arXiv research

Gauss map of complete minimal surfaces avoids certain hypersurfaces.

In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^3 and R^4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto and Ru for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a c…

Complete minimal surfaces with any genus found in a specific 3-manifold.

Minimal Legendrian surfaces found in 5D sphere.

We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal resu…

New minimal surfaces found with Cantor ends in convex domains.

Sharp upper bound found for stable minimal surfaces.

We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal …

In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of C^k convergence on compact sets, for any k. As a consequence of the above densi…

In this article, we study the modified defect relations of the Gauss map of complete minimal surfaces in $\mathbb R^3$ and $\mathbb R^4$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto~[J. Differential Geometry \textbf{29} (1989), 245-262] for (the whole) complete minimal surfaces…

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of fin…

Study proves conditions for area-minimizing surfaces in a specific space.

In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets).

The paper classifies and characterizes special surfaces in a 3D space.

In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^m on annular ends. This work is a continuation of previous work of Dethloff-Ha. We thus give an improvement of the results on annular ends of complete minimal surfaces of Jin-Ru.

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a consequence, complete minimal surfaces with arbitrary conformal structure and wh…

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resol…

We give several applications of a lemma on completeness used by Osserman to show the meromorphicity of Weierstrass data for complete minimal surfaces with finite total curvature. Completeness and weak completeness are defined for several classes of surfaces which admit singular points. The completeness lemma is a usefu…

The paper proves uniform approximation for minimal surfaces with applications to a Mittag-Leffler theorem.

The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagran…

The stability and the index of complete one-sided minimal surfaces of certain three-dimensional Riemannian manifolds with positive scalar curvature are studied.

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 …

Examples of complete minimal surfaces properly embedded in H^2 x R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces embedded in H^2 x R, not necessarily proper, which are invariant by a vertical tra…

Consider a convex domain B of space. We prove that there exist complete minimal surfaces which 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 an…

In this paper we give new existence results for complete non-orientable minimal surfaces in $\mathbb{R}^3$ with prescribed topology and asymptotic behavior.

The study proves theorems about minimal and H-surfaces in 3D space.

Minimal surfaces can't have certain epitrochoid geodesics.

Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz reflection extends to a complete minimal surface. Studying the group of Schwarz refl…

We construct complete nonorientable minimal surfaces whose Gauss map omits two points of the projective plane. This result proves that Fujimoto's theorem is sharp in nonorientable case.

Study proves rigidity of minimal hypersurfaces in specific manifolds.

Study of minimal surfaces in 4D with specific ends.

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a …

Study proves no minimal surfaces can be contained in certain half-spaces or cones.

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 immer…

Solves minimal surface problem in H^2xR.

The paper studies the space of Gauss maps of complete minimal surfaces and their homotopy types.

The study proves a strong parametric h-principle for minimal surfaces.

We study the topology of (properly) immersed complete minimal surfaces $P^2$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of…

In [15] Robert Osserman proved that the image of the Gauss map of a complete, non flat minimal surface in R^3 with finite total curvature miss at most 3 points. In this paper we prove that the Gauss map of such a minimal immersions omit at most 2 points. This is a sharp result since the Gauss map of the catenoid omits …

We prove that, given a compact Riemann surface $Σ$ and disjoint finite sets $\varnothing\neq E\subsetΣ$ and $Λ\subsetΣ$, every map $Λ\to \mathbb{R}^3$ extends to a complete conformal minimal immersion $Σ\setminus E\to \mathbb{R}^3$ with finite total curvature. This result opens the door to study optimal hitting problem…

Proves existence of minimal surfaces of arbitrary genus with two ends.

We investigate the geometric properties of hyperbolic affine flat, affine minimal surfaces in the equiaffine space $\mathbb{A}^3$. We use Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of such surfaces and…

In this article, we propose some conditions on the modified defect relations of the Gauss map of a complete minimal surface $M$ to show that $M$ has finite total curvature.

We give a complete topological classification of minimal surfaces in Euclidian three-space.

The study constructs minimal surfaces in a product space with specific properties.

Study on minimal surfaces with constraints on index and branching order.

Study on Gauss images of specific minimal surfaces with finite curvature.