Complete Minimal Surfaces Research Articles

Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions

We study complete minimal surfaces in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy W:=14∫|H→|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}: =\\frac{1}{4} \\int |\\vec H|^2$$\\end{document}. In codimension one, we prove that the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly m-3=W4π-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m-3=\\frac{\\mathcal {W}}{4\\pi }-3$$\\end{document}. We also consider several geometric properties—for example, the property that all m asymptotic planes meet at a single point—of these minimal surfaces and explore their relation to the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Morse index of their inverted surfaces.

The Gauss Images of Complete Minimal Surfaces of Genus Zero of Finite Total Curvature

This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We focus on the number of omitted values and the total weight of the totally ramified values of their Gauss maps. In particular, we construct new complete minimal surfaces of finite total curvature whose Gauss maps have 2 omitted values and 1 totally ramified value of order 2, that is, the total weight of the totally ramified values of their Gauss maps are 5/2(=2.5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5/2\\,(=2.5)$$\\end{document} in Euclidean 3-space and Euclidean 4-space, respectively. Moreover we discuss several outstanding problems in this study.

Modified defect relation for Gauss maps of minimal surfaces with hypersurfaces of projective varieties in the subgeneral position

AbstractIn this paper, we establish some modified defect relations for the Gauss map of a complete minimal surface into a ‐dimension projective subvariety with hypersurfaces of in ‐subgeneral position with respect to . In particular, we give the upper bound for the number if the image intersects each hypersurface a finite number of times and is nondegenerate over , where , that is, the image of is not contained in any hypersurface of degree with . Our results extend and generalize the previous results for the case of the Gauss map and hyperplanes in a projective space. The results and the method of this paper have been applied by some authors to study the unicity problem of the Gauss maps sharing families of hypersurfaces.

Minimal Legendrian surfaces in the tangent sphere bundle of 𝕊³

In this paper we study minimal Legendrian surfaces Σ \Sigma immersed in tangent sphere bundle T 1 S 3 T_1{\mathbb {S}}^3 . We classify (1) totally geodesic Legendrian surfaces, (2) closed minimal Legendrian surfaces of genus smaller than or equal to one and complete minimal Legendrian surfaces with non-negative Gauss curvature.

Research on Hyperbolic Complete Minimal Surfaces Based on Special Power Series

Research on Hyperbolic Complete Minimal Surfaces Based on Special Power Series

Modified defect relation of Gauss maps on annular ends of minimal surfaces for hypersurfaces of projective varieties in subgeneral position

Let A be an annular end of a complete minimal surface S in Rm and let V be a k-dimension projective subvariety of Pn(C)(n=m−1). Let g be the generalized Gauss map of S into V⊂Pn(C). In this paper, we establish a modified defect relation of g on the annular end A for q hypersurfaces {Qi}i=1q of Pn(C) in N-subgeneral position with respect to V. Our result implies that the image g(A) cannot omit all q hypersurfaces Q1,…,Qq if g is nondegenerate over Id(V) and q>(2N−k+1)(M+1)(M+2d)2d(k+1), where M=HV(d)−1 and d is the least of common multiple of deg⁡Q1,…,deg⁡Qq. As our best knowledge, it is the first time the value distribution of the Gauss map on an annular end of a minimal surfaces with hypersurface targets is studied, in particular the product into sum inequality for holomorphic curves on Riemann surfaces with hypersurfaces targets is presented. This our result has been used to study the unicity of the gauss maps in the recent work of C. Lu and X. Chen [14].

On the Index of Willmore spheres

We consider unbranched Willmore surfaces in Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index—the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three‑space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either two or three. For $m=4$, we prove that $d=3$. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.

Area and Spectrum Estimates for Stable Minimal Surfaces

This paper mainly concerns the area growth and bottom spectrum of complete stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. When the ambient manifold is the Euclidean space, by an elementary argument, it is shown directly from the stability inequality that the area of such minimal surfaces grows exactly as the Euclidean plane. Consequently, such minimal surfaces must be flat, a well-known result due to Fisher-Colbrie and Schoen as well as do Carmo and Peng. In the case of general ambient manifold, explicit area growth estimate is also derived. For the bottom spectrum, a self-contained argument involving positive Green’s function is provided for its upper bound estimates. The argument extends to stable minimal hypersurfaces in a complete manifold of dimension up to six with sectional curvature bounded from below.

Classification of Complete Regular Minimal Surfaces in ℝn with Total Curvature −6π

In this paper, we classify the complete regular orientable minimal surfaces in Rn with total curvature −6π and give a method to construct a series of complete non-holomorphic minimal surfaces with total curvature −6π. Specially, we give a simplified classification in another method if the surfaces lie in R4.

GENUS MINIMAL k-NOIDS AND SADDLE TOWERS IN

AbstractFor each $k\geq 3$ , we construct a $1$ -parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb {H}^2\times \mathbb {R}$ with genus $1$ and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb {H}^2\times \mathbb {R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi $ . Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb {H}^2\times \mathbb {R}$ by the action of a hyperbolic or parabolic translation.

The Classification of Branched Willmore Spheres in the 3-Sphere and the 4-Sphere

We extend the classification of Robert Bryant of Willmore spheres in $S^3$ to variational branched Willmore spheres $S^3$ and show that they are inverse stereographic projections of complete minimal surfaces with finite total curvature in $\mathbb{R}^3$ and vanishing flux. We also obtain a classification of variational branched Willmore spheres in $S^4$, generalising a theorem of Sebastian Montiel. As a result of our asymptotic analysis at branch points, we obtain an improved $C^{1,1}$ regularity of the unit normal of variational branched Willmore surfaces in arbitrary codimension. We also prove that the width of Willmore sphere min-max procedures in dimension $3$ and $4$, such as the sphere eversion, is an integer multiple of $4\pi$.

Distribution value of algebraic curves and the Gauss maps on algebraic minimal surfaces

In this paper, we first establish a degeneracy second main theorem for algebraic curves from compact complex Riemann surfaces into projective varieties intersecting hypersurfaces in subgeneral position. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in [Formula: see text] with finite total curvature in the case where the Gauss map may be algebraic degenerate. Our results generalize and improve the previous results in the field.

Nonorientable minimal surfaces with catenoidal ends

Starting from the pioneering work by Meeks, complete nonorientable minimal surfaces with finite total curvature have been studied by many researchers. However, it seems that there are no known examples all of whose ends are embedded except for Kusner’s flat-ended N-noids. In this paper, we show the existence of a 1-parameter family of complete $${\mathbf{Z}}_N$$ -invariant conformal minimal immersions from finitely punctured real projective planes into $${\mathbf{R}}^3$$ , each of which has $$N+1$$ catenoidal ends, for any odd integer $$N\ge 3$$ . This family gives a deformation from an $$(N+1)$$ -noid with N catenoidal ends and a planar end to Kusner’s flat-ended N-noid. We also give a nonexistence result for such surfaces for any even integer $$N\ge 2$$ .

Modified defect relations of the Gauss map of a complete minimal surface on annular ends

Modified defect relations of the Gauss map of a complete minimal surface on annular ends

Correction to: Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

Correction to: Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

Interpolation by Complete Minimal Surfaces Whose Gauss Map Misses Two Points

Let $M$ be an open Riemann surface and let $\Lambda\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $\Lambda$ and whose generalized Gauss map $M\to\mathbb{CP}^{n-1}$, $n\ge 3$, avoids $n$ hyperplanes of $\mathbb{CP}^{n-1}$ located in general position. In case $n=3$, we obtain complete nonflat conformal minimal immersions whose Gauss map $M\to\mathbb{S}^2$ omits two (antipodal) values of the sphere. This result is deduced as a consequence of an interpolation theorem for conformal minimal immersions $M\to\mathbb{R}^n$ into the Euclidean space $\mathbb{R}^n$, $n\ge 3$, with $n-2$ prescribed components.

New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ

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 $\mathbb{R}^n$.

Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

We prove that, given a compact Riemann surface $$\Sigma $$ and disjoint finite sets $$\varnothing \ne E\subset \Sigma $$ and $$\Lambda \subset \Sigma $$ , every map $$\Lambda \rightarrow \mathbb {R}^3$$ extends to a complete conformal minimal immersion $$\Sigma \setminus E\rightarrow \mathbb {R}^3$$ with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in $$\mathbb {R}^3$$ with finite total curvature. To this respect we provide, for each integer $$r\ge 1$$ , a set $$A\subset \mathbb {R}^3$$ consisting of $$12r+3$$ points in an affine plane such that if A is contained in a complete nonflat orientable immersed minimal surface $$X:M\rightarrow \mathbb {R}^3$$ , then the absolute value of the total curvature of X is greater than $$4\pi r$$ . In order to prove this result we obtain an upper bound for the number of intersections of a complete immersed minimal surface of finite total curvature in $$\mathbb {R}^3$$ with a straight line not contained in it, in terms of the total curvature and the Euler characteristic of the surface.

A Note on a Unicity Theorem for the Gauss Maps of Complete Minimal Surfaces in Euclidean Four-space

AbstractThe classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

We investigate complete minimal submanifolds $$f: M^3\rightarrow \mathbb {H}^n$$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in Dajczer et al. (Math Z 287: 481–491, 2017 and Comment Math Helv, to appear, 2017), respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of $$\mathbb {H}^n$$ .