Related papers: Complete Flat Cone Metrics on Punctured Surfaces

Let G be a n-dimensional Lie group (n>2) with a bi-invariant Riemannian metric. We prove that if a surface of constant Gaussian curvature in G can be expressed as the product of two curves, then it must be flat. In particular, we can…

We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length…

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In…

It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite…

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a…

In this paper we describe all possible reduced complete intersection sets of points on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of…

We introduce the notion of pseudo-cones of metric spaces as a generalization of both of the tangent cones and the asymptotic cones. We prove that the Assouad dimension of a metric space is bounded from below by that of any pseudo-cone of…

We get sharp degree bound for generic smoothness and connectedness of the space of conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on Hypersurfaces.

We prove that there exists a metric of positive curvature in a three-sphere which admits a given torus knot as a closed geodesic.We also sketch a construction of a metric in a four sphere, very likely of positive curvature, which admits a…

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

This paper introduces path triangulation of points in a bounded, simply connected surface region, replacing ordinary triangles in a Delaunay triangulation with path triangles from homotopy theory. A {\bf path triangle} has a border that is…

We prove that the deformation space of geodesic triangulations of a flat torus is homotopy equivalent to a torus. This solves an open problem proposed by Connelly et al. in 1983, in the case of flat tori. A key tool of the proof is a…

Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a…

We define singular points of the first kind and singular points of the second kind as singular points of mappings between surfaces. Typical examples of these singular points are fold singular points and cusp singular points, respectively.…

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial…

We provide a classification of positive integral friezes on marked bordered surfaces in the style of Conway and Coxeter. More precisely, we prove that positive integral friezes are in one-to-one correspondence with ideal triangulations…

The moduli space of flat SU(2) connections on a punctured surface, having prescribed holonomy around the punctures, is a compact smooth manifold if the prescription is generic. This paper gives a direct, elementary proof that the trace of…

We propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of regularity $H^s$, $s\in(3/2,2)$. We highlight its close relationship to the (generalized) tangent-point energies and employ it to show that this…

A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of…