Dimensions of very ample pluricanonical subsystems for compact quotients of bounded symmetric domains

A compact complex manifold X obtained by taking quotient of a bounded symmetric domain has an ample canonical line bundle. We prove that the dimension of very ample pluricanonical subsystem is strictly bigger than 2n, where n is the dimension of X. Received: 23 June 2000 / Revised version: 30 March 2001     

A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces

A submanifold in a symmetric space is called equifocal if it has a globally flat abelian normal bundle and its focal data is invariant under normal parallel transportation. This is a generalization of the notion of isoparametric submanifolds in Euclidean spaces. To each equifocal submanifold, we can associate a Coxeter group, which is determined by the focal data at one point. In this paper we prove that an equifocal submanifold in a simply connected compact symmetric space is a non-trivial product of two such submanifolds if and only if its associated Coxeter group is decomposable. As a consequence, we get a similar splitting result for hyperpolar group actions on compact symmetric spaces. These results are an application of a splitting theorem for isoparametric submanifolds in Hilbert spaces by Heintze and Liu.

     

Bloch spaces on bounded symmetric domains in complex Banach spaces

We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Weakly symmetric spaces and bounded symmetric domains

LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K _s G/K, whereK _s is the semisimple part ofK=K _s ·Z _K ⁰ andZ _K ⁰ is the connected center ofK. The natural action ofG onM extends to an action ofG ¹=G×Z _K ⁰ . We prove as the main result thatM is weakly symmetric with respect toG ¹ and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S ¹ and determine a one-parameter family of Riemannian metrics onM invariant underG ¹. Furthermore,M is irreducible with respect to . As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.Research partially supported by an NSF grant. The author wishes to thank the International Erwin Schroedinger Institute for its hospitality during the preparation of this paper.     

Bloch constants of bounded symmetric domains

Let and be two irreducible bounded symmetric domains in the complex spaces and respectively. Let be the Euclidean metric on and the Bergman metric on . The Bloch constant is defined to be the supremum of , taken over all the holomorphic functions and , and nonzero vectors . We find the constants for all the irreducible bounded symmetric domains and . As a special case we answer an open question of Cohen and Colonna.

     

Compressions of infinite-dimensional bounded symmetric domains

\noindent If D is a bounded symmetric domain in a complex Banach space Z, then the identity component G of its group of biholomorphic automorphisms permits a natural embedding into a complex Banach—Lie group H acting partially on Z. A typical model is the action of the group PSL(2,C) by Moebius transformations. In this paper we show that the interior S ⁰ of the compression semigroup S := { h ∈ H: h.D \subeq D } has a polar decomposition in the sense that S ⁰ = G \exp(W_G^0), where W_G \subeq ig is a closed convex invariant cone and the polar map G \times W_G^0 → S^0 is a diffeomorphism. March 31, 2000     

Boundary structure of bounded symmetric domains

Let D be a bounded symmetric domain realized as the open unit ball of a complex Banach space E and denote for every by s _a the symmetry of D about a. We show that for every boundary point the locally uniform limit s _c := lim _{c → c} s _a exists as holomorphic map s _c :D→E and has values in the boundary of D. Received: 7 August 1999 / Revised version: 3 November 1999     

Harmonic maps of bounded symmetric domains

Research supported in part by NNSFC, SFECC and ICTP     

Geodesics of bounded,symmetric domains

Commentarii Mathematici Helvetici -     

Sobolev spaces on bounded symmetric domains

We obtain a formula for the Sobolev inner product in standard weighted Bergman spaces of holomorphic functions on a bounded symmetric domain in terms of the Peter–Weyl components in the Hua–Schmidt decomposition, and use it to clarify the relationship between the analytic continuation of these standard weighted Bergman spaces and the Sobolev spaces on bounded symmetric domains.     

Angular derivatives on bounded symmetric domains

In this paper we generalise the classical Julia-Wolff-Carathéodory theorem to holomorphic functions defined on bounded symmetric domains. This work comprises part of the Ph.D. thesis [13] of the first author who gratefully acknowledges the support of Forbairt Basic Research grant SC/97/614.     

Determining boundary sets of bounded symmetric domains

We consider bounded symmetric domains in complex Banach spaces. It is known that each of these domains can be realized as open unit ballD of a uniquely determined complex Banach spaceE and that every biholomorphic automorphismg ofD extends holomorphically to the closure ofD inE. We study subsetsS of (and in particular of the boundary ϖD) such that every automorphismg is already uniquely determined by its values onS. We also consider subsetsS with the analogue topological property: For every sequence (g _n ) of automorphisms converging uniformly onS tog the convergence is already uniform onD. Supported by Acción Integrada Hispano-Alemana Supported by Xunta de Galicia, project Xuga 20702 B 90     

Berezin transform on real bounded symmetric domains

Let be a bounded symmetric domain in a complex vector space with a real form and be the real bounded symmetric domain in the real vector space . We construct the Berezin kernel and consider the Berezin transform on the -space on . The corresponding representation of is then unitarily equivalent to the restriction to of a scalar holomorphic discrete series of holomorphic functions on and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the -space.

     

Homogeneous multiplication operators on bounded symmetric domains

Let D=G/K be an irreducible bounded symmetric domain of dimension d and let be the analytic continuation of the weighted Bergman spaces of holomorphic functions on D. We consider the d-tuple M=(M₁,…,M_d) of multiplication operators by coordinate functions and consider its spectral properties. We find those parameters ν for which the tuple M is subnormal and we answer some open questions of Bagchi and Misra. In particular, we prove that when D=B^d is the unit ball in , then B^d is a k-spectral set of M if and only if is the Hardy space or a weighted Bergman space.