We propose a discrete surface theory in that unites the most prevalent versions of discrete special parametrizations. This theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. The theo…
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48 results for “surface theory”
New surface observables yield 2-knot invariants in nonabelian theories.
problem Developing new invariants for nonabelian theories.
method Introducing surface observables in BF theory and Yang-Mills theory.
result Surface observables induce new 2-knot invariants and electric fluxes.
Explains the history and challenges of minimal surfaces.
problem Understanding the regularity of minimal surfaces.
method Historical overview and technical analysis.
result Outlines the evolution and current state of minimal surfaces.
Random surfaces with long systoles created from graph theory ideas.
problem Finding surfaces with long systoles.
method Two constructions inspired by graph theory.
result Proved a new lower bound on systole length.
A Discrete Surface Theorymath.DG
In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily discretization or approximation of smooth surfaces. The Gauss curvature and the mean curvature of discrete surfaces are defined which satisfy properties corresponding…
Two algorithms use normal surfaces to detect unknots and prove knots.
problem Detecting and proving the unknot and knottedness of links.
method Normal surface theory algorithms and split-link algorithm.
result Figure-eight knot is proven to be knotted.
Unified approach to totally ramified values in various surface theories.
problem Totally ramified values in value distribution theory, normal family theory, and Gauss maps of surfaces.
method Bloch--Ros principle applied to various surface theories.
result Unified approach to phenomena concerning totally ramified values.
Modernizes classical theory linking isothermic surfaces to Bonnet pairs.
problem Classical theory of isothermic surfaces and Bonnet pairs.
method Identifies derivatives of Bonnet pairs with retraction form of isothermic surfaces.
result Modern account and identification of retraction form.
Standardizes surfaces in 3D handlebodies using Morse theory.
problem Classifying and identifying incompressible surfaces in 3D handlebodies.
method Applied Morse theory to provide a standard form and a condition for incompressibility.
result Developed a practical algorithm to determine incompressibility of surfaces in standard form.
Ozawa solution describes surface deformation from Davey-Stewartson II equation.
problem Surface deformation ruled by the Ozawa solution of Davey-Stewartson II equation.
method Soliton deformation of surfaces ruled by the Ozawa solution.
result Explicit singularity of deformed surface at blow-up moment of Ozawa solution.
A derivation of first variation formulas from the strain-displacement relations in thin shell theorymath.DG
Paper derives formulas for surface variations in shell theory.
problem Deriving first variation formulas for surfaces in thin shell theory.
method Using strain-displacement relations from thin shell theory.
result Provides formulas for linear Weingarten surfaces as stationary points.
Geometrically interprets virtual knotoids in thickened surfaces.
problem No specific problem stated; focuses on geometric interpretation.
method Geometric interpretation of virtual knotoids as arcs in thickened surfaces.
result Shows virtual knotoid theory as a generalization of classical knotoid theory.
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauff…
Quaternionic reformulation simplifies surface curvature theory.
problem Prescribed extrinsic curvature of surfaces.
method Quaternionic reformulation of Labourie's theory.
result Simpler proofs and higher-dimensional generalization.
The paper explores stable surfaces in Einstein-Maxwell theory, proving mass bounds and nonexistence results.
problem Exploring stable surfaces in static Einstein-Maxwell space-time.
method Using mean-stable surfaces theory to prove properties of lapse functions and mass bounds.
result Proves ADM mass is bounded by Hawking quasi-local mass.
Scattering theory for harmonic one-forms on Riemann surfaces.
problem Understanding harmonic one-forms on Riemann surfaces.
method Constructing scattering theory through boundary value problems and integral operators.
result Explicit expression for the scattering matrix and proof of unitarity.
This paper contains some results about Teichmüller spaces of non-orientable surfaces (Klein surfaces). We prove several theorems giving isomorphisms between deformation spaces of Klein surfaces. These results show the similarity between the deformation theory of Klein surfaces, and the theory of Riemann surfaces. We al…
Scattering theory on Riemann surfaces II: The scattering matrix and generalized period mappingsmath.DG
Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.
problem Scattering theory for harmonic one-forms on Riemann surfaces.
method Construction of scattering theory from boundary value problems involving systems of curves and jump problems. Explicit expression for scattering matrix using Schiffer operators.
result Unitary scattering matrix and general association of polarizing Lagrangian spaces.
This paper introduces Alexandrov's theory of singular surfaces and their curvature.
problem Understanding the geometry of singular surfaces with intrinsic metrics and curvature.
method Develops the theory of Alexandrov surfaces, focusing on their convergence and stability properties.
result Classifies compact Alexandrov surfaces using the conformal viewpoint introduced by Reshetnyak.
Extends Heegaard Floer theory to surfaces of dimension one.
problem Developing a theory for surfaces of dimension one.
method 2-representation theory of gl(1|1)^+ and tensor product construction.
result Extension of Heegaard Floer theory to dimension one.
We show that the theory of isothermic surfaces in $\E^3$ -- one of the oldest branches of differential geometry -- can be reformulated within the modern theory of completely integrable (soliton) systems. This enables one to study the geometry of isothermic surfaces in $\E^3$ by means of powerful spectral methods availa…
Extends Kummer's theory to singular surfaces for line congruences.
problem Applying Kummer's theory to singular surfaces for line congruences.
method Analyzing the equation of principal surfaces and developable surfaces for normal congruences.
result The multiplicative factor for the principal surfaces is associated with the singular set of .
Weierstrass-type representations have been used extensively in surface theory to create surfaces with special curvature properties. In this paper we give a unified description of these representations in terms of classical transformation theory of -surfaces.
Nudos y Superficiesmath.HO
Introduces knot theory via surface perspectives.
problem Understanding knots through surface geometry.
method Explains isotopies, Reidemeister moves, and Seifert surfaces.
result Introduces a group structure on knots.
In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of embedded triply periodic minimal surfaces, we seek examples of triply periodic polyhedral surfaces that have an identifia…
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
In this paper, based on the theory of surfaces in the four-dimensional Euclidean space which generalizes the theory of surfaces in three-dimensional Euclidean space, beside other results, we will give a characterization of points on particular kind of these surfaces.
Bound critical points for minimal Radó functions.
problem Counting interior critical points for minimal Radó functions.
method Bounding critical points in terms of boundary data and domain Euler characteristic.
result Bound the number of interior critical points.
Theory for capillary surfaces in 3-manifolds with smooth boundary.
problem Existence and multiplicity of capillary surfaces with given mean curvature and contact angle.
method Min-max theory applied to capillary surfaces in 3-manifolds.
result Existence of nontrivial, smooth, almost properly embedded surfaces with constant mean curvature and contact angle.
Tollefson described a variant of normal surface theory for 3-manifolds, called Q-theory, where only the quadrilateral coordinates are used. Suppose is a triangulated, compact, irreducible, boundary-irreducible 3-manifold. In Q-theory, if contains an essential surface, then the projective solution space has an e…
Paper generalizes Bloch-Ros principle to various surface classes.
problem Understanding the relationship between normal family theory, value distribution theory, and surface theory.
method Formulation and generalization of Bloch-Ros principle to different surface classes.
result Effective criterion for determining Gaussian curvature estimates for various surface classes.
The paper studies a new class of affine maximal surfaces with singularities.
problem Understanding the properties of affine maximal surfaces with singularities.
method Defining a new subclass of affine maximal surfaces and applying Euclidean minimal surface theory.
result Affine maxfaces satisfy an Osserman-type inequality and do not contain non-trivial improper affine fronts.
Study of holomorphic curves and surfaces using singularity theory.
problem Understanding the geometry of holomorphic curves and complex surfaces.
method Application of singularity theory to holomorphic curves and surfaces.
result Definition of geometric invariants for curves and surfaces.
The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.
The theory of complete surfaces of (nonzero) constant mean curvature in $\RR^3$ has progressed markedly in the last decade. This paper surveys a number of these developments in the setting of Alexandrov embedded surfaces; the focus is on gluing constructions and moduli space theory, and the analytic techniques on which…
A method to describe Riemann surfaces using graph profiles is proposed.
problem Describing Riemann surfaces in a graph-theoretic framework.
method Graph profiles to represent Riemann surfaces, establishing necessary and sufficient conditions for their existence.
result A criterion for the existence of Riemann surfaces with a given signature.
Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering the surf…
Introduces Möbius structures and hyperbolic ends for -surfaces in hyperbolic space.
problem Understanding -surfaces in hyperbolic space.
method Studies Möbius structures and hyperbolic ends.
result Applications to -surfaces in hyperbolic space.
The curve graph's model theory reveals its central role in surface study.
problem Why is the curve graph central in surface and mapping class group studies?
method Developed a bridge between model theory, topology, and group theory; bi-interpreted curve graph with mapping class group.
result Proved the curve graph's first-order theory is -stable and has quantifier elimination.
We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known …
Estimates the rational homological dimension of Riemann surfaces with boundary and marked points.
problem Estimating the homological dimensions of Riemann surfaces with boundary and marked points.
method Developed an estimate for the rational homological dimension of Riemann surfaces with possible boundary and marked points.
result Provided an estimate for the rational homological dimension of Riemann surfaces with boundary and marked points.
By carrying out a rational transformation on the base curve of the Seiberg-Witten curve for supersymmetric pure -gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure -ga…
Small sets of systoles fill hyperbolic surfaces of large genus.
problem Constructing hyperbolic surfaces with minimal systole sets.
method Theory of Coxeter groups combined with number theory.
result Cardinality of systole sets is in o(g/ ln g) for large genus.
In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Rieman…
This paper introduces a homology theory for links in I-bundles over an orientable surface. The theory is unique in that the elements of the chain groups are surfaces instead of diagrams. It is then shown this theory yields the same results as the homology theory constructed by Asaeda, Przytycki and Sikora.
Constructs a family to handle unstable fibers on complex surfaces.
problem Handling unstable fibers on complex surfaces.
method Uses Teichmüller theory to construct a degenerating family over the moduli space.
result Any fibered complex surface with unstable fibers can be pulled back from the constructed family.
We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul an…
New model calculates Wilson surfaces in higher gauge theory.
problem Calculating Wilson surfaces in higher gauge theory.
method Derived geometric framework, topological coadjoint orbit model, functional integral framework.
result Strong evidence that model underlies Wilson surfaces partition function.