A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Characterizes polygonal surfaces in pseudo-hyperbolic spaces.
This paper studies deformations of hyperbolic surfaces with special structures.
The paper proves rigidity of bordered polyhedral surfaces using variational principles.
In this paper, we study Alexandrov-embedded r-noids with genus 1 and horizontal ends. Such minimal surfaces are of two types and we build several examples of the first one. We prove that if a polygon bounds an immersed polygonal disk, it is the flux polygon of an r-noid with genus 1 of the first type. We also study the…
The abstract proves spherical surface decompositions with conical singularities.
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, t…
A nice trick for studying the billiard flow in a rational polygon is to unfold the polygon along the trajectories. This gives rise to a translation or half-translation surface tiled by the original polygon, or equivalently an Abelian or quadratic differential. Veech surfaces are a special class of translation surfaces …
We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.
Researchers calculate complexity of billiard paths in regular polygons.
New proof confirms surfaces can be divided into polygons.
A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. W…
Survey on discrete curvature concepts for polygons and polyhedral surfaces.
Algorithm constructs algebraic curves from translation surfaces.
New proof for weak mixing in polygonal billiards.
Given a closed polygon P having n edges, embedded in R^d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface in R^d having P as its geometric boundary. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert suface constru…
Lecture notes introduce Abelian differentials and their flat surfaces, focusing on families and Teichmüller dynamics.
Study strip deformations of hyperbolic polygons with decorated vertices.
We introduce a new method for estimating the growth of various quantities arising in dynamical systems. We apply our method to polygonal billiards on surfaces of constant curvature. For instance, we obtain power bounds of degree two plus epsilon in length for the number of billiard orbits between almost all pairs of po…
Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine inv…
The paper studies right-angled links on higher genus surfaces.
Gordon and Wilton recently proved that the double D of a free group F amalgamated along a cyclic subgroup C of F contains a surface group if a generator w of C satisfies a certain 3-manifold theoretic condition, called virtually geometricity. Wilton and the author defined the polygonality of w which also guarantees the…
We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.
Algorithm finds Dirichlet domains for hyperbolic surfaces.
We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to de…
We construct harmonic diffeomorphisms from the complex plane onto any Hadamard surface whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in over domains of bounded by ideal geodesic polygons and show the existence of a se…
New findings on strong convexity in triangulations of convex polygons.
Study calculates slope gaps on polygon surfaces, finding non-unimodal distributions.
We prove the existence of complete minimal surfaces of genus g>1 which minimize the total curvature for their genus. Our method is first to identify this (Weierstrass high dimensional period) problem with the problem of finding a particular type of polygonal arc in the complex domain: the arc alternates between horizon…
We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant on the space of those isometric deformations which, for conv…
This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface i…
Dancing polygons and rolling balls linked via a special geometric distribution.
Simplified proof for dimension reduction of polygonal curves.
Estimates KVol on surfaces with geometric constraints.
We give a short proof of the contractibility of the space of geodesic triangulations with fixed combinatorial type of a convex polygon in the Euclidean plane. Moreover, for any , we show that there exists a space of geodesic triangulations of a polygon with a triangulation, whose -th homotopy group is not trivi…
Uniqueness of circle packings on certain translation surfaces is proven.
A surface with boundary is randomly generated by gluing polygons along some of their sides. We show that its genus and number of boundary components asymptotically follow a bivariate normal distribution.
We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the exis…
Exact diameter found for some Riemann surfaces.
A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection …
Deroin and Tholozan's representations are mapped to complex projective space via action-angle coordinates.
We borrow a classical construction from the study of rational billiards in dynamical systems known as the "unfolding construction" and show that it can be used to study the automorphism group of a Platonic surface. More precisely, the monodromy group, or deck group in this case, associated to the cover of a regular pol…
The paper calculates period matrices for specific algebraic curves.
Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of . The first examples appearing in this context are vertical geodesic planes and Scherk…
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
Study intersections of curves on translation surfaces, focusing on regular polygons and their Teichmüller disks.
We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.
We study closed geodesics on hyperbolic surfaces, and give bounds for their angles of intersection and self-intersection, and for the sides of the polygons that they form, depending only on the lengths of the geodesics