Related papers: What is an Almost Normal Surface
Let $\Omega$ be a planar Jordan domain. We consider double-dome-like surfaces $\Sigma$ defined by graphs of functions of $dist( \cdot ,\partial \Omega)$ over $\Omega$. The goal is to find the right conditions on the geometry of the base…
Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there…
Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of…
In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space $E^3$. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal…
Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) with automorphism groups of the same order, and in many cases…
This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can…
We develop an equivariant min-max theory as proposed by Pitts-Rubinstein in 1988 and then show that it can produce many of the known minimal surfaces in $\mathbb{S}^3$ up to genus and symmetry group. We also produce several new infinite…
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…
A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds contains robustly an open set. Introduced by Bonatti and D{\'i}az in the 90s, blenders turned out to have many powerful…
The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R^3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron…
The origin of quasiconformal mappings, like that of conformal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality,…
The notion of an unexpected curve in the plane was introduced in 2018, and was quickly generalized in several directions in a flurry of mathematical activity by many authors. In this expository paper we first describe some of the main…
Quasi-homogeneous surfaces, or Gizatullin surfaces, are normal affine surfaces such that there exists an open orbit of the automorphism group with a finite complement. If the action of the automorphism group is transitive, the surface is…
As a generalization of slant Riemannian maps (Sahin), semi-slant Riemannian maps (Park), almost h-slant submersions (Park 2012), and almost h-semi-slant submersions (Park 2011), we introduce the notion of almost h-semi-slant Riemannian maps…
This paper introduces and investigates a new class of almost normal spaces, referred to as almost SC*-normal spaces, which are defined using SC*-open sets. Building on the work of A. Chandrakala and K. Bala Deepa Arasi, we explore several…
We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with non-empty boundary which has a quadrilateral in each tetrahedron. While this condition…
This paper is devoted to topological phenomena in normal metals with rather complicated Fermi surface. The results of the article are based on the deep topological theorems concerning the geometry of non-compact plane sections of level…
The topological index of a surface was previously introduced by the first author as the topological analogue of the index of an unstable minimal surface. Here we show that surfaces of arbitrarily high topological index exist.
In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means of number theory and hyperbolic geometry. First, we show the existence of infinitely many semi-arithmetic Riemann surfaces of various shapes and prove…
A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and…