Interpolation by Maximal Surfaces

On the duality between rotational minimal surfaces and maximal surfaces

An affine translation surface is a graph of a function introduced by Liu and Yu in 2013. The article considers the spacelike affine translation surfaces in the Minkowski space with density establishing the Lagrange’s equation type for -maximal surface, classifying -maximal spacelike affine translation surfaces. The result obtains two parameters and . From that, the Calabi – Bernstein theorem in this space is not true because two function and are defined on

A maximal surface in a 3-dimensional Lorentzian manifold is a space-like surface with zero mean curvature. One of the most relevant results in the context of global geometry of maximal surfaces is the well-known Calabi–Bernstein theorem, which states that the only entire maximal graphs in the 3-dimensional Lorentz–Minkowski space, \({\mathbb{R}}_{1}^{3}\), are the space-like planes. This result can also be formulated in a parametric version, stating that the only complete maximal surfaces in \({\mathbb{R}}_{1}^{3}\) are the space-like planes. In this chapter, we review about the Calabi–Bernstein theorem, summarizing some of the different extensions and generalizations of it made by several authors in recent years, and describing also some recent results obtained by the authors for maximal surfaces immersed in Lorentzian product spaces.

We give criteria for cuspidal butterflies and cuspidal ( ) singularities in terms of the Weierstrass data for maximal surfaces, and also show non-existence of cuspidal ( ) singularities on maximal surfaces. Moreover, we show duality between these singularities considering the conjugate maximal surfaces each other.

Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces

As in the case of minimal surfaces in the Euclidean 3-space, the reflection principle for maximal surfaces in the Lorentz-Minkowski 3-space asserts that if a maximal surface has a spacelike line segment L, the surface is invariant under the $$180^\circ$$ -rotation with respect to L. However, such a reflection property does not hold for lightlike line segments on the boundaries of maximal surfaces in general. In this paper, we show some kind of reflection principle for lightlike line segments on the boundaries of maximal surfaces when lightlike line segments are connecting shrinking singularities. As an application, we construct various examples of periodic maximal surfaces with lightlike lines from tessellations of $$\mathbb {R}^2$$ .

We investigate semi-discrete maximal surfaces with singularities in Minkowski 3-space. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space admit a Weierstrass-type representation and they generally have singularities. In this paper, we first describe semi-discrete isothermic maximal surfaces in Minkowski 3-space and give a Weierstrass-type representation for them determined from integrable system principles. Furthermore, we show that semi-discrete isothermic maximal surfaces admit associated one-parameter families of deformations whose mean curvature remains identically 0. Finally we give a criterion that naturally describes the unified scheme of the “singular set” for these semi-discrete maximal surfaces, including the associated family.

Maximal surfaces in Lorentzian Heisenberg space

A compact Klein surface X=D/Γ , where D denotes the hyperbolic plane and Γ is a surface NEC group, is said to be q-trigonal if it admits an automorphism ϕ of order 3 such that the quotient X/<ϕ > has algebraic genus q. In this paper we obtain for each q the automorphism groups of q-trigonal planar Klein surfaces, that is surfaces of topological genus 0 with k≥ 3 boundary components. We also study the surfaces in this family, which have an automorphism group of maximal order (maximal surfaces). It will be done from an algebraic and geometrical point of view.

Prescribing singularities of maximal surfaces via a singular Björling representation formula

abstract: We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\Sp(4,\R)$-symmetric space. We describe a homeomorphism between the "Hitchin component" of wild $\Sp(4,\R)$-Higgs bundles over $\CP^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\h^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\R^4$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\h^{2,2}$ associated to $\Sp(4,\R)$-Hitchin representations along rays of holomorphic quartic differentials.

We show that any element of the universal Teichmuller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdSn+1, any subset E of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In AdS3, if E is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.

Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, we discuss singularities and several global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, our examples are embedded outside a compact set. Moreover, we deform such examples to CMC-1 faces (mean curvature one surfaces with admissible singularities in de Sitter 3-space) and obtain “cousins of those maximal surfaces.

The purpose of this paper is to briefly outline the uniqueness of the helicoid and Enneper's surface among maximal surfaces in the Lorentz‐Minkowski space R13.The purpose of this paper is to briefly outline the uniqueness of the helicoid and Enneper's surface among maximal surfaces in the Lorentz‐Minkowski space R13.