Examples Of Minimal Surfaces Research Articles

The Second Variation for Null-Torsion Holomorphic Curves in the 6-Sphere

In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces. This class of minimal surfaces is quite rich: By a theorem of Bryant, extended by Rowland, every closed Riemann surface may be conformally embedded in the round 6-sphere as a null-torsion holomorphic curve. In this work, we study the second variation of area for compact null-torsion holomorphic curves \(\Sigma \) of genus g and area \(4\pi d\), focusing on the spectrum of the Jacobi operator. We show that if \(g \le 6\), then the multiplicity of the lowest eigenvalue \(\lambda _1 = -2\) is equal to 4d. Moreover, for any genus, we show that the nullity is at least \(2d + 2 - 2g\). These results are likely to have implications for the deformation theory of asymptotically conical associative 3-folds in \({\mathbb {R}}^7\), as studied by Lotay.

Characterizations of a helicoid and a catenoid

In the present article, we consider a parametric surface generated by the Frenet frame of a curve, and study the minimality condition for the surface. As a result, we give characterizations of a helicoid and a catenoid. Finally we show some examples of minimal surfaces generated by a circle and a helix.

Some examples of algebraic surfaces with canonical map of degree 20

In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.

Examples of Mori dream surfaces of general type with pg = 0

In this paper we study effective, nef and semiample cones of minimal surfaces of general type with pg=0. We provide examples of minimal surfaces of general type with pg=0 and 2≤K2≤9 which are Mori dream spaces. On these examples we also give explicit description of their effective cones with all negative curves. We also present non-minimal surfaces of general type with pg=0 that are not Mori dream surfaces.

Enneper representation of minimal surfaces in the three-dimensional Lorentz–Minkowski space

In this paper, we will give an Enneper-type representation for spacelike and timelike minimal surfaces in the Lorentz–Minkowski space $${\mathbb {L}}^{3}$$ , using the complex and the paracomplex analysis (respectively). Then, we exhibit various examples of minimal surfaces in $${\mathbb {L}}^{3}$$ constructed via the Enneper representation formula that it is equivalent to the Weierstrass representation obtained by Kobayashi (for spacelike immersions) and by Konderak (for the timelike ones).

Invariant surfaces in Sol3 with constant mean curvature and their computer graphics

Abstract In Sol3 space there are three uniparametric groups of isometries. In this work we study constant mean curvature surfaces invariant by one of these groups. We analyze the geometric properties of these surfaces by means of their computer graphics. We construct explicit examples of minimal surfaces and we shall relate them with recent examples of spheres with constant mean curvature.

Sym–Bobenko formula for minimal surfaces in Heisenberg space

We give an immersion formula, the Sym–Bobenko formula, for minimal surfaces in the 3-dimensional Heisenberg space. Such a formula can be used to give a generalized Weierstrass type representation and construct explicit examples of minimal surfaces.

The classification of isotrivially fibred surfaces with p g = q = 2

An isotrivially fibred surface is a smooth projective surface endowed with a morphism onto a curve such that all the smooth fibres are isomorphic to each other. The first goal of this paper is to classify the isotrivially fibred surfaces with p g = q = 2 completing and extending a result of Zucconi. As an important byproduct, we provide new examples of minimal surfaces of general type with p g = q = 2 and K 2 = 4, 5 and a first example with K 2 = 6.

VOLUME FORMS AND MINIMAL SURFACES OF ROTATION IN FINSLER SPACES WITH (α, β)-METRICS

In this paper, the relation between the volume form of (α, β)-metric [Formula: see text] and that of Riemannian metric α is given. Minimal submanifolds in Finsler spaces with (α, β)-metrics are studied. Finally, an ordinary differential equation that characterizes minimal surfaces of revolution and an example of minimal surface of rotation are given.

Standard isotrivial fibrations with [formula omitted], II

A smooth, projective surface S is called a standard isotrivial fibration if there exists a finite group G which acts faithfully on two smooth projective curves C and F so that S is isomorphic to the minimal desingularization of T ≔ ( C × F ) / G . Standard isotrivial fibrations of general type with p g = q = 1 have been classified in [F. Polizzi, Standard isotrivial fibrations with p g = q = 1 , J. Algebra 321 (2009),1600–1631] under the assumption that T has only Rational Double Points as singularities. In the present paper we extend this result, classifying all cases where S is a minimal model. As a by-product, we provide the first examples of minimal surfaces of general type with p g = q = 1 , K S 2 = 5 and Albanese fibration of genus 3. Finally, we show with explicit examples that the case where S is not minimal actually occurs.

Standard isotrivial fibrations with [formula omitted

A smooth, projective surface S of general type is said to be a standard isotrivial fibration if there exists a finite group G acting faithfully on two smooth projective curves C and F so that S is isomorphic to the minimal desingularization of T : = ( C × F ) / G . If T is smooth then S = T is called a quasi-bundle. In this paper we classify the standard isotrivial fibrations with p g = q = 1 which are not quasi-bundles, assuming that all the singularities of T are rational double points. As a by-product, we provide several new examples of minimal surfaces of general type with p g = q = 1 and K S 2 = 4 , 6 .

A representation formula for curves in C3 with preset infinitesimal arc length

We consider an algebraic representation formula for meromorphic curves in C3 with preset infinitesimal arc length, i. e., a differential operator M assigning to triples (f; h; d) of meromorphic functions meromorphic curves ? = (?1; ?2; ?3)T such that d is the infinitesimal arc length of ?, in this way obtaining the complete solution of the differential equation ?021+?022 + ?023=d2 in terms of derivatives of f; h; d only and without integrations. Computer algebra systems are an excellent tool to handle formulas of this type. We give simple Mathematica code and apply it to work out some examples, graphics as well as algebraic expressions of complex curves with special properties. For the case d = 0 of null curves, we give some graphical examples of minimal surfaces constructed in this way, showing deformations and symmetries. We give an expression for the curvature k of ? in terms of the Schwarzian derivative of f and for the case d = 1 a simple differential relation for f and h equivalent to the condition ?=1.

ON LINEAR WEINGARTEN SURFACES

In this paper, we study surfaces in Euclidean 3-space that satisfy a Weingarten condition of linear type as κ1 = mκ2 + n, where m and n are real numbers and κ1 and κ2 denote the principal curvatures at each point of the surface. We investigate the existence of such surfaces parametrized by a uniparametric family of circles. We prove that the only surfaces that exist are surfaces of revolution and the classical examples of minimal surfaces discovered by Riemann. The latter situation only occurs in the case (m, n) = (-1, 0).

A Weierstrass Representation Formula for Minimal Surfaces in ℍ3 and ℍ2 × ℝ

We give a general setting for constructing a Weierstrass representation formula for simply connected minimal surfaces in a Riemannian manifold. Then, we construct examples of minimal surfaces in the three dimensional Heisenberg group and in the product of the hyperbolic plane with the real line.

Minimal Surfaces in the Heisenberg Group

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot-Caratheodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.

Generalized Snell's Law for Weighted Minimal Surface in Heterogeneous Media

The weighted minimal surface problem in heterogeneous media is studied in this paper. The solution to the weighted minimal surface problem is continuous but the derivatives have a jump across the interface where the medium property is discontinuous. The jump condition of the derivatives derived in this paper generalized the Snell's law in geometric optics to weighted minimal surfaces of co-dimension one in any dimensional space. A numerical method based on the gradient flow and the maximum principal preserving immersed interface method is developed to solve this nonlinear elliptic interface problem with jump conditions. Numerical computations are presented to verify both the analysis and the numerical algorithm. 1. Introduction. The minimal surface problem, that is, the problem of finding the surface of the least area among all surfaces having fixed boundary data, has been extensively studied. A recent workshop on minimal surfaces presented the latest re- search on minimal surface applications in chemistry and biology (5). Many phenomena that occur in nature relate to this problem which has been a motivation for devel- oping new mathematical theories and techniques to solve the problem analytically and numerically. Minimal surfaces were shown to be important in various chemical micro-structures and their corresponding phase transitions (5). Computer graphics and image analysis use minimal surfaces frequently for boundary detection, and to construct surfaces that are visually appealing (2), (15). Soap films and other mem- branes passing through a fixed boundary provide mechanical examples of minimal surfaces (14). A related concept is the idea of capillary surfaces, which result from surface tension in liquids. These surfaces are closely related to minimal surfaces (6). For a precise mathematical description of the minimal surface problem we refer, for example, to the classical treatises (7) and (16). The minimal surface problem can be described in two different ways, using the parametric or the non-parametric formulation. In the non-parametric setting the candidate surfaces are graphs of functions, while in the parametric setting the surfaces are treated as boundaries of sets (7). The former is usually seen in more physically- based treatments of the problem, whereas the later provides an excellent framework for the mathematical analysis of minimal surfaces. When the medium is homogeneous, the energy density at each point is constant, and therefore the surface energy is equivalent to the surface area. This is the standard minimal surface problem. In this paper, we consider the weighted minimal surface problem in a heterogeneous medium in which the energy density is piecewise smooth. For example this is the case for capillary interfaces in porous media or composite materials. In particular we derive a jump condition for the weighted minimal surface at the interface between two different media. The jump condition can be regarded as a generalized Snell's law which describes the refraction of minimal surfaces instead of light rays in geometric optics.

Null $2$-type surfaces in $E^3$ are circular cylinders

Abstract In this article we prove that open portions of circular cylinders are theonly surfaces in E z which are constructed from eigenfunctions of Δ witheigenvalue 0 and an eigenvalue λ (=£0). 1. Introduction. Let M be a connected (not necessary compact) surface in a Euclidean 3-space E 3 . Denote by Δ the Laplacian of M associated with the induced metric.Then the position vector x and the mean curvature vector H of M in E 3 satisfy(1.1) Ax = -2H. This formula yields the following well-known result: A surface M in E* isminimal if and only if all coordinate functions of E 3 , restricted to M, areharmonic functions, that is,(1.2) Ax=0. In other words, minimal surfaces are constructed from eigenfunctions of Δ witheigenvalue zero.According to the famous Douglas and Rado's solutions to the Plateau prob-lem there exist ample examples of minimal surfaces in E 3 . The study ofminimal surfaces in E 3 has attracted many mathematicians for many years (cf.[3]).On the other hand, it is easy to see that circular cylinders in