Related papers: Domes over curves

We prove that, if $\Gamma$ is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of $\Gamma$ of order at least $6$, or the number of vertices of $\Gamma$ is bounded above by an absolute…

Nandakumar asked whether there is a tiling of the plane by pairwise non-congruent triangles of equal area and equal perimeter. Here a weaker result is obtained: there is a tiling of the plane by pairwise non-congruent triangles of equal…

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,…

Let $\Gamma$ be a closed Jordan curve, and $f$ the conformal mapping that sends the unit disc $\mathbb{D}$ onto the interior domain of $\Gamma$. If $\log f'$ belongs to the Dirichlet space $\mathcal{D}$, we call $\Gamma$ a Weil-Petersson…

If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex…

We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…

Jets frames, that is a generalisation of ordinary frames on a manifold, are described in a language similar to that of gauge theory. This is achieved by constructing the Cartan geometry of a manifold with respect to the diffeomorphism…

Illumination complexes are examples of 'flat polyhedral complexes' which arise if several copies of a convex polyhedron (convex body) Q are glued together along some of their common faces (closed convex subsets of their boundaries). A…

We introduce the regularized integrals for decorated graphs on elliptic curves, which produces an almost holomorphic function on upper half plane. Then we give the graph version of holomorphic anomaly equation to study the anti-holomorphic…

We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can…

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain…

We provide new results and new proofs of results about the torsion of curves in $\mathbb{R}^3$. Let $\gamma$ be a smooth curve in $\mathbb{R}^3$ that is the graph over a simple closed curve in $\mathbb{R}^2$ with positive curvature. We give…

We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a…

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…

This paper details an algorithm for unfolding a class of convex polyhedra, where each polyhedron in the class consists of a convex cap over a rectangular base, with several restrictions: the cap's faces are quadrilaterals, with vertices…

We consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly…

Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we…

A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each…

Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene…

A widely believed conjecture predicts that curves of bounded geometric genus lying on a variety of general type form a bounded family. One may even ask whether the canonical degree of a curve $C$ in a variety of general type is bounded from…