Related papers: Face-simple minimal quadrangulations of surfaces
In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any…
We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other…
A few pages in Siegel describe how, starting with a fundamental polygon for a compact Riemann surface, one can construct a symplectic basis of its homology. This note retells that construction, specializing to the case where the surface is…
In the paper, we construct, for $\lambda>0$, complete embedded and non-convex $\lambda$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $\lambda$-hypersurfaces share a common conclusion on the planar…
Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than 180 degrees. In this paper we prove that the opposite statement is also true, namely that planar…
Twenty years ago, N. Kapouleas introduced a singular perturbation construction known as "doubling", which produces sequences of high-genus minimal surfaces converging to a given minimal surface with multiplicity two. Doubling constructions…
This paper is concerned with the surface embedding of matching extendable graphs. There are two directions extending the theory of perfect matchings, that is, matching extendability and factor-criticality. In solving a problem posed by…
A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from…
We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are…
In this paper, we prove that every confomal minimal immersion of an open Riemann surface into $\mathbb{R}^n$ for $n\ge 5$ can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open…
In this paper, we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by the period. These surfaces are constructed by adding a handle to the toroidal half-plane layers defined by H.…
Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and…
We present two versions of a method for generating all triangulations of any punctured surface in each of these two families: (1) triangulations with inner vertices of degree at least 4 and boundary vertices of degree at least 3 and (2)…
We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a…
In this paper, we show that any biharmonic simple rotational surface in the four-dimensional Euclidean space is minimal. The proof is based on reducing the biharmonic equation to a system of ordinary differential equations for the profile…
The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied,…
We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how…
An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…
Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.
Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic…