arXiv research

We study proper holomorphic maps between bounded symmetric domains $D$ and $Ω$. In particular, when $D$ and $Ω$ are of the same rank $\ge 2$ such that all irreducible factors of $D$ are of rank $\ge 2$, we prove that any proper holomorphic map from $D$ to $Ω$ is a totally geodesic holomorphic isometric embedding with r…

The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.

Study proper holomorphic maps between specific domains, proving rigidity under certain conditions.

Proves rigidity of maps between balls with Hölder boundary continuity.

We prove that proper pseudo-holomorphic maps between strictly pseudoconvex regions in almost complex manifolds extend to the boundary. The key point is that the Jacobian is far from zero near the boundary, and the proof is mainly based on an almost complex analogue of the scaling method. We also establish a link betwee…

In this paper we survey results on the existence of holomorphic embeddings and immersions of Stein manifolds into complex manifolds. Most results pertain to proper maps into Stein manifolds. We include a new result saying that every continuous map $X\to Y$ between Stein manifolds is homotopic to a proper holomorphic em…

The main goal of this paper is to prove that a connected bounded geometry complete Kahler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. This also provides a different proof of the theorem of Gromov and Schoen that, for a connected compact Kahler manifold whose f…

The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the…

Let $ \B^{n+1} \subset \C^{n+1}$ be the unit ball in a complex Euclidean space, and let $ Σ^n = \partial \B^{n+1} = S^{2n+1}$. Let $ f: Σ^n \hook Σ^{N}$ be a local CR immersion.If $N-n<2n-1$, the asymptotic vectors of the second fundamental form of $f$ at each point form a subspace of the holomorphic tangent space of…

Proves stability of lcK spaces under holomorphic mappings.

For a holomorphic one-form $\mathbfξ$ on a weakly 1-complete manifold $X$ with certain properties, we discussed the connectivity of the pair $(\hat{X}, F^{-1}(z))$, where $π: \hat{X} \to X$ is a covering map and $dF=π^*\mathbfξ$. We also discussed the criteria about when such a manifold $X$ admits a proper holomorphic …

We give some results concerning the smoothness of the image of a real-analytic submanifold in complex space under the action of a finite holomorphic mapping. For instance, if the submanifold is not contained in a proper complex subvariety, we give a necessary and sufficient condition guaranteeing that its image is smoo…

Hua domain, named after Chinese mathematician Loo-Keng Hua, is defined as a domain in $\mathbb{C}^{n}$ fibered over an irreducible bounded symmetric domain $Ω\subset \mathbb{C}^{d}\;(d<n)$ with the fiber over $z\in Ω$ being a $(n-d)$-dimensional generalized complex ellipsoid $Σ(z)$. In general, a Hua domain is a nonhom…

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic m…

In this paper (Math. Res. Lett. 13 (2006). No 4, 509-523), the authors established a pseudo-normal form for proper holomoprhic mappings between balls in complex spaces with degenerate rank. This then was used to give a complete characterization for all proper holomorphic maps with geometric rank one, which, in particul…

Analytic linearization and holomorphic extensions for proper groupoids.

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the…

We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of C^2, and a complete proper holomorphic embedding into a ball of C^3.

In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…

Study numerically flat bundles on Fujiki manifolds using algebraic groups.

Proper holomorphic isometries between Bergman domains are biholomorphisms.

Let $(X,J)$ be an almost-complex manifold. In \cite{li-zhang} Li and Zhang introduce $H^{(p,q),(q,p)}_J(X)_{\rr}$ as the cohomology subgroups of the $(p+q)$-th de Rham cohomology group formed by classes represented by real pure-type forms. Given a proper, surjective, pseudo-holomorphic map between two almost-complex ma…

This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter …

We prove that for any open Riemann surface $N$ and finite subset $Z\subset \mathbb{S}^1=\{z\in\mathbb{C}\,|\;|z|=1\},$ there exist an infinite closed set $Z_N \subset \mathbb{S}^1$ containing $Z$ and a null holomorphic curve $F=(F_j)_{j=1,2,3}:N\to\mathbb{C}^3$ such that the map $Y:Z_N\times N\to \mathbb{R}^2,$ $Y(v,P)…

We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $\mathcal{C}^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.

Constructs a space for stable holomorphic submersions over a fixed base.

Every nonflat conformal minimal surface is homotopic to a proper one.

We construct knotted proper holomorphic embeddings of the unit disc in C^2.

Let $X$ be a CR manifold with transversal, proper CR $G$-action. We show that $X/G$ is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorises uniquely over a holomorphic map on $X/G$. We then use this result and complex …

A principal pair consists of a holomorphic principal $G$-bundle together with a holomorphic section of an associated Kaehler fibration. Such objects support natural gauge theoretic equations coming from a moment map condition, and also admit a notion of stability based on Geometric Invariant Theory. The Hitchin--Kobaya…

Constructs continuous families of minimal surfaces and holomorphic immersions.

The symplectic vortex equations admit a variational description as global minimum of the Yang-Mills-Higgs functional. We study its negative gradient flow on holomorphic pairs $(A,u)$ where $A$ is a connection on a principal $G$-bundle $P$ over a closed Riemann surface $Σ$ and $u: P \rightarrow X$ is an equivariant map …

In this paper we prove that every open Riemann surface properly embeds in the Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve, where $SL_2(\mathbb{C})$ is endowed with its standard contact structure. As a consequence, we derive the existence of proper, weakly complete, flat fronts in the real …

This is the first part of a trilogy where we apply the theory of virtual manifold/orbifolds developed by the first named author and Tian to study the Gromov-Witten moduli spaces. In this paper, we resolve the main analytic issue arising from the lack of differentiability of $PSL(2, \C)$-action on spaces of $W^{1, p}$-m…

For proper surjective holomorphic maps from K"ahler manifolds to analytic spaces, we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by Nakano semi-positive vector bundles by means of the higher direct image sheaves, by using the theory of harmonic integrals developed by Takegoshi…

Simple approaches to the proofs of the L^2 Castelnuovo-de Franchis theorem and the cup product lemma which give new versions are developed. For example, suppose u and v are two linearly independent closed holomorphic 1-forms on a bounded geometry connected complete Kaehler manifold X with v in L^2. According to a versi…

The paper develops theory for holomorphic null curves in SL2(C).

Study curvature of direct image bundles in deformations of maps.

The study characterizes and constructs polynomial harmonic morphisms on spheres.

Proper superminimal surfaces in hyperbolic 4-space can be approximated by conformal immersions.

Classifies $π_1$-injective maps between non-compact surfaces.

Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.

Authors create stable proper biharmonic maps from unit ball to spheres.

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theor…

Proves properties of complex algebraic varieties and local systems.

We prove that given an open Riemann surface $N,$ there exists an open domain $M\subset N$ homeomorphic to $N$ which properly holomorphically embeds in $\mathbb{C}^2.$ Furthermore, $M$ can be chosen with hyperbolic conformal type. In particular, any open orientable surface admits a complex structure properly holomorphic…

Reduces proper actions to simpler core actions for analysis.

We explicitly classify all pairs $(M,G)$, where $M$ is a connected complex manifold of dimension $n\ge 2$ and $G$ is a connected Lie group acting properly and effectively on $M$ by holomorphic transformations and having dimension $d_G$ satisfying $n^2+2\le d_G<n^2+2n$. These results extend -- in the complex case -- the…