Related papers: Regular Polygon Surfaces

The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph on two vertices. A graph $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. We prove that every polyhedral graph (i.e. 3-connected planar graph)…

A Platonic surface is a Riemann surface that underlies a regular map and so we can consider its vertices, edge-centres and face-centres. A symmetry (anticonformal involution) of the surface will fix a number of simple closed curves which we…

A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in the 3-sphere to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this…

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for…

We say that a $2$-dimensional CW complex is a multibranched surface if we remove all points whose open neighborhoods are homeomorphic to the $2$-dimensional Euclidean space, then we obtain a $1$-dimensional complex which is homeomorphic to…

We prove that a closed embedded minimal surface in the round three-sphere which satisfies the symmetries of a Lawson surface and has the same genus is congruent to the Lawson surface.

If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…

A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$,…

Let $\Sigma$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $\Sigma$ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the…

Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-\alpha$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then,…

Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic…

A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees…

Let $M$ be a simply connected homogeneous three-manifold with isometry group of dimension $4$, and let $\Sigma$ be any compact surface of genus zero immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic…

We show that a realization of a closed connected PL-manifold of dimension n-1 in n-dimensional Euclidean space (n>2) is the boundary of a convex polyhedron (finite or infinite) if and only if the interior of each (n-3)-face has a point,…

We show that for every $\epsilon>0$, there exists a compact lamination by $\epsilon$-holomorphic surfaces in the complex projective plane, minimal, and that carries hyperbolic holonomy. We call $\epsilon$-holomorphic a real 2-dimensional…

We revisit the geometric foundations of mesh representation through the lens of Plane-based Geometric Algebra (PGA), questioning its efficiency and expressiveness for discrete geometry. We find how $k$-simplices (vertices, edges, faces,…

Given a smooth curve $\gamma$ in some $m$-dimensional surface $M$ in $\mathbb{R}^{m+1}$, we study existence and uniqueness of a flat surface $H$ having the same field of normal vectors as $M$ along $\gamma$, which we call a flat…

We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the…

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…

We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces…