Hyperbolic imbedding and spaces of continuous extensions of holomorphic maps

LetX be a complex subspace of a complex spaceY. We show that hyperbolic imbeddedness ofX inY is characterized by relative compactness in the compact-open topology of certain spaces of continuous extensions of holomorphic maps from the punctured diskD* toX and fromM -A toX whereM is a complex manifold andA is a divisor onM with normal crossings. We apply these characterizations to obtain generalizations and extensions of theorems of Kobayashi, Kiernan, Kwack, Noguchi and Vitali forD and for higher dimensions. Relative compactness ofX inY is not assumed.     

Iterations of holomorphic maps of infinite dimensional homogeneous domains

In this paper regions of variability are determined and a distortion theorem is proved for iterations of holomorphic maps of bounded symmetric homogeneous domains inJ ^*-algebras.     

Extensions of homomorphisms to dense extensions of semigroups

Gluskin [2] has shown that if α is an isomorphism of a weakly reductive semigroup S onto a semigroup T, if V is a dense extension of S and T is densely embedded in W then α extends uniquely to an isomorphism of V into W. Here we consider the problem of extending epimorphisms and as a consequence of a few simple observations obtain as the main theorem a homomorphism of Ω(S), for any semisimple semigroup S, into the product of the translational hulls of the principal factors of S. A few consequences are considered.     

Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations

We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in \(\mathbb {P}^2\).     

Iteration of holomorphic maps in Hilbert spaces

Previous work (Gong-ning Chen, J. Math. Anal. Appl.98 (1984), 305–313) on iteration of holomorphic maps of Cⁿ is continued. The purpose of this note is to extend results given in the above mentioned reference to the case of complex Hilbert spaces. Other comments are appended.     

Fibrewise extensions, shanin compactification and extensions of fibrewise maps

We introduce an alternative definition of fibrewise uniformity and discuss consequences deduced from new axioms. By modifying James’ definition of fibrewise uniform structure, which is a slightly strengthened one, we define a new fibrewise uniformity which is symmetric in global and realizes 1-1 correspondence between fibrewise entourage uniformities and fibrewise covering uniformities. Moreover, we obtain a characterization of the fibrewise completion of fibrewise generalized uniform space as a fibrewise extension of a fibrewise space. As an application of the fibrewise completion theory, we show that there exists a fibrewise Shanin compactification of a fibrewise space. Finally, we study extendability of fibrewise maps from dense subspaces. That is, for a fibrewise space X, A ? X dense in X and a fibrewise continuous map f: A → Y, when can f be extended to the whole space X? Many characterization theorems of extendable fibrewise continuous maps are given.     

On the dimension of infinite covers

We prove the following theorem and some generalizations. . Let be a connected CW complex which satisfies Poincaré duality of dimension . For any subgroup of , let denote the cover of corresponding to . If has infinite index in , then is homotopy equivalent to an -dimensional CW complex.

     

Sharp dimension estimates of holomorphic functions and rigidity

Let be a complete noncompact Kähler manifold of complex dimension with nonnegative holomorphic bisectional curvature. Denote by the space of holomorphic functions of polynomial growth of degree at most on . In this paper we prove that

for all , with equality for some positive integer if and only if is holomorphically isometric to . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.