Minimal topology on surface homeomorphisms proven.
problem Proving the compact-open topology is minimal for surface homeomorphisms.
method Combining Hausdorff group topology properties and automatic continuity results.
result Compact-open topology is unique Hausdorff separable group topology on surface homeomorphisms.
Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.
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.
Topology classifies bipolar surfaces; they are not embedded.
problem Classifying bipolar surfaces in the 5-sphere.
method Topological classification and area bounds calculation.
result Bipolar surfaces are not embedded.
Minimal surfaces match symmetries and topology exactly.
problem Matching symmetries and topology in minimal surfaces.
method Proved congruence of surfaces with specific symmetries and topology.
result Lawson surfaces are uniquely determined by their symmetries and topology.
In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal surfaces.
Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.
problem Topology of collapsed spaces with lower Ricci bounds
method Prove that collapsed spaces are topological surfaces
result Collapsed spaces are topological surfaces
Researchers found multiple surfaces with same topological and symmetry properties.
problem Determining unique free boundary minimal surfaces from topology and symmetry.
method Provided pairs of non-isometric surfaces with same genus, boundary components, and symmetry group.
result Infinitely many pairs of surfaces with same topology and symmetry group exist.
Optimizes material distribution on surfaces using topological derivatives.
problem Optimal distribution of two materials on smooth submanifolds in Rd. method Topological derivative approach for shape optimization constrained by PDEs.
result Numerical solution of topology optimization problem on surfaces.
New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
String topology coproduct and Turaev cobracket computed for surfaces.
problem Computing string topology coproduct and cobracket for surfaces.
method Algorithm to compute coproduct of cyclic words in terms of generators of the fundamental group.
result String cobracket is the negative of Turaev cobracket.
The disk complex of a surface in a 3-manifold is used to define its {\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface H with top…
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
Compactness proven for CMC surfaces with bounded topology and boundary length.
problem Proving compactness of CMC surfaces with specific constraints.
method Graphical Ck compactness proof for surfaces with bounded topology, area, and boundary length. result Space of free boundary CMC surfaces is compact in the Ck graphical sense away from a finite set of points. Study shows certain surface homeomorphisms groups can't be precompact.
problem Understanding precompactness of groups of surface homeomorphisms.
method Analyzing subgroup properties and transitivity of homeomorphisms.
result No subgroup of surface homeomorphisms can be Roelcke precompact.
In this paper we study topological surfaces as gridded surfaces in the 2-dimensional scaffolding of cubic honeycombs in Euclidean and hyperbolic spaces.
Finite-type surfaces have a topological Hopf property.
problem Characterizing surfaces with a topological Hopf property.
method Using topological analogs of the Hopf property.
result Infinite-type surfaces do not have the Hopf property.
We show that except for n=2 if a bridge surface for a knot is an index n topologically minimal surface, then after a perturbation it is still topologically minimal with index at most n+1.
We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.
Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.
problem Distribution of nearly geodesic surfaces in hyperbolic 3-manifolds.
method Invariant measures on the Grassmann bundle G(M) derived from limits of random minimal surfaces.
result Topological limiting measures are totally scarring if M contains a totally geodesic subsurface, while geometrical limiting measures are not.
Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.
problem Minimal topological generating sets of mapping class groups consisting of torsion elements.
method Investigation of minimal topological generating sets for Map(S(n)) consisting entirely of torsion elements, with special attention to involutions. result Minimal topological generating sets for Map(S(n)) consisting of torsion elements are found for various n. Surface groups are uniquely identified by their profinite completions.
problem Identifying surface groups among limit groups.
method Using profinite completions and topology.
result The set of surface words is closed in the profinite topology.
This paper finds minimal sets of generators for mapping class groups of specific surfaces.
problem Finding minimal sets of generators for mapping class groups of infinite-type surfaces.
method Analyzing specific surfaces S(n) to determine minimal sets of generators. result Minimal sets of generators for Map(S(n)) are identified for n≥8 (3 elements), n≥3 (4 elements), and S(1) (2 elements). We present and prove a topological characterization of geodesic laminations on hyperbolic surfaces of finite type.
Overview of infinite surface mapping class groups.
problem Understanding mapping class groups of infinite surfaces.
method Survey of recent research findings.
result Recent developments in mapping class groups of infinite surfaces.
Constructs surfaces with specific topologies and curvatures.
problem Creating surfaces with prescribed genus and ends.
method Using a family of constant mean curvature surfaces constructed in \cite{Kleene}, resolving tangency points over catenoidal necks.
result Complete embedded surfaces with freely prescribed genus and ends.
Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.
problem Understanding topological constraints for stable free boundary CMC surfaces in negatively curved settings.
method Established intrinsic area-length-topology inequalities via a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator.
result Explicit topological restrictions for stable free boundary CMC surfaces, showing low genus and few boundary components.
Strong rigidity proven for non-compact surfaces.
problem Proving rigidity of non-compact surfaces.
method Generalized result showing proper homotopy to homeomorphism.
result All non-compact orientable surfaces, except plane and punctured plane, are topologically rigid.
We give a complete topological classification of minimal surfaces in Euclidian three-space.
A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with a translation structure with a topology, which we call the immersive topology be…
Study on 3D surfaces and tangles formed by Poncelet triangles.
problem Geometric and topological properties of 3D surfaces and tangles.
method Exploration of Poncelet triangles and associated points.
result Properties of 3D surfaces and tangles formed by Poncelet triangles.
This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
Manufacturing infinite sets of knotted and unknotted surfaces in 4-manifolds.
problem Creating infinite sets of knotted and unknotted surfaces in 4-manifolds.
method Recent constructions of inequivalent smooth structures.
result Infinite sets of pairwise smoothly non-isotopic nullhomologous 2-tori and spheres.
Study on surfaces minimizing elastic energy with boundary constraints.
problem Finding stable configurations of surfaces with elastic boundaries and surface energy.
method Investigation of critical surfaces with mean curvature and spontaneous curvature, coupled to boundary elastic energy.
result Characterization and minimization of surface energy for specific topological shapes.
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
Characterizes curves with short representatives on hyperbolic surfaces.
problem Inequalities on lengths of curves on hyperbolic surfaces.
method Characterization of topological types of curves and multicurves with short representatives.
result Characterizes which topological types of curves and multicurves always have a short representative.
Compact Special Weingarten surfaces with planar convex boundaries are disks.
problem Characterizing Special Weingarten surfaces with specific boundary conditions.
method Proved a Ros-Rosenberg theorem in the context of Special Weingarten surfaces.
result Compact Special Weingarten surfaces with planar convex boundaries are topological disks.
Study topological components of surface group representations into SL(2,R) and PSL(2,R).
problem Understanding the structure of representation spaces of surface groups.
method Utilized the signature formula to determine connected components of representation spaces.
result Determined the number of connected components for different types of holonomies.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Study on OI surfaces with unique geometric properties.
problem Characterizing and classifying ortho-integral surfaces.
method Analyzing geodesic arcs and cosh-length properties.
result Infinitely many commensurability classes of OI surfaces arise as topologies vary.
New minimal surfaces in spheres with complex topologies from capillarity.
problem Constructing minimal surfaces in spheres with rich topologies.
method General construction of embedded minimal and constant mean curvature surfaces in Sn using capillary hypersurfaces. result Non-trivial sphere bundles over various base spaces, including Stiefel manifolds and complex quadrics.
We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of Kneser and Haken, and more recent theorems of Rubinstein and Stocking, and is t…
Optimal curves minimize crossings on surfaces.
problem Minimizing crossings on congruence surfaces.
method Examining systoles and their intersections.
result Modular systoles have minimal crossing numbers.
Topological model created for HOMFLY-PT polynomial from link diagrams.
problem Constructing categorifications for HOMFLY-PT polynomial.
method Explicit Lagrangian submanifolds on Heegaard surfaces.
result Invariants derived from link diagrams are given by intersections of submanifolds.
In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in Rn with finite Willm…
Study surfaces in 4-manifolds with cyclic fundamental group.
problem Characterize surfaces with infinite cyclic knot group.
method Algebraic topological criteria and homeomorphism proofs.
result Ambiently isotopic surfaces with the same genus have a homeomorphism.