Complete Minimal Surfaces Research Articles

Modified defect relations of the Gauss map and the total curvature of a complete minimal surface

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.

Complete minimal surfaces densely lying in arbitrary domains of ℝn

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 proof can be adapted to give analogous results for non-orientable minimal surfaces in $\mathbb{R}^n$ $(n\ge 3)$, complex curves in $\mathbb{C}^n$ $(n\ge 2)$, holomorphic null curves in $\mathbb{C}^n$ $(n\ge 3)$, and holomorphic Legendrian curves in $\mathbb{C}^{2n+1}$ $(n\in\mathbb{N})$.

The Gauss Map of Algebraic Complete Minimal Surfaces Omits Hypersurfaces in Subgeneral Position

The first aim of this paper is to show a second main theorem for algebraic curves into the n-dimensional projective space sharing hypersurfaces in subgeneral position. We then use it to study the value distribution of the generalized Gauss map of the complete (regular) minimal surfaces in \(\mathbb {R}^{m}\) with finite total curvature, as well as the unicity problem, sharing hypersurfaces in subgeneral position. Our results generalize and complete previous results in this area, especially the works of Chern and Osserman (J. Anal. Math. 19, 15–34, 1967), Jin and Ru (Differ. Geom. Appl. 25: 701–712, 2007).

Finite topology self-translating surfaces for the mean curvature flow in [formula omitted

Finite topology self-translating surfaces for the mean curvature flow constitute a key element in the analysis of Type II singularities from a compact surface because they arise as limits after suitable blow-up scalings around the singularity. We prove the existence of such a surface M⊂R3 that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface S(t)=M+tez evolves by mean curvature flow, or equivalently, that M satisfies the equation HM=ν⋅ez where HM denotes mean curvature, ν is a choice of unit normal to M, and ez is a unit vector along the z-axis. This surface M is in correspondence with the classical three-end Costa–Hoffman–Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded complete minimal surfaces with finite total curvature.

New minimal surfaces in the hyperbolic space

We obtain new complete minimal surfaces in the hyperbolic space H3, by using Ribaucour transformations. Starting with the family of spherical catenoids in H3 found by Mori (1981), we obtain 2- and 3-parameter families of new minimal surfaces in the hyperbolic space, by solving a non trivial integro-differential system. Special choices of the parameters provide minimal surfaces whose parametrizations are defined on connected regions of R2 minus a disjoint union of Jordan curves. Any connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of H3 is a closed curve. The geometric properties of the surfaces regarding the ends, completeness and symmetries are discussed.

Analytic extension of Jorge-Meeks type maximal surfaces in Lorentz-Minkowski 3-space

The Jorge-Meeks $n$-noid ($n\ge 2$) is a complete minimal surface of genus zero with $n$ catenoidal ends in the Euclidean 3-space $\boldsymbol{R}^3$, which has $(2\pi/n)$-rotation symmetry with respect to its axis. In this paper, we show that the corresponding maximal surface $f_n$ in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ has an analytic extension $\tilde f_n$ as a properly embedded zero mean curvature surface. The extension changes type into a time-like (minimal) surface.

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

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 results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space.

HELICOIDAL MINIMAL SURFACES IN A CONFORMALLY FLAT 3-SPACE

Abstract. In this work, we introduce the complete Riemannian man-ifold F 3 which is a three-dimensional real vector space endowed with aconformally flat metric that is a solution of the Einstein equation. Weobtain a second order nonlinear ordinary differential equation that char-acterizes the helicoidal minimal surfaces in F 3 . We show that the helicoidis a complete minimal surface in F 3 . Moreover we obtain a local solutionof this differential equation which is a two-parameter family of functionsλ h,K 2 explicitly given by an integral and defined on an open interval.Consequently, we show that the helicoidal motion applied on the curvedefined from λ h,K 2 gives a two-parameter family of helicoidal minimalsurfaces in F 3 . 1. IntroductionIn the monograph entitled “Ricci Curvature of Seminar on Differential Ge-ometry” edited by Yau (see Problem Section in [13]), there is a problem whichis to find the necessary and sufficient conditions for a symmetric tensor T ij ona compact manifold so that one can find a metric g

Minimal surfaces with two ends which have the least total absolute curvature

In this paper, we consider complete non-catenoidal minimal surfaces of finite total curvature with two ends. A family of such minimal surfaces with least total absolute curvature is given. Moreover, we obtain a uniqueness theorem for this family from its symmetries.

Ramification of the Gauss map of complete minimal surfaces in Rmon annular ends

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 complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement of the results on annular ends of complete minimal surfaces of Kao.

Ramification of the Gauss map and the total curvature of a complete minimal surface

In this article, we study the relations between the ramifications of the Gauss map and the total curvature of a complete minimal surface. More precisely, we introduce some conditions on the ramifications of the Gauss map of a complete minimal surface M to show that M has finite total curvature.

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

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; for $n\ge 5$ we find embeddings with these properties.

Non-integrated defect relations for the Gauss maps of complete minimal surfaces with finite total curvature

In this article, we give the non-integrated defect relations in the sense of Fujimoto for the Gauss maps of complete minimal surfaces with finite total curvature in R3,R4. These are some strict improvements of previous results for the ramifications and modified defect relations of the Gauss maps of complete minimal surfaces with finite total curvature.

Counting minimal surfaces in quasi-Fuchsian three-manifolds

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many stable ones. In this paper, for any prescribed integer $N > 0$, we construct a quasi-Fuchsian manifold which contains at least $2^N$ such minimal surfaces. As a consequence, there exists some simple closed Jordan curve on $S_{\infty }^2$ such that there are at least $2^N$ disk-type complete minimal surfaces in $\mathbb {H}^3$ sharing this Jordan curve as the asymptotic boundary.

On the spectrum of bounded immersions

In this article, we investigate some of the relations between the spectrum of a non-compact, extrinsically bounded submanifold $\varphi : M^m \to N^n$ and the Hausdorff dimension of its limit set $\lim \varphi$. In particular, we prove that if $\varphi : M^2 \to \mathbb{R}^3$ is a (complete) minimal surface immersed into an open, bounded, strictly convex subset $\Omega$ with $C^3$-boundary, then $M$ has discrete spectrum, provided that $\mathcal{H}_{\Psi} (\lim \varphi \cap \Omega) = 0$, where $\mathcal{H}_{\Psi}$ is the Hausdorff measure of order $\Psi(t) = t^2 \lvert \log t \rvert$. Our main theorem, Thm. 2.4, applies to a number of examples recently constructed, by many authors, in the light of Nadirashvili’s discovery of complete bounded minimal disks in $\mathbb{R}^3$, as well as to solutions of Plateau’s problems for non-rectifiable Jordan curves, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. On the other hand, we present a simple criterion, called the ball property, whose fulfilment guarantees the existence of elements in the essential spectrum. As an application, we show that some of the examples of Jorge-Xavier and Rosenberg-Toubiana of complete minimal surfaces between two planes have essential spectrum $\sigma_{\mathrm{ess}}(-\Delta) = [0, \infty)$.

A Schoen theorem for minimal surfaces in [formula omitted

In this paper we prove that a complete minimal surface immersed in H2×R, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H2×R. We also prove that a minimal complete end E with finite total curvature is properly immersed and that the Gaussian curvature of E is locally bounded in terms of the geodesic distance to its boundary.

Uniform approximation by complete minimal surfaces of finite total curvature in ℝ³

We prove that any compact minimal surface in R 3 \mathbb {R}^3 can be uniformly approximated by complete minimal surfaces of finite total curvature in R 3 \mathbb {R}^3 . 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 regions of finite conformal type. We deal only with the orientable case.

Four Classes of Surfaces with Constant Mean Curvature in S 3 × R and H 3 × R

Deforming rotation surfaces with constant mean curvature in S3 and H3 to S3 × R and H3 × R respectvely, we give four classes of surfaces with mean curvature vector of constant length in S3 × R and H3 × R. We have complete minimal surfaces in S3 × R and H3 × R. Also we obtain minimal 2-tori in S3 × S1, some of which are embedded.

Ramification of the Gauss map of complete minimal surfaces in ℝ 3 and ℝ 4 on annular ends

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 complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement of the results on annular ends of complete minimal surfaces of Kao.

Complete Non-Orientable Minimal Surfaces in ℝ3 and Asymptotic Behavior

Abstract In this paperwe give new existence results for complete non-orientable minimal surfaces in ℝ3 with prescribed topology and asymptotic behavior