Minimal Surfaces In R3 Research Articles

Calabi-Bernstein type results for critical points of a weighted area functional in [formula omitted] and [formula omitted]

In this paper we prove some Calabi-Bernstein type and non-existence results concerning complete [φ,e→3]-minimal surfaces in R3 whose Gauss maps lie on compacts subsets of open hemispheres of S2. We also give a general non-existence result for complete spacelike [φ,e→3]-maximal surfaces in L3 and, in particular, we obtain a Calabi-Bernstein type result when φ˙ is bounded.

A Weierstrass type representation for translating solitons and singular minimal surfaces

In this paper we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in R3. As application we study when the euclidean Gauss map has a harmonic argument and solve a general Cauchy problem in this class of surfaces.

The embedded Calabi–Yau conjecture for finite genus

Suppose that M is a complete, embedded minimal surface in R3 with an infinite number of ends, finite genus, and compact boundary. We prove that the simple limit ends of M have properly embedded representatives with compact boundary, genus zero, and constrained geometry. We use this result to show that if M has at least two simple limit ends, then M has exactly two simple limit ends. Furthermore, we demonstrate that M is properly embedded in R3 if and only if M has at most two limit ends if and only if M has a countable number of limit ends.

A family of genus one minimal surfaces with two catenoid ends and one Enneper end

In this paper we construct a family of complete immersed minimal surfaces in R3 of genus one with two embedded catenoid-type ends, one Enneper-type end and total Gauss curvature −16π. The proof of the existence of this one-parameter family of minimal surfaces, was obtained using the Weierstrass representation, the theory of elliptic functions and explicitly solving the period problem.

The Morse index of a triply periodic minimal surface

In the previous work, the first author established an algorithm to compute the Morse index and the nullity of an n-periodic minimal surface in Rn. In fact, the Morse index can be translated into the number of negative eigenvalues of a real symmetric matrix and the nullity can be translated into the number of zero-eigenvalue of a Hermitian matrix. The two key matrices consist of periods of the abelian differentials of the second kind on a minimal surface, and the signature of the Hermitian matrix gives a new invariant of a minimal surface. On the other hand, H family, rPD family, tP family, tD family, and tCLP family of triply periodic minimal surfaces in R3 have been studied in physics, chemistry, and crystallography. In this paper, we first determine the two key matrices for the five families explicitly. As its applications, by numerical arguments, we compute the Morse indices, nullities, and signatures for the five families.

Geodesics in minimal surfaces

Abstract—In this paper, we consider connected minimal surfaces in R3 with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper’s surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one 1-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature k g 1 and k g 2 of the coordinates curves satisfy αk g 1 + βk g 2 = 0, α, β ∈ R.

A remark on limits of triply periodic minimal surfaces of genus 3

In this paper, we consider limits of triply periodic minimal surfaces in R3. We prove that some important examples of singly or doubly periodic minimal surfaces can be obtained as limits of triply periodic minimal surfaces. Moreover, we give a mathematical proof of the existence of one-parameter family of triply periodic minimal surfaces which is defined in chemistry.

Approximation theory for nonorientable minimal surfaces and applications

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An existence theorem for non-orientable minimal surfaces in R3, with arbitrary conformal structure, properly projecting into a plane. 3. An existence result for non-orientable minimal surfaces in R3 with arbitrary conformal structure and Gauss map omitting one projective direction.

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation − Δu = u − u 3 in ℝ2 has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem −u″ =u − u 3. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ℝ3.

Nonlinear sigma model, Zakharov-Shabat method, and new exact forms of the minimal surfaces in ℝ3

General formulas for the construction of exact solutions of the equation of the minimal surface in $R^3$, which appears in various physical problems, have been derived by the Zakharov-Shabat "dressing" method. Particular examples are considered.

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.

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

Complete Laguerre minimal surfaces in R3

We study from a global point of view the Laguerre minimal surfaces in R3. In particular, we generalize in a natural way the class of Laguerre minimal surfaces (introducing what we call Generalized Laguerre minimal surfaces) and obtain a conformal representation formula for this new class of surfaces. Besides, we study the completeness of the Laguerre metric and classify the flat Laguerre minimal surfaces. Finally, we solve the Björling problem for Generalized Laguerre minimal surfaces and, as an application, we classify the rotational ones.

Branched Willmore spheres

Abstract In this paper we classify branched Willmore spheres with at most three branch points (including multiplicity), showing that they may be obtained from complete minimal surfaces in ℝ3 with ends of multiplicity at most three. This extends the classification result of Bryant. We then show that this may be applied to the analysis of singularities of the Willmore flow of non-Willmore spheres with Willmore energy 𝒲 ( f ) ≤ 16 π $ \mathcal {W} (f) \le 16\pi $ .

Exotic Minimal Surfaces

We prove a general fusion theorem for complete orientable minimal surfaces in ℝ3 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 surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries.

On the equation of minimal surface in R3: various representations, properties of exact solutions, conservation laws

Various representations of the equation of minimal surface in $$ {\mathbb{R}^3} $$ are considered. Properties of exact solutions are studied, and a procedure of construction the corresponding conservation laws is suggested. Links between the solutions of this equation and those of the elliptic version of the Monge–Ampere equation are found. Bibliography: 19 titles.

An end-to-end construction for singly periodic minimal surfaces

We construct families of properly embedded singly periodic minimal surfaces in R3 with Scherk-type ends and arbitrary finite genus in the quotient. The construction follows by gluing small perturbations of pieces of already known minimal surfaces: Scherk minimal surfaces, Costa-Hoffman-Meeks surfaces and KMR examples.

Laguerre minimal surfaces in ℝ3

Laguerre geometry of surfaces in ℝ3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space ℝ3. We show that any Laguerre minimal surface in ℝ3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in ℝ3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space ℝ03.

Half-space type theorems in warped product spaces with one-dimensional factor

This work states some half-space type theorems in a warped product space of the form I ×ρM, where \({I \subseteq {\bf R}}\) is an open interval and M is either a compact n-manifold, or a complete simply connected surface with constant curvature c ≤ 0. Such theorems generalize the classical half-space theorem for minimal surfaces in R3, obtained by Hoffmann and Meeks (Invent Math 101:373–377, 1990), and recent results for surfaces contained in a slab of R ×ρM, obtained by Dajczer and Alias (Comment Math Helvetici 81:653–663, 2006).

COMPLETE MINIMAL CYLINDERS PROPERLY IMMERSED IN THE UNIT BALL

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.