In this paper,we prove that a proper μ holomorphic mapping f:D 1→D 2 between bounded domains with some convexity,such that f satisfies some growth condition,extends smoothly to bD 1-{z:U(z)=0}.
In this paper,we study a family of Hartogs domains fibred over Hermitian symmetric manifolds being a unit ball in C^(m).The aim of the present study is to establish the rigidity results about proper holomorphic mappin...In this paper,we study a family of Hartogs domains fibred over Hermitian symmetric manifolds being a unit ball in C^(m).The aim of the present study is to establish the rigidity results about proper holomorphic mappings between two equidimensional Hartogs domains over Hermitian symmetric manifolds.In particular,we can fully determine its biholomorphic equivalence and automorphism group.展开更多
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 typeⅡ,(z,ζ)∈D×C~N and KD(z,z)is the Bergman kernel of D.Recently,Seo ob...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 typeⅡ,(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_(1,s)and obtain a classification of proper holomorphic mappings between D_(1,s)and D'_(1,s').展开更多
It is proved that every proper holomorphic self-mapping of some kinds of Generalized Hartogs Triangles is an automorphism.and its explicit expression is given.
Sufficient conditions were given to assert that between any two Banach spaces over K, Fredholm mappings share at least one .value in a specific open ball. The proof of the result is constructive and based upon continu...Sufficient conditions were given to assert that between any two Banach spaces over K, Fredholm mappings share at least one .value in a specific open ball. The proof of the result is constructive and based upon continuation methods.展开更多
The authors discuss the proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem.
In 1969, Ky Fan^[3] proved that for any continuous function f from a compact convex subset M of a normed linear space X into X, there exists x E M such that ｜｜f（x） - x｜｜ = dist（f（x）,M）. Since then, there hav...In 1969, Ky Fan^[3] proved that for any continuous function f from a compact convex subset M of a normed linear space X into X, there exists x E M such that ｜｜f（x） - x｜｜ = dist（f（x）,M）. Since then, there have appeared several generalizations, extensions and applications of this result. This paper also deals with some extensions and generalizations of this result when the underlying spaces are convex metric spaces.展开更多
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} = ...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 .展开更多
In this paper,we first introduce the notion of n-generalized Hartogs triangles.Then,we characterize proper holomorphic mappings between some of these domains,and describe their automorphism groups.
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 domai...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.展开更多
In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in ...In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.展开更多
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 cl...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.展开更多
In 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ? 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:=...In 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ? 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:= rank(Ωg’) ? 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 number 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 rational 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 complex 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.展开更多
Let a connected compact Lie group G act on a connected symplectic orbifold of orbifold fundamental group Г. If the action preserves the symplectic structure and there is a G-equivariant and mod-Г proper momentum map...Let a connected compact Lie group G act on a connected symplectic orbifold of orbifold fundamental group Г. If the action preserves the symplectic structure and there is a G-equivariant and mod-Г proper momentum map for the lifted action on the universal branch covering orbifold, and if the lifted G-action commutes with that of Г, then the symplectic convexity theorem is still true for this kind of lifted Hamiltonian action.展开更多
文摘In this paper,we prove that a proper μ holomorphic mapping f:D 1→D 2 between bounded domains with some convexity,such that f satisfies some growth condition,extends smoothly to bD 1-{z:U(z)=0}.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12271411,11901327)。
文摘In this paper,we study a family of Hartogs domains fibred over Hermitian symmetric manifolds being a unit ball in C^(m).The aim of the present study is to establish the rigidity results about proper holomorphic mappings between two equidimensional Hartogs domains over Hermitian symmetric manifolds.In particular,we can fully determine its biholomorphic equivalence and automorphism group.
基金the National Natural Science Foundation of China(Grant Nos.11801187,11871233 and 11871380)。
文摘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 typeⅡ,(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_(1,s)and obtain a classification of proper holomorphic mappings between D_(1,s)and D'_(1,s').
基金Project supported by the National Natural Science Foundation of China(No.19631010)
文摘It is proved that every proper holomorphic self-mapping of some kinds of Generalized Hartogs Triangles is an automorphism.and its explicit expression is given.
基金Project supported by D.G.E.S. Pb 96-1338-CO 2-01 and the Junta de Andalucia
文摘Sufficient conditions were given to assert that between any two Banach spaces over K, Fredholm mappings share at least one .value in a specific open ball. The proof of the result is constructive and based upon continuation methods.
基金the National Natural Science Foundation of China (No. 10571135)the Doctoral Program Foundation of the Ministry of Education of China (No. 20050240711)
文摘The authors discuss the proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem.
基金Supported by University Grants Commission, India(F. 30-238/2004(SR))
文摘In 1969, Ky Fan^[3] proved that for any continuous function f from a compact convex subset M of a normed linear space X into X, there exists x E M such that ｜｜f（x） - x｜｜ = dist（f（x）,M）. Since then, there have appeared several generalizations, extensions and applications of this result. This paper also deals with some extensions and generalizations of this result when the underlying spaces are convex metric spaces.
基金Supported by National Natural Science Foundation of China(Grant Nos.11801572,11688101)。
文摘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 .
基金supported by the National Natural Science Foundation of China(Grant No.11871333)。
文摘In this paper,we first introduce the notion of n-generalized Hartogs triangles.Then,we characterize proper holomorphic mappings between some of these domains,and describe their automorphism groups.
基金a CERG of the Research Grants Council of Hong Kong,China.
文摘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.
基金supported by National Natural Science Foundation of China (Grant No.10701007)
文摘In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.
文摘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.
基金This research is partially supported by a Competitive Earmarked Research Grant of the Research Grants Council of Hong Kong,China
文摘In 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ? 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:= rank(Ωg’) ? 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 number 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 rational 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 complex 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.
文摘Let a connected compact Lie group G act on a connected symplectic orbifold of orbifold fundamental group Г. If the action preserves the symplectic structure and there is a G-equivariant and mod-Г proper momentum map for the lifted action on the universal branch covering orbifold, and if the lifted G-action commutes with that of Г, then the symplectic convexity theorem is still true for this kind of lifted Hamiltonian action.
