Sui-Chung Ng

On holomorphic isometric embeddings of the unit disk into polydisks

We study the classification of holomorphic isometric embeddings of the unit disk into polydisks. As a corollary of our results, we can give a complete classification when the target is the 2-disk and the 3-disk. We also prove that the holomorphic isometric embeddings between polydisks are induced by those of the unit disk into polydisks.

Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products

Abstract Let X be the quotient of an irreducible bounded symmetric domain Ω by a lattice. In order to characterize algebraic correspondences on X commuting with exterior Hecke correspondences, Clozel–Ullmo studied certain germs of measure-preserving maps from (Ω; 0) into its Cartesian products, proving that such maps are totally geodesic when dim(X) = 1. Here we prove total geodesy when dim(Ω) ≧ 2 by methods of analytic continuation. For Bn, n ≧ 2, total geodesy follows then from Alexanders theorem. When rank(Ω) ≧ 2, we deduce total geodesy from Alexander-type theorems, especially from a new Alexander-type theorem involving Reg(∂Ω) in place of the Shilov boundary.

On proper holomorphic mappings among irreducible bounded symmetric domains of rank at least 2

Proper holomorphic mappings among domains on Euclidean spaces is a classical topic in Several Complex Variables. The literature can date back to the earliest results like the theorem of H. Alexander [1] which says that any proper holomorphic self-map of the complex unit n-ball is a biholomorphism if n ≥ 2. Since then, the study of the proper holomorphic mappings between complex unit balls of different dimensions has become a very popular topic in the field. Many important inputs from various perspectives have been made, like Algebraic Geometry, Chern-Moser Theory, Segre variety and Bergman kernel, etc. It is apparent by now that the complexity of the problem grows with the codimension and one in general must impose certain regularity assumptions on the proper maps in order to give any satisfactory classification.

On compact splitting complex submanifolds of quotients of bounded symmetric domains

We study compact complex submanifolds S of quotient manifolds X = Ω/Γ of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms, and we are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifolds S ⊂ X, i.e., under the assumption that the tangent sequence over S splits holomorphically. We prove results of two types. The first type of results concerns S ⊂ X which are characteristic complex submanifolds, i.e., embedding Ω as an open subset of its compact dual manifold M by means of the Borel embedding, the non-zero (1, 0)-vectors tangent to S lift under a local inverse of the universal covering map π: Ω → X to minimal rational tangents of M. We prove that a compact characteristic complex submanifold S ⊂ X is necessarily totally geodesic whenever S is a splitting complex submanifold. Our proof generalizes the case of the characterization of totally geodesic complex submanifolds of quotients of the complex unit ball Bn obtained by Mok (2005). The proof given here is however new and it is based on a monotonic property of curvatures of Hermitian holomorphic vector subbundles of Hermitian holomorphic vector bundles and on exploiting the splitting of the tangent sequence to identify the holomorphic tangent bundle TS as a quotient bundle rather than as a subbundle of the restriction of the holomorphic tangent bundle TX to S. The second type of results concerns characterization of total geodesic submanifolds among compact splitting complex submanifolds S ⊂ X deduced from the results of Aubin (1978) and Yau (1978) which imply the existence of Kähler-Einstein metrics on S ⊂ X. We prove that compact splitting complex submanifolds S ⊂ X of sufficiently large dimension (depending on Ω) are necessarily totally geodesic. The proof relies on the Hermitian-Einstein property of holomorphic vector bundles associated to TS, which implies that endomorphisms of such bundles are parallel, and the construction of endomorphisms of these vector bundles by means of the splitting of the tangent sequence on S. We conclude with conjectures on the sharp lower bound on dim(S) guaranteeing total geodesy of S ⊂ X for the case of the type-I domains of rank 2 and the case of type-IV domains, and examine a case which is critical for both conjectures, viz. on compact complex surfaces of quotients of the 4-dimensional Lie ball, equivalently the 4-dimensional type-I domain dual to the Grassmannian of 2-planes in ℂ4.