein Neighborhood Bases of Embedded Strongly Pseudoconvex Domains and Approximation of Mappings

In this paper we construct a Stein neighborhood basis for any compact subvariety A with strongly pseudoconvex boundary bA and Stein interior A \ bA in a complex space X. This is an extension of a well known theorem of Siu. When A is a complex curve, our result coincides with the result proved by Drinovec-Drnovšek and Forstnerič. We shall adapt their proof to the higher dimensional case, using also some ideas of Demailly’s proof of Siu’s theorem. For embedded strongly pseudoconvex domain in a complex manifold we also find a basis of tubular Stein neighborhoods. These results are applied to the approximation problem for holomorphic mappings. Research supported by grants ARRS (3311-03-831049), Republic of Slovenia.     

Stein compacts in Levi-flat hypersurfaces

We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface and holomorphic convexity of compact sets in , or bounded in part by . Applications include extendability of Cauchy-Riemann functions, solvability of the -equation, approximation of Cauchy-Riemann and holomorphic functions, and global regularity of the -Neumann operator.

     

On the homology of locally compact spaces with ends

We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original graph-theoretical definition could be captured in the usual language for homology theories, so as to make it applicable to more general spaces. In this paper we provide such a general topological framework: we define a homology theory which satisfies the usual axioms, but which maintains the special role for ends that has made this homology work so well for graphs.     

Improved Bohr’s inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $\left|g^{\prime }\left(z\right)\right|\le \left|h^{\prime }\left(z\right)\right|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.