The Mittag-Leffler theorem for proper minimal surfaces and directed meromorphic curves

Working over perfect ground fields of arbitrary characteristic, I classify minimal normal del Pezzo surfaces containing a nonrational singularity. As an application, I determine the structure of 2-dimensional anticanonical models for proper normal algebraic surfaces. The anticanonical ring may be non-finitely generated. However, the anticanonical model is either a proper surface, or a proper surface minus a point.

We study the Bonnet problem for surfaces in 4-dimensional space forms, namely, to what extent a surface is determined by the metric and the mean curvature. Two isometric surfaces have the same mean curvature if there exists a parallel vector bundle isometry between their normal bundles that preserves the mean curvature vector fields. We deal with the structure of the moduli space of congruence classes of isometric surfaces with the same mean curvature and with properties inherited on a surface by this structure. The study of this problem led us to a new conformally invariant property, called isotropic isothermicity, that coincides with the usual concept of isothermicity for surfaces lying in totally umbilical hypersurfaces, and is related to lines of curvature and infinitesimal isometric deformations that preserve the mean curvature vector field. The class of isotropically isothermic surfaces includes the one of surfaces with a vertically harmonic Gauss lift and particularly the minimal surfaces, and overlaps with that of isothermic surfaces without containing the entire class. We show that if a simply connected surface is not proper Bonnet, which means that the moduli space is a finite set, then it admits either at most one, or exactly three Bonnet mates. For simply connected proper Bonnet surfaces, the moduli space is either 1-dimensional with at most two connected components diffeomorphic to the circle, or the 2-dimensional torus. We prove that simply connected Bonnet surfaces lying in totally geodesic hypersurfaces of the ambient space as surfaces of non-constant mean curvature always admit Bonnet mates that do not lie in any totally umbilical hypersurface. Such surfaces either admit exactly three Bonnet mates, or they are proper Bonnet with moduli space the torus. We show that isotropic isothermicity characterizes the proper Bonnet surfaces, and we provide relevant conditions for non-existence of Bonnet mates for compact surfaces. Moreover, we study compact surfaces that are locally proper Bonnet, and we prove that the existence of a uniform substructure on the local moduli spaces characterizes surfaces with a vertically harmonic Gauss lift that are neither minimal, nor superconformal. In particular, we show that the only compact, locally proper Bonnet surfaces with moduli space the torus, are those with nonvanishing parallel mean curvature vector field and positive genus.

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to $\mathbb{R}^n$ is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable conformal minimal surfaces in $\mathbb{R}^n$. We also give the first known example of a properly embedded non-orientable minimal surface in $\mathbb{R}^4$; a Mobius strip. All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in $\mathbb{R}^n$ with any given conformal structure, complete non-orientable minimal surfaces in $\mathbb{R}^n$ with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in $p$-convex domains of $\mathbb{R}^n$.

An interesting class of submanifolds of Hermitian manifolds is the class of slant submanifolds which are submanifolds with constant Wirtinger angle. In (1-4,7,8) slant submanifolds of complex projective and complex hyperbolic spaces have been investigated. In particular, it was shown that there exist many proper slant surfaces in CP 2 and in CH 2 and many proper slant minimal surfaces in C 2 .I n contrast, in the first part of this paper we prove that there do not exist proper slant minimal surfaces in CP 2 and in CH 2 . In the second part, we present a general con- struction procedure for obtaining the explicit expressions of such slant submani- folds. By applying this general construction procedure, we determine the explicit expressions of special slant surfaces of CP 2 and of CH 2 . Consequently, we are able to completely determine the slant surface which satisfies a basic equality. Finally, we apply the construction procedure to prove that special -slant isometric immersions of a hyperbolic plane into a complex hyperbolic plane are not unique in general.

Smooth projective varieties with small invariants have received renewed interest in recent years, primarily due to the fine study of the adjunction mapping. Now, through the effort of several mathematicians, a complete classification of smooth surfaces in ${\Bbb P}^4$ has been worked out up to degree $10$, and a partial one is available in degree $11$. On the other side, recently Ellingsrud and Peskine have proved Hartshorne's conjecture that there are only finitely many families of smooth surfaces in ${\Bbb P}^4$, not of general type. It is believed that the degree of the smooth, non-general type surfaces in ${\Bbb P}^4$ should be less than or equal to $15$. The aim of this paper is to provide a series of examples of smooth surfaces in ${\Bbb P}^4$, not of general type, in degrees varying from $12$ up to $14$, and to describe their geometry. By using mainly syzygies and liaison techniques, we construct the following families of surfaces: \begin{enumerate} \item[] minimal proper elliptic surfaces of degree $12$ and sectional genus $\pi=13$; \item[] two types of non-minimal proper elliptic surfaces of degree $12$ and sectional genus $\pi=14$; \item[] non-minimal $K3$ surfaces of degree $13$ and sectional genus $16$; and \item[] non-minimal $K3$ surfaces of degree $14$ and sectional genus $19$. 1991 Mathematics Subject Classification: 14M07, 14J25, 14J26, 14J28, 14C05.

This chapter begins with the development of some of the principal tools used in the book. It continues with the first major collection of new results on oriented minimal surfaces in Rn and holomorphic null curves in Cn for any n ≥ 3, obtained since 2012 by complex analytic methods. The main results of the chapter include Runge, Mergelyan, and Carleman type approximation theorems for conformal minimal surfaces, analogues of the Weierstrass and Mittag-Leffler interpolation theorems, general position theorems for minimal surfaces and null curves, the construction of proper minimal surfaces with arbitrary conformal structure in Euclidean spaces, and the homotopy theory for the space of conformal minimal immersions from a given open Riemann surface into Rn.

In this paper, we give examples of proper embedded Jenkins-Serrin type minimal surfaces with a reflective symmetry. That is, for each\( n \geq 2 \), we prove that there is a unique constant\( \phi _n, 0 \leq \phi _n < \pi \), and there exists a family of properly embedded minimal surfaces \( M(n, \phi), \phi_n < \phi < \pi\). Each \( M(n, \phi) \) is bounded by 2n parallel straight lines such that the interior of \( M(n, \phi) \) is the union of a minimal graph \( G(n,\phi) \) and its reflection. Each \( M(n,\phi) \)is invariant under \( D_{n}\times\mathbb{Z}_{2} \), where \( D_{n} \) is the general dihedral group, and\( \mathbb{Z}_{2} \) is generated by a reflection keeping each of the boundary lines invariant. The graph \( G(n,\phi) \) is over an non-convex bounded domain with a Jenkins-Serrin type capillary boundary values.¶¶Moreover, for each \( n\geq 2, M(n, \phi) \) can be put in a bigger family of immersed minimal surfaces, \( 0\leq \phi \leq \pi \). For \( 0 < \phi < \pi, M(n, \phi) \)has the same symmetric property and is bounded by parallel straight lines.¶¶When \( n\geq 2, G(n, \pi) \) is the Jenkins-Serrin graph over a domain bounded by a regular 2n-gon;\( M(2, 0) \) is a catenoid while for \( n\geq 3,M(n, 0) \) is the Jorge-Meeks n-noid; for\( 0 < \phi < \pi, M(2, \phi) \) is the KMR surface discovered by Karcher, and Meeks and Rosenberg.¶¶Thus in the moduli space of properly immersed minimal surfaces,\( M(n, \phi), n \geq 2, 0 \leq \phi \leq \pi \), is a connected path connecting the catenoid or Jorge-Meeks n-noid to the Jenkins-Serrin graph.

Consider a convex domain $B$ of $\\mathbbR^3$ . We prove that there exist complete minimal surfaces that are properly immersed in $B$ . We also demonstrate that if $D$ and $D'$ are convex domains with $D$ bounded and the closure of $D$ contained in $D'$ , then any minimal disk whose boundary lies in the boundary of $D$ can be approximated in any compact subdomain of $D$ by a complete minimal disk that is proper in $D'$ . We apply these results to study the so-called type problem for a minimal surface: we demonstrate that the interior of any convex region of $\\mathbbR^3$ is not a universal region for minimal surfaces, in the sense explained by Meeks and Pérez in [9].

In this chapter, the Riemann-Hilbert modification technique is applied to the construction of complete proper minimal surfaces from bordered Riemann surfaces into minimally convex domains in Rn for any n≥3. This class of domains contains all convex domains as well as many non-convex ones. In dimension n=3 it coincides with the class of mean-convex domains, and it is the largest class for which our results hold. A major role is played by minimal plurisubharmonic functions whose key property is that they restrict to subharmonic functions on any conformal minimal surface, and they are the largest class of functions having this property.

The Calabi-Yau conjecture is one of the main problems in the global theory of complete minimal surfaces in R3. Francisco Martin and Santiago Morales have constructed complete proper minimal surfaces in convex bodies of R3. In this paper, we modify their technique in the cylindrical case, and construct a complete minimal cylinder properly immersed in the unit ball.

SLANT IMMERSIONS OF COMPLEX SPACE FORMS AND CHEN'S INEQUALITY

A slant immersion is an isometric immersion from a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. In this paper we establish the existence and uniqueness theorem for slant immersions into complex-space-forms. By applying this result, we prove in this paper several existence and nonexistence theorems for slant immersions. In particular, we prove the existence theorems for slant surfaces with prescribed mean curvature or with prescribed Gaussian curvature. We also prove the non-existence theorem for flat minimal proper slant surfaces in non-flat complex space forms.

Let D be a regular, strictly convex bounded domain of \mathbb{R}^3 , and consider a Jordan curve \Gamma \subset \partial D . Then, for each \varepsilon&gt;0 , we obtain the existence of a complete proper minimal immersion \psi_\varepsilon \colon \mathbb{D} \rightarrow D satisfying that the Hausdorff distance \delta^H(\psi_\varepsilon(\partial \mathbb{D}), \Gamma) &lt; \varepsilon , where \psi_\varepsilon(\partial \mathbb{D}) represents the limit set of the minimal disk \psi_\varepsilon(\mathbb{D}) .

Existence of proper minimal surfaces of arbitrary topological type

In this article on minimal surfaces, considered in close connection with meromorphic curves, Nevanlinna's main theorems for meromorphic functions are generalized. Besides the counting function and the characteristic function, the visibility function is introduced, which has a physical significance explained in the article and plays an essential role for minimal surfaces in , , so that in this case the analog of the first main theorem is the equivisibility theorem. Moreover, an analog of Nevanlinna's second main theorem is obtained, and some properties of the characteristic function are explained that connect it with topological properties, the character of the singularities and the growth of a minimal surface.Bibliography: 7 titles.