ON THE ASYMPTOTIC BEHAVIOR OF HOLOMORPHIC ISOMETRIES OF THE POINCARE DISK INTO BOUNDED SYMMETRIC DOMAINS

In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometrics up to normalizing constants with respect to the Bergman metric, showing in particular that the graph 170 of any germ of holomorphic isometry of the Poincar6 disk A into an irreducible bounded symmetric domain Ω belong to C^N in its Harish-Chandra realization must extend to an affinealgebraic subvariety V belong to C × C^N = C^N＋1, and that the irreducible component of V ∩ （△ × Ω） containing V0 is the graph of a proper holomorphic isometric embedding F ： A→ Ω. In this article we study holomorphie isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δ△. Starting with the structural equation for holomorphic isometrics arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphie isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ‖σ‖ must vanish at a general boundary point either to the order 1 or to the order 1/2, called a holomorphie isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.     

Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains

The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] that a proper holomorphic mapping f：Ω→Ω′from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ′is necessarily totally geodesic provided that r′：=rank（Ω′）≤rank（Ω）：=r.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f：Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of （Ω→Ω′） of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds

Given a bounded symmetric domain Ω the author considers the geometry of its totally geodesic complex submanifolds S ? Ω.In terms of the Harish-Chandra realization Ω and taking S to pass through the origin 0 ∈Ω,so that S=E ∩Ωfor some complex vector subspace of Cⁿ,the author shows that the orthogonal projectionρ:Ω→ E maps Ω onto S,and deduces that S?Ω is a holomorphic isometry with respect to the Caratheodory metric.His first theorem gives a new derivation of a result of Yeung’s deduced...     

THE EXISTENCE OF μ-HOLOMORPHIC SEPARATING FUNCTION ON BOUNDED SMOOTH DOMAINS

The Complex analysis of strongly pseudoconvex domains in C~n is rather well known. In this paper it is proved that for a bounded smoothly domain Ω there is a new complex structure on it under which Ω will locally become a strongly convex even though the point on bΩ is not a pseudoconvex point from the view of the original complex structure. Particularly if Ω is a weakly pseudoconvex domain, the μ can be made sufficiently close to the original complex structure. Therefore a lot of properties of strongly pseudoconvex domains will become true on weakly pseudoconvex domains, or general domains. For example, it is proved that there is a μ-holomorphic separating function which is holomorphic under the new complex structure.     

On compact splitting complex submanifolds of quotients of bounded symmetric domains

We study compact complex submanifolds S of quotient manifolds X = ?/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 ? 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 : ? → 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 Bnobtained 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 TSas a quotient bundle rather than as a subbundle of the restriction of the holomorphic tangent bundle TXto 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 K¨ahler-Einstein metrics on S ? X. We prove that compact splitting complex submanifolds S ? X of sufficiently large dimension(depending on ?) 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 rank2 and the case of type-IV domains, and examine a case which is critical for both conjectures, i.e., 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 C~4.     

A Vanishing Theorem on a Class of Hartogs Domain

In this paper, we consider the d-boundedness of the Bergman metric and a vanishing theorem of L²-cohomology on a class of Hartogs domain, whose base domain is the production of two irreducible bounded symmetric domains of the first type, by using the Bergman kernel function,invariant function, holomorphic automorphism group and so on.     

The Degree of Proper Holomorphic MappingsBetween Special Domains in C^n

Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f .     

Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

The Hartogs domain over homogeneous Siegel domain D_N,s(s > 0) is defined by the inequality ■, where D is a homogeneous Siegel domain of type Ⅱ,(z, ζ) ∈ D × C^N and KD(z, z) is the Bergman kernel of D. Recently, Seo obtained the rigidity result that proper holomorphic mappings between two equidimensional domains D_N,s and D’_N’,s’ are biholomorphisms for N ≥ 2. In this article, we find a counter-example to show that the rigidity result is not true for D...     

The Khler-Einstein metric for some Hartogs domains over symmetric domains

We study the complete Kahler-Einstein metric of a Hartogs domainΩbuilt on an irreducible bounded symmetric domainΩ, using a power Nμof the generic norm ofΩ.The generating function of the Kahler-Einstein metric satisfies a complex Monge-Ampere equation with a boundary condition. The domainΩis in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X∈[0,1[. This allows us to reduce the Monge-Ampere equation to an ordinary differential equation with a limit condition. This equation can be explicitly solved for a special valueμ0 ofμ.. We work out the details for the two exceptional symmetric domains. The special valueμ0 seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.     

CR submanifolds in a sphere and their Gauss maps

The relationship between CR submanifolds in a sphere and their Gauss maps are investigated.Let V be the image of a sphere by a rational holomorphic map F with degree two in another sphere.It is show that the Gauss map of V is degenerate if and only if F is linear fractional.     

Proper Holomorphic Maps from Domains in C2 with Transverse Circle Action

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space,where the domain is of finite type and admits a transverse circle action.The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.     

KAEHLER SUBMANIFOLDS IN A LOCALLY SYMMETRIC BOCHNER-KAEHLER MANIFOLD

This paper gives some sufficient conditions for a compact Kaehler submanifold M~n in a locally symmetric Bochner-Kaehler manifold ~(n p) to be totally geodesic. The conditions are given by inequalities which are established between. the sectional curvature(resp, holomorphic sectional curvature) of M~n and the Ricci curvature of ~(n p). In particular, similar results in the case where ~(n p) is a complex projective spathe are contained.     

本刊英文版Vol.39(2023),No.1论文摘要（英文）

<正>A Vanishing Theorem on a Class of Hartogs Domain Cheng Chen ZHONG An WANG Li Shuang PAN Abstract In this paper,we consider the d-boundedness of the Bergman metric and a vanishing theorem of L2-cohomology on a class of Hartogs domain,whose base domain is the production of two irreducible bounded symmetric domains of the first type,by using the Bergman kernel function,invariant function,holomorphic automorphism group and so on.     

Explicit formula of holomorphic automorphism group on complex homogeneous bounded domains

We known that the maximal connected holomorphic automorphism group Aut (D)(0) is a semi-direct product of the triangle group T(D) and the maximal connected isotropic subgroup Iso(D)(0) of a fixed point in the complex homogeneous bounded domain D and any complex homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain D(VN,F). In this paper, we give the explicit formula of any holomorphic automorphism in T(D(VN, F)) and Iso(D(VN,F))(0), where G(0) is the unit connected component of the Lie group G.     

Existence and Nonexistence of Global Classical Solutions to Porous Medium and Plasma Equations with Singular Sources

Let Ω be a bounded or unbounded domain in R^n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching phenomenon and the existence of global classical solution to the above problem are established. Also, the life span of the quenching solution is estimated or evaluated for some domains.     

Rational holomorphic maps from B~2 into B~N with degree 2 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Rational proper holomorphic maps from the unit ball in C2 into the unit ball CN with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.     

Characterization of Bounded Symmetric Domain by Carathèodory and Eisenmann-Kobayashi Volume Elements

1 IntroductionClassification of bounded domain is an important problem in several complex variables.The best result was got by E. Cartan. He proved that any boundd symmetric domain must bebiholimorphic to one of the following domains or their topological …     

Jackson＇s Theorem on Bounded Symmetric Domains

Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson＇s theorem to several complex variables：Eκ（f,X）≤ω（1/k,f,X）, where Eκ（f,X） is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω（1/k,f,X） is the corresponding modulus of continuity.     

On holomorphic immersions into Khler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f:X→M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain