Proper Holomorphic Mappings Research Articles

Local continuous extension of proper holomorphic maps: Low-regularity and infinite-type boundaries

We prove a couple of results on local continuous extension of proper holomorphic maps F:D→Ω, D,Ω⊊Cn, making local assumptions on ∂D and ∂Ω. The first result allows us to have much lower regularity, for the patches of ∂D,∂Ω that are relevant, than in earlier results. The second result (and a result closely related to it) is in the spirit of a result by Forstnerič–Rosay. However, our assumptions allow ∂Ω to contain boundary points of infinite type.

Geometric pushforward in Hodge filtered complex cobordism and secondary invariants

We construct a functorial pushforward homomorphism in geometric Hodge filtered complex cobordism along proper holomorphic maps between arbitrary complex manifolds. This significantly improves previous results on such transfer maps and is a much stronger result than the ones known for differential cobordism of smooth manifolds. This enables us to define and provide a concrete geometric description of Hodge filtered fundamental classes for all proper holomorphic maps. Moreover, we give a geometric description of a cobordism analog of the Abel–Jacobi invariant for nullbordant maps which is mapped to the classical invariant under the Hodge filtered Thom morphism. For the latter we provide a new construction in terms of geometric cycles.

A rigidity result for proper holomorphic maps between balls

In this note, we prove a rigidity result for proper holomorphic maps between unit balls that have many symmetries and which extend to C 2 \mathcal {C}^2 -smooth maps on the boundary.

Regularity of CR maps into uniformly pseudoconvex hypersurfaces and applications to proper holomorphic maps

We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a minimal manifold into such a foliated target is either generically smooth or geometrically highly constrained, and to show generic smoothness of sufficiently regular CR transversal CR maps between pseudoconvex hypersurfaces. As an application, we discuss boundary regularity of proper holomorphic maps into bounded symmetric domains.

Three-Jets Determinations of Normalized Proper Holomorphic Maps from $$ \mathbb {H}_n$$ into $$\mathbb {H}_{3n-2}$$

Three-Jets Determinations of Normalized Proper Holomorphic Maps from $$ \mathbb {H}_n$$ into $$\mathbb {H}_{3n-2}$$

On the Proper Holomorphic Mappings between Equidimensional Hartogs Domains over Hermitian Symmetric Manifolds

On the Proper Holomorphic Mappings between Equidimensional Hartogs Domains over Hermitian Symmetric Manifolds

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

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

Rigidity of proper holomorphic mappings between certain nonequidimensional unbounded non-hyperbolic domains

The Fock-Bargmann-Hartogs domain Dn,m(μ)(μ>0) in Cn+m is defined by the inequality ‖w‖2<e−μ‖z‖2, where (z,w)∈Cn×Cm, which is an unbounded non-hyperbolic domain in Cn+m. In this article, we prove that any proper holomorphic mapping from Dn,m(μ) to Dn,m+1(μ)(m≥3) is necessarily the standard linear embedding up to their automorphisms and therefore is a totally geodesic isometric embedding with respect to their Bergman metrics.

Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

We show that if E is a closed convex set in {mathbb {C}}^n(n>1) contained in a closed halfspace H such that Ecap bH is nonempty and bounded, then the concave domain Omega = {mathbb {C}}^n{setminus } E contains images of proper holomorphic maps f:Xrightarrow {mathbb {C}}^n from any Stein manifold X of dimension <n, with approximation of a given map on closed compact subsets of X. If in addition 2dim X+1le n then f can be chosen an embedding, and if 2dim X=n, then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties.

EMBEDDING BORDERED RIEMANN SURFACES IN STRONGLY PSEUDOCONVEX DOMAINS

We show that every bordered Riemann surface, M, with smooth boundary bM admits a proper holomorphic map M → Ω into any bounded strongly pseudoconvex domain Ω in Cn, n &gt; 1, extending to a smooth map f : M → Ω which can be chosen an immersion if n ≥ 3 and an embedding if n ≥ 4. Furthermore, f can be chosen to approximate a given holomorphic map M → Ω on compacts in M and interpolate it at finitely many given points in M.

On CR maps from the sphere into the tube over the future light cone

We determine all local smooth or formal CR maps from the unit sphere S3⊂C2 into the tube T:=C×iR3⊂C3 over the future light cone C:={x∈R3:x12+x22=x32,x3>0}. This result leads to a complete classification of proper holomorphic maps from the unit ball in C2 into Cartan's classical domain of type IV in C3 that extend smoothly to some boundary point. Up to composing with CR automorphisms of the source and target, the classification consists of four algebraic maps. Two maps among them were known earlier in the literature. They can be generalized to higher dimensional cases and were shown to be “rigid” when the source dimension is at least 4 in a recent paper by Xiao and Yuan. Two newly discovered quadratic polynomial maps provide counterexamples to a conjecture appeared in the same paper for the case of dimension two.

Proper mappings between indefinite hyperbolic spaces and type I classical domains

In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [Proper holomorphic maps between bounded symmetric domains, Springer, Tokyo, 2015, pp. 207–219] and himself.

Proper Holomorphic Mappings Between n-generalized Hartogs Triangles

Proper Holomorphic Mappings Between n-generalized Hartogs Triangles

Transformation Formula for the Reduced Bergman Kernel and its Application

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

Algebraic properties of Hermitian sums of squares, II

We study real bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) in n n complex variables for which r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 is the squared norm of a holomorphic polynomial mapping. Such polynomials are the focus of the Sum of Squares Conjecture, which describes the possible ranks for the squared norm r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 and has important implications for the study of proper holomorphic mappings between balls in complex Euclidean spaces of different dimension. Questions about the possible signatures for r ( z , z ¯ ) r(z,\bar {z}) and the rank of r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 can be reformulated as questions about polynomial ideals. We take this approach and apply purely algebraic tools to obtain constraints on the signature of r r .

Reducing submodules of Hilbert modules and Chevalley-Shephard-Todd theorem

Let G be a finite pseudoreflection group, Ω⊆Cn be a bounded domain which is a G-space and H⊆O(Ω) be an analytic Hilbert module possessing a G-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator Mθ on H, where {θi}i=1n is a homogeneous system of parameters associated to G and θ=(θ1,…,θn) is a polynomial map of Cn. We show that it admits a family {PϱH:ϱ∈Gˆ} of non-trivial joint reducing subspaces, where Gˆ is the set of all equivalence classes of irreducible representations of G. We prove a generalization of Chevalley-Shephard-Todd theorem for the algebra O(Ω) of holomorphic functions on Ω. As a consequence, we show that for each ϱ∈Gˆ, the multiplication operator Mθ on the reducing subspace PϱH can be realized as multiplication by the coordinate functions on a reproducing kernel Hilbert space of C(deg⁡ϱ)2-valued holomorphic functions on θ(Ω). This, in turn, provides a description of the structure of joint reducing subspaces of the multiplication operator induced by a representative of a proper holomorphic map from a domain Ω in Cn which is factored by automorphisms G⊆Aut(Ω).

On the rigidity of proper holomorphic mappings for the Bergman-Hartogs domains

In this paper, we mainly discuss the proper holomorphic mappings of the Bergman-Hartogs domains raised by Roos, which can also be regarded as a natural generalization of Cartan-Hartogs domains. We show that any proper holomorphic mapping between two equidimensional Bergman-Hartogs domains over bounded circular homogeneous domains is a biholomorphism. As an application of our result, we can firstly obtain the rigidity of the proper holomorphic mappings for the Cartan-Hartogs domains. Secondly, we are able to describe the biholomorphism between two Bergman-Hartogs domains over any bounded circular homogeneous domains, and thus we can completely determine its automorphism group without fibre's restriction.

Multiplication operators on the Bergman space by proper holomorphic mappings

Suppose that f:=(f1,…,fd):Ω1→Ω2 is a proper holomorphic map between two bounded domains in Cd. We show that the multiplication operator (tuple) Mf=(Mf1,…,Mfd) on the Bergman space A2(Ω1) admits a non-trivial minimal joint reducing subspace, say M and the restriction of Mf to M is unitarily equivalent to the Bergman operator on A2(Ω2). A number of interesting consequences of this result have been observed.

On holomorphic mappings of strictly pseudoconvex domains

We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with boundaries of class $C^2$. In the second part of the paper we establish an extension of the Wong-Rosay theorem to piecewise smooth strictly pseudoconvex domains. Bibliography: 37 titles.

Rigidity of Proper Holomorphic Maps Among Generalized Balls with Levi-Degenerate Boundaries

In this paper, we study a broader type of generalized balls which are domains on the complex projective spaces with possibly Levi-degenerate boundaries. We prove rigidity theorems for proper holomorphic mappings among them by exploring the structure of the moduli spaces of projective linear subspaces, which generalize some earlier results for the ordinary generalized balls with Levi-nondegenerate boundaries.