New method to parametrize infinite Riemann surfaces with bounded triangulations.
problem Parametrizing infinite Riemann surfaces with bounded triangulations.
method Introducing bounded ideal triangulations and proving real-analyticity of the parametrization.
result Real-analytic parametrization of Teichmüller spaces for infinite surfaces with bounded triangulations.
Bounding shears in ideal triangulations on hyperbolic surfaces.
problem Bounding shears in ideal triangulations on hyperbolic surfaces.
method Showing an ideal triangulation with bounded shear parameters on hyperbolic surfaces.
result An upper bound on shear parameters depends logarithmically on the surface's topology.
Sharp upper bound found for stable minimal surfaces.
problem Bounding the diameter of stable minimal surfaces.
method Analyzing three-dimensional Riemannian manifolds with specific curvature conditions.
result Sharp upper bound for the diameter of stable minimal surfaces.
Compactness proven for CMC surfaces with bounded topology and boundary length.
problem Proving compactness of CMC surfaces with specific constraints.
method Graphical C k C^k C k compactness proof for surfaces with bounded topology, area, and boundary length. result Space of free boundary CMC surfaces is compact in the C k C^k C k graphical sense away from a finite set of points. New surfaces defy traditional pants decompositions.
problem Creating surfaces without bounded pants decompositions.
method Constructing quasiconformally homogeneous hyperbolic Riemann surfaces.
result Found surfaces with infinite type and compact boundary components that defy pants decompositions.
Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.
problem Finding area constraints for surfaces with constant mean curvature in hyperbolic 3-manifolds.
method Established an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and genus bounds.
result Area bound implies compactness for such surfaces, with specific proportional bounds for Bryant surfaces.
Sharp bounds found for minimal surface solutions.
problem Finding bounds for minimal surface solutions.
method Analyzing minimal surface equation with specific boundary conditions.
result Sharp bounds established for solutions over certain domains.
The paper classifies group-actions on surfaces of small genus, focusing on bounding and geometrically bounding cases.
problem Classifying group-actions on surfaces of small genus, particularly focusing on bounding and geometrically bounding cases.
method Analyzing large group-actions on surfaces of genus 3, distinguishing between bounding and geometrically bounding cases.
result Identifies which large group-actions on surfaces of genus 3 are bounding or geometrically bounding.
Harmonic maps intersect all minimal surfaces with bounded curvature.
problem Intersection of harmonic maps with minimal surfaces.
method Nonconstant conformal harmonic maps intersecting bounded curvature minimal surfaces.
result Harmonic maps intersect every nonflat properly embedded minimal surface of bounded curvature.
Enneper's wire, the image of the circle of radius R R R under Enneper's surface, bounds exactly three minimal surfaces for R R R between 1 and 3 \sqrt 3 3 , and these three surfaces depend continuously on R R R . The other two surfaces (besides Enneper's surface) are absolute minima of area among disk-type surfaces bounded by En…
This paper proves properties of CAT(κ) surfaces with bounded curvature.
problem Properties of CAT(κ) surfaces with bounded curvature.
method Proof of bounded curvature, approximation by smooth surfaces.
result CAT(κ) surfaces have bounded curvature.
Triangulates surfaces with bounded energy using diffeomorphisms.
problem Triangulating surfaces with bounded Kolasinski--Menger energy.
method Uses bounded distortion diffeomorphisms of subsets of a plane.
result Triangulation with bounded number of triangles.
In this paper we obtain a bound on the number of isometry classes of finite area hyperbolic surfaces which are length isospectral to a given surface depending only on the topological type of the surface and the length of the shortest closed geodesic on the surface. This will follow from a more general bound applying to…
Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
The theorem connects surface mapping groups to fundamental groupoids.
problem Mapping class groups of bounded surfaces.
method Proving isomorphism between mapping class groups and fundamental groupoid automorphisms.
result Mapping class groups of bounded surfaces are isomorphic to fundamental groupoid automorphisms fixing boundary loops.
The paper calculates the index and nullity of Fraser-Sargent surfaces and provides bounds.
problem Computing the index and nullity of Fraser-Sargent surfaces.
method Analytical and numerical methods to compute index and nullity; provides bounds.
result Lower and upper bounds on the index of Fraser-Sargent surfaces inside the ball.
Improves bounds on surface decompositions.
problem Surface decomposition bounds
method Improved theorem of Bers for surfaces with boundary and closed surfaces.
result Slight improvement on best known bound for closed surfaces.
The paper proves bounded curvature for Kähler Ricci shrinker surfaces.
problem Understanding the curvature of Kähler Ricci shrinker surfaces.
method Proved bounded sectional curvature using earlier work.
result Complete classification of all Kähler Ricci shrinker surfaces.
Sharp bounds for charged Hawking mass in electrostatic space-times.
problem Bounding charged Hawking mass in electrostatic space-times.
method Proving sharp lower bounds and upper bounds for the charged Hawking mass.
result Sharp lower bounds for the charged Hawking mass of stable surfaces in electrostatic space-times.
This note shows surfaces in stable 3D data are bounded by area and diameter.
problem Bounding stable surfaces in 3D initial data sets.
method Demonstrating area and diameter bounds for stable, marginally outer trapped surfaces.
result Stable surfaces in 3D data sets are bounded by both area and diameter.
Optimizes bounds for multiple T-singularities on surfaces.
problem Bounding T-singularities on non-rational projective surfaces with many singularities.
method Analyzes combinatorial configurations and classifies them to find optimal bounds.
result Classifies all combinatorial configurations leading to high bounds, proving their non-existence gives optimal bounds.
New examples show no upper bounds on link volumes on incompressible surfaces.
problem Finding upper bounds on volumes of links on incompressible surfaces.
method Examined weakly generalised alternating and fully augmented links on incompressible surfaces.
result Found infinite families of links on incompressible surfaces with no upper bounds on volume.
The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.
problem Characterizing geodesics on hyperbolic surfaces.
method One-parameter Allen-Cahn min-max constructions.
result Every geodesic occurs with multiplicity one and provides uniform sharp lower bounds.
Paper bounds surface diameter and solves Plateau-Douglas problem.
problem Bounding the diameter of compact surfaces and solving the Plateau-Douglas problem.
method Geometric argument based on Topping's diameter bound for closed surfaces.
result Explicit nonexistence criterion for the Plateau-Douglas problem.
The study provides polynomial bounds for essential surfaces in various 3-manifolds.
problem Bounding the number of isotopy classes of embedded essential surfaces in 3-manifolds.
method Restricting to alternating link complements in 3-sphere, then extending results to other classes of cusped 3-manifolds.
result Explicit polynomial bounds on all embedded essential surfaces in 3-manifolds.
Lower bounds on geodesic length with few intersections on hyperbolic surfaces.
problem Finding the minimum length of geodesics with at least 2 intersections.
method Analyzing geodesics on hyperbolic surfaces with at least 2 self-intersections.
result The minimum length of such geodesics is 2 log ( 5 + 2 6 ) 2\log(5+2\sqrt6) 2 log ( 5 + 2 6 ) , and this bound is sharp. New bounds on genus and area for CMC surfaces in 3-manifolds.
problem Bounding genus and area of CMC surfaces in 3-manifolds.
method Local degeneration of minimal surfaces and index-area bounds.
result Genus and area of CMC surfaces are bounded by index and area.
Minimal Delaunay triangulations on hyperbolic surfaces have linear number of vertices.
problem Finding the minimum number of vertices in Delaunay triangulations of hyperbolic surfaces.
method Analyzing the genus g g g of hyperbolic surfaces to derive bounds on the number of vertices. result The number of vertices in minimal Delaunay triangulations of hyperbolic surfaces is linear in the genus g g g . Improved bounds on geodesic lengths in Riemannian surfaces.
problem Finding precise lengths of geodesics in Riemannian surfaces.
method Proved curvature-free linear length bounds on geodesics.
result Length of k e x t t h k^ ext{th} k e x t t h -shortest geodesic is at most 8 k d 8kd 8 k d . Proves energy quantization for surfaces with bounded index.
problem Energy quantization for Willmore surfaces with bounded index.
method Translated the question to the conformal Gauss map's perspective and showed convergence in specific regions.
result Conformal Gauss map converges to a light-like geodesic in De Sitter space in neck or collar regions.
This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact, orientable 3--manifold in terms of the quadrilaterals in its cell decomposition---different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minima…
We prove two explicit bounds for the multiplicities of Steklov eigenvalues σ k σ_k σ k on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index k k k of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given smooth Rie…
Polynomial bound on surfaces in hyperbolic 3-manifolds.
problem Bounding the number of surfaces in hyperbolic 3-manifolds.
method Using polynomial functions of the volume of the manifold and the Euler characteristic.
result An upper bound for the number of compact essential surfaces is a polynomial function of the volume of the manifold.
Sharp bounds found on nonabelian quotients of surface braid groups.
problem Finding the smallest nonabelian quotients of surface braid groups.
method Sharp lower bounds and classification of quotients.
result Quotients of minimum order are either symmetric groups or 2-step nilpotent p-groups.
The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.
problem Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces.
method Analyzing flat surfaces and K3 surfaces, using asymptotics and stability conditions.
result Exact leading term in the asymptotics of the number of semistable Mukai vectors.
Sharp lower bound on fold singularities self-intersections.
problem Finding a lower bound on the number of self-intersections of fold singularities.
method Established a sharp lower bound on the number of self-intersections of the boundary of an immersed surface, then applied this to fold singularities.
result Sharp lower bound on the number of self-intersections of fold singularities.
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
Geometric bounds for low Steklov eigenvalues on hyperbolic surfaces with boundaries.
problem Finding lower bounds for low Steklov eigenvalues of hyperbolic surfaces with geodesic boundaries.
method Analysis of eigenfunction behavior on an adapted thick-thin decomposition for hyperbolic surfaces with geodesic boundaries.
result Sharp geometric lower bounds for low Steklov eigenvalues that depend on the shortest multi-geodesic disconnecting the surfaces.
Study systoles and diameters on hyperbolic surfaces, finding an upper bound for their ratio.
problem Understanding the relationship between systoles and diameters on hyperbolic surfaces.
method Exploring the inequality between systoles and diameters, deducing an upper bound for their ratio.
result The ratio of systoles and diameters has a genus-dependent upper bound.
The paper proves geometric bordisms for specific hyperbolic surfaces.
problem Proving geometric bordisms for Accola-Maclachlan, Kulkarni, and Wiman surfaces.
method Explicit geodesic embeddings and geometric proofs for specific surfaces.
result The surfaces bound geometrically compact hyperbolic 3-manifolds.
Surface bundles' signatures tied to their Euler characteristics.
problem Relating signatures and Euler characteristics of surface bundles.
method Using bounded cohomology and Morita's results.
result The inequality ∣ 3 σ ( E ) ∣ ≤ ∣ χ ( E ) ∣ |3σ(E)| \leq |χ(E)| ∣3 σ ( E ) ∣ ≤ ∣ χ ( E ) ∣ for surface bundles. Study shows infinite-dimensional third bounded cohomology for non-orientable surfaces.
problem Understanding the third bounded cohomology of non-orientable surfaces.
method Analyzing measure-preserving homeomorphisms of non-orientable surfaces.
result Third bounded cohomology is infinite-dimensional.
New bounds on hyperbolic surfaces' properties using linear programming.
problem Finding bounds on various geometric and spectral properties of hyperbolic surfaces.
method Adapted linear programming methods from sphere packings to hyperbolic surfaces.
result Obtained new upper and lower bounds on multiple properties of hyperbolic surfaces.
Improved bounds for eigenfunctions on hyperbolic surfaces found.
problem Establishing improved bounds for eigenfunctions of magnetic Laplacians.
method Using explicit eigenstates called magnetic zonal states.
result Explicit eigenstates called magnetic zonal states found.
Upper bound found for Steklov eigenvalue of a surface of revolution.
problem Finding an upper limit for Steklov eigenvalues of a specific surface.
method Analyzing a surface of revolution with boundary conditions of two spheres.
result An upper bound for the first Steklov eigenvalue is derived and shown to be sharp in some cases.
We bound the size of d d d -dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces.
The study quantifies distances between certain hyperbolic surfaces and bounds their number.
problem Quantifying distances and bounding numbers of hyperbolic surfaces.
method Parametrization of Teichmüller space by length functions.
result Two coverings of arithmetic surfaces are $P( rac{1}{g})$ apart, with P P P depending only on the surface. We show that immersed minimal surfaces of R 3 \mathbb{R}^{3} R 3 with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map is injective the whole surface is a wide component. Prior to these results it wa…