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 theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map.
As a consequence we give an approximate solution to a Plateau problem for divergent Jordan curves in the Euclidean spaces.
Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartan's method to the question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth $9$-dimensional real analytic real hypersurface $M^{9}\subset\mathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}\times U(2)$. We then provide an example where one of the functions does not identic…
We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into $\math…
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 embedding provided that $\mathrm{dim} Y > 2\mathrm{dim}\, X$ and we allow a homotopic deformation of the Stein structure on $X$.
For any subvariety of a compact holomorphic symplectic Kaehler manifold, we define the number W(X), which we call Wirtinger number. We show that $W(X)\leq 1$, and the equality is reached if and only if the subvariety $X\subset M$ is trianalytic, i. e. compactible with the hyperkaehler structure on M. For a sequence $X_1 \arrow X_2 \arrow ... X_n\arrow M$ of immersions of simple holomorphic symplectic manifolds, we show that $W(X_1) \leq W(X_2) \leq >... \leq W(X_n)$.
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 values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into $\mathbb{R}^n$ if the prescription of values is proper, and one-to-one if $n\ge 5$ and the prescription of values is one-to-one. We may also prescribe the flu…
In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,ξ)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We…
In the previous paper, the authors constructed a complete holomorphic immersion of the unit disk D into C^2 whose image is bounded. In this paper, we shall prove existence of complete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology, whose image is bounded in C^2. To construct such immersions, we apply the method used by F. J. Lopez to perturb the genus zero example changing its genus. As an analogue the above construction, we also give a new method to construct complete bounded minimal immersions (resp. weakly complete maximal surface) with arbitrary ge…
For a complex Lie group $G$ with a real form $G_0\subset G$, we prove that any Hamiltionian automorphism $ϕ$ of a coadjoint orbit $\mathcal O_0$ of $G_0$ whose connected components are simply connected, may be approximated by holomorphic $\mathcal O_0$-invariant symplectic automorphism of the corresponding coadjoint orbit of $G$ in the sense of Carleman, provided that $\mathcal O$ is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.
This paper introduces a complex representation for spacelike surfaces in the Lorentz-Minkowski space $L^4$, based in two complex valued functions which can be assumed to be holomorphic or anti-holomorphic. When the immersion is contained in quadrics of $L^4$, the representation then allows us to obtain interesting partial differential equations with holomorphic or anti-holomorphic parameters, within which we find the partial Riccati Equation. Using then theory of holomorphic complex functions we construct explicitly new local solutions for those PDEs together with its associated geometric solu…
We approach the study of totally real immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1. We introduce a notion of first and second fundamental form, we prove that they satisfy a similar version of the classic Gauss-Codazzi equations, and conversely that solutions of Gauss-Codazzi equations are immersion data of some equivariant map. This study has some interesting geometric consequences: 1) it provides a formalism to study immersions of surfaces into SL(2,C) and into the space of geodesics of H^3; 2) it generalizes the classical theory of…
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 $ Σ^n$ of codimension at most 1. We exploit the successive derivatives of this relation and show that a linearly full local CR immersion $ f: Σ^n \hook Σ^{N}$, $ N \leq 3n-2$, can only occur when $ N = n, 2n$, or $ 2n+1$. Together with the recent classification of the rational p…
In this paper, we show that for every conformal minimal immersion $u:M\to \mathbb{R}^3$ from an open Riemann surface $M$ to $\mathbb{R}^3$ there exists a smooth isotopy $u_t:M\to\mathbb{R}^3$ $(t\in [0,1])$ of conformal minimal immersions, with $u_0=u$, such that $u_1$ is the real part of a holomorphic null curve $M\to \mathbb{C}^3$ (i.e. $u_1$ has vanishing flux). Furthermore, if $u$ is nonflat then $u_1$ can be chosen to have any prescribed flux and to be complete.
Let $M$ be an open Riemann surface. It was proved by Alarcón and Forstnerič (arXiv:1408.5315) that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. In this paper, we prove the following much stronger result in this direction: for any $n\geq 3$, the inclusion $ι$ of the space of real parts of nonflat null holomorphic immersions $M\to\mathbb C^n$ into the space of nonflat conformal minimal immersions $M\to \mathbb R^n$ satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equiva…
This thesis mainly treats two developments of the classical theory of hypersurfaces inside pseudo-Riemannian space forms. The former - a joint work with Francesco Bonsante - consists in the study of immersions of smooth manifolds into holomorphic Riemannian space forms of constant curvature -1 (including SL(2,C) with a multiple of its Killing form): this leads to a Gauss-Codazzi theorem, it suggests an approach to holomorphic transitioning of immersions into pseudo-Riemannian space forms, a trick to construct holomorphic maps into the PSL(2,C)-character variety, and leads to a restatement of B…
Following earlier work of Loftin-McIntosh, we study minimal Lagrangian immersions of the universal cover of a closed surface (of genus at least 2) into CH2, with prescribed data of a conformal structure plus a holomorphic cubic differential. We show existence and non-uniqueness of such minimal Lagrangian immersions. We also establish the surface area with respect to the induced metric as a Weil-Petersson potential function for the space of holomorphic cubic differentials on the Riemann surface.
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.
A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and asymptotic behavior of the spectral curve. If this curve has finite genus the Dirichlet energy of a map from a 2-torus to the 2-sphere or the Willmore energy of an immersion from a 2-torus into the 4-sphere is given by the residue of a specific meromorphic differential on the cur…
A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean three- or four-space. The Weierstrass representation, the spin transform, the Darboux transforms, surfaces of parallel mean curvature vector, families of flat connections associated with a harmonic map from a Riemann surface to a sphere are explained. The degree of the spinor bundle associated with a conformal immersion is calculated. Analogues of a polar surface…
Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved for conformal minimal immersions $M\to\mathbb{R}^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic through conformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identify the path connected components of the space of all conformal minimal immersio…
In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann-Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi-Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^n$ and in minimally convex domains of $\mathbb{R}^n…
This paper gives a construction for all minimal immersions $f$ of the Poincaré disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $ρ$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal harmonic map and therefore has a corresponding Higgs bundle. We identify the structure of these Higgs bundles and show how each is determined by properties of the map, including the induced metric and a holomorphic cubic differential on the surface. We show that the moduli space …
Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ of continuous maps from $M$ to the sphere $\mathbb S^{4n-1}$. If $M$ has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of $\mathscr L(M,\mathbb C^{2n+1})$ in terms of the homotopy groups of $\mathbb S^{4n-1}$. It follows that …
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 family of non-minimal conformal biharmonic immersions of cylinder into $\r^3$ and some examples of conformal biharmonic immersions of 4-dimensional Euclidean space into sphere and hyperbolic space thus provide many simple examples of proper biharmonic maps with rich geometric mea…
Consider a very ample line bundle $ E \to X$ over a compact complex manifold, endowed with a hermitian metric of curvature $-i ω$, and the space $\mathcal{O}(E)$ of its holomorphic sections. The Fubini--Study map associates with positive definite inner products $\langle \, , \rangle$ on $\mathcal{O}(E)$ functions FS$(\langle \, ,\rangle) \in \mathcal{H}_ω=\{u \in C^{\infty}(X):ω+i\partial\overline{\partial} u >0\}$. We prove that FS is an injective immersion, but its image in general is not closed in $\mathcal{H}_ω$. To obtain a closed range, FS has to be extended to certain degenerate inner p…
We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a Kähler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense o…
An equivariant minimal surface in $\mathbb{CH}^n$ is a minimal map of the Poincaré disc into $\mathbb{CH}^n$ which intertwines two actions of the fundamental group of a closed surface $Σ$: a Fuchsian representation on the disc and an irreducible action by isometries on $\mathbb{CH}^n$. The moduli space of these can been studied by relating it to the nilpotent cone in each moduli space of $PU(n,1)$-Higgs bundles over the conformal surface corresponding to the map. By providing a necessary condition for points on this nilpotent cone to be smooth this article shows that away from the points corre…
We investigate the existence of a holomorphic and isometric immersion in the complex projective space for the complete Ricci-flat Kaehler metrics constructed by M. B. Stenzel on the cotangent bundle of a compact, rank one, globally symmetric space.
In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology. Moreover, we show that every domain in $\mathbb{R}^n$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface. Our method of p…
In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves in SL_2(C) that are related to Bryant surfaces. We establish a basic structure theorem for the set of all A-immersions of a bordered Riemann surface, and we prove several approximation and desingularization theorems. Assuming that A is irreducible and is not contained in any hyper…
Let $f\colon M^{2n}\to\mathbb{R}^{2n+4}$ be an isometric immersion of a Kaehler manifold of complex dimension $n\geq 5$ into Euclidean space with complex rank at least $5$ everywhere. Our main result is that, along each connected component of an open dense subset of $M^{2n}$, either $f$ is holomorphic in $\mathbb{R}^{2n+4}\cong\mathbb{C}^{n+2}$ or it is in a unique way a composition $f=F\circ h$ of isometric immersions. In the latter case, we have that $h\colon M^{2n}\to N^{2n+2}$ is holomorphic and $F\colon N^{2n+2}\to\mathbb{R}^{2n+4}$ belongs to the class, by now quite well understood, of n…
We study minimal immersions of closed surfaces (of genus $g \ge 2$) in hyperbolic 3-manifolds, with prescribed data $(σ, tα)$, where $σ$ is a conformal structure on a topological surface $S$, and $αdz^2$ is a holomorphic quadratic differential on the surface $(S,σ)$. We show that, for each $t \in (0,τ_0)$ for some $τ_0 > 0$, depending only on $(σ, α)$, there are at least two minimal immersions of closed surface of prescribed second fundamental form $Re(tα)$ in the conformal structure $σ$. Moreover, for $t$ sufficiently large, there exists no such minimal immersion. Asymptotically, as $t \to 0$…
We show that generic rank conditions on the second fundamental form of an isometric immersion $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with low codimension $p$ implies that the submanifold has to be minimal. If $M^{2n}$ if simply connected, this amounts to the existence of a one-parameter associated family of isometric minimal immersions unless $f$ is holomorphic.
We prove that for any open Riemann surface $N,$ natural number $n\geq 3,$ non-constant harmonic map $h:N\to \mathbb{R}^{n-2}$ and holomorphic 2-form $H$ on $N,$ there exists a weakly complete harmonic map $X=(X_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ with Hopf differential $H$ and $(X_j)_{j=3,\ldots,n}=h.$ In particular, there exists a complete conformal minimal immersion $Y=(Y_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ such that $(Y_j)_{j=3,\ldots,n}=h.$ As a consequence of these results, complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss m…
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic line bundles over "discrete surfaces", by which we mean the vertex sets of triangulated surfaces with bi-colored set of faces. The resulting theory of discrete conformality is simultaneously Moebius invariant and based on linear equations. In the special case of maps into the 2-sp…
We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of "finite type", that is, has a spectral curve of finite genus, or is of "holomorphic type" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in the 4-sphere can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin's approach to the study of harmonic tori in the 3-sphere.
In this paper we prove that every bordered Riemann surface M admits a complete proper null holomorphic embedding into a ball of the complex Euclidean $3$-space $\mathbb{C}^3$. The real part of such an embedding is a complete conformal minimal immersion $M\to \mathbb{R}^3$ with bounded image. For any such $M$ we also construct proper null holomorphic embeddings $M\to \mathbb{C}^3$ with a bounded coordinate function; these give rise to properly embedded null curves $M\to SL_2(\mathbb{C})$ and to properly immersed Bryant surfaces $M\to \mathbb{H}^3$ in the hyperbolic $3$-space. In particular, we …
We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle of this family over the spectral curve describes the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has finite genus the kernel bundle can be extended to the compa…
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 $Y:Z_N\times N\to \mathbb{R}^2,$ $Y(v,P)=Re(v(F_1,F_2)(P)),$ is proper. In particular, $Re(vF):N \to\mathbb{R}^3$ is a proper conformal minimal immersion properly projecting into $\mathbb{R}^2=\mathbb{R}^2\times\{0\}\subset\mathbb{R}^3,$ for all $v \in Z_N.$
In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of $\mathbb{R}^n$ which extend continuously up to the boundary; f…
