Related papers: Minimal surfaces from circle patterns: Geometry fr…
A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known…
How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analyzing models of discrete…
In this paper we will construct a Weierstrass type representation for minimal surfaces in 4-dimensional Lorentzian Damek-Ricci spaces and we give some examples of such surfaces.
Using the fact that any minimal strongly regular surface carries locally canonical principal parameters, we obtain a canonical representation of these surfaces, which makes more precise the Weierstrass representation in canonical principal…
Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an…
We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry…
We construct harmonic diffeomorphisms from the complex plane $C$ onto any Hadamard surface $M$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $M\times R$ over…
Weierstrass-type representations have been used extensively in surface theory to create surfaces with special curvature properties. In this paper we give a unified description of these representations in terms of classical transformation…
We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two…
We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or…
We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a…
Harmonic mappings have long intrigued researchers due to their intrinsic connection with minimal surfaces. In this paper, we investigate shearing of two distinct classes of univalent conformal mappings which are convex in horizontal…
It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes…
Hexagonal circle patterns with constant intersection angles are introduced and studied. It is shown that they are described by discrete integrable systems of Toda type. Conformally symmetric patterns are classified. Circle pattern analogs…
We present a definition of discrete channel surfaces in Lie sphere geometry, which reflects several properties for smooth channel surfaces. Various sets of data, defined at vertices, on edges or on faces, are associated with a discrete…
It is pointed out that despite of the non-linearity of the underlying equations, there do exist rather general methods that allow to generate new minimal surfaces from known ones.
Glickenstein \cite{Glickenstein} and Glickenstein-Thomas \cite{GT} introduced the discrete conformal structures on surfaces in an axiomatic approach and studied its classification. In this paper, we give a full classification of the…
In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…
A discretisation of differential geometry using the Whitney forms of algebraic topology is consistently extended via the introduction of a pairing on the space of chains. This pairing of chains enables us to give a definition of the…
We construct embedded closed minimal surfaces in the round three-sphere, resembling two parallel copies of the Clifford torus, joined by m^2 small catenoidal bridges symmetrically arranged along a square lattice of points on the torus.