Characterizes Coxeter groups with specific boundary shapes.
problem Identifying Coxeter groups with Sierpiński or Menger curve boundaries.
method Combining results from the literature on Gromov boundaries and Coxeter groups.
result Complete characterizations of hyperbolic Coxeter groups with Sierpiński or Menger curve boundaries.
Study shows boundary curves of free boundary minimal surfaces are circles.
problem Characterizing boundary curves of free boundary minimal surfaces.
method Holomorphic techniques applied to the unit ball in Euclidean 3-space.
result Intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
Classifies ancient convex curves in convex domains.
problem Ancient convex curve shortening flows on convex domains.
method Classification of convex ancient solutions.
result Ancient convex curves in convex domains classified.
Describes curves on surfaces with punctures and boundaries.
problem Representing multiple curves on surfaces with punctures and boundaries.
method Using geometric intersection numbers with embedded curves.
result Each multiple curve can be uniquely described.
Study on curve shortening flow with boundary conditions, proving convergence or contraction.
problem Analyzing curve shortening flow with free boundaries.
method Introduced a reflected chord-arc profile and obtained chord-arc estimates.
result Proved that flows either converge to a critical chord or contract to a round half-point.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
We prove two theorems on the removal of singularities on the boundary of a pseudo-holomorphic curve. In one theorem, we need no apriori assumption on the area of the curve. The proof uses a doubling argument with the goal of converting curves with boundary to curves without boundary. Our method is new and geometric and…
Estimates radius and volume of curved spaces with convex boundaries.
problem Estimating the radius and volume of curved spaces with convex boundaries.
method Analyzes Alexandrov spaces with strictly convex boundaries, using Base-Angle and volume estimates.
result Estimates for radius and volume of curved spaces with convex boundaries.
Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simp…
Study on curve shortening flow with Neumann boundary conditions, showing singularities and limits.
problem Analyzing singularities in curve shortening flow with Neumann boundary conditions.
method Criterion for initial curves leading to singularities, proof of singularity type, rescaling analysis.
result Existence of type II singularities and limit shapes for convex initial curves.
The paper classifies algebraic curves in 4-balls and their boundaries.
problem Understanding algebraic curves in 4-dimensional balls and their boundaries.
method Analyzing algebraic curves in complex 2-space and their intersections with 4-balls.
result Classification of algebraic curves with up to 5 crossings.
Primitive curves in handlebodies form a connected complex.
problem Understanding the structure of curves in handlebodies.
method Defining and analyzing primitive curves and constructing sequences between them.
result The primitive curve complex for a handlebody is connected.
The paper studies the connectedness of a graph's boundary for surfaces.
problem Understanding the topology of the Gromov boundary of fine curve graphs for surfaces.
method Proved a bounded geodesic image theorem, used to show linear connectivity of the Gromov boundary.
result The Gromov boundary of fine curve graphs for surfaces is linearly connected.
Simple closed curves in ε-boundaries separate sets in the plane.
problem Separating sets with simple closed curves in ε-boundaries.
method Analyzing ε-boundaries of planar sets and proving the existence of simple closed curves.
result Simple closed curves in ε-boundaries separate sets in the plane.
The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations…
Formula derived for mean curvature of a special surface with boundary curve.
problem Understanding the properties of constant mean curvature discs with analytic boundary.
method Analysis of holomorphic quadratic Hopf differential and normal curvature.
result Formula for mean curvature as a weighted average of boundary curve normal curvature.
Study curve shortening flow in high dimensions with boundary constraints.
problem Understanding the behavior of curves in high-dimensional spaces with boundary conditions.
method Used curvature and higher-derivative estimates, Stahl-type maximum principle, and blow-up analysis.
result Flow converges to a shrinking semicircle model or has only semicircle boundary singularities in low entropy regimes.
New method for surfaces with boundary using hyperbolic metrics.
problem Uniformization of surfaces with boundary.
method Representing conformal structures by hyperbolic metrics with boundary curves of constant positive geodesic curvature.
result Boundary curves cannot collapse as conformal structure degenerates.
The boundary of certain hyperbolic groups is like a Menger curve.
problem Characterizing boundaries of hyperbolic Coxeter groups.
method Analyzing the nerve of hyperbolic right-angled Coxeter groups.
result Many triangulations and disks have boundaries homeomorphic to the Menger curve.
Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
Constructs a universal Cannon-Thurston map for a new curve complex.
problem Mapping class groups and their boundaries.
method Using Birman exact sequence, proves hyperbolicity, constructs map.
result Universal Cannon-Thurston map to surviving curve complex boundary.
Study on holomorphic curves in 6-sphere with boundary conditions.
problem Characterizing holomorphic curves in nearly-Kähler 6-manifolds with boundary conditions.
method Complex-geometric methods, including second variation formula for area.
result Obtained rigidity results for reflection-invariant holomorphic curves and topological lower bounds for Morse index.
Negative-curvature surfaces uniquely define boundary distances.
problem Defining distances on curved surfaces at infinity.
method Renormalized distance calculation for negatively-curved surfaces.
result Negative-curvature surfaces are boundary distance rigid.
New groups with Menger curve boundaries found.
problem Finding non-hyperbolic groups with specific boundary shapes.
method Using Coxeter groups with complete graph nerves and embedding theorems.
result First non-hyperbolic CAT(0) groups with Menger curve boundaries discovered. Study the geometry of nonseparating curves on surfaces with punctures.
problem Investigate the geometry of nonseparating curves on surfaces with punctures.
method Study finite covers and lifts of nonseparating curves, analyze bicorn curves and laminations.
result Lifts of nonseparating curves span quasiconvex subgraphs in the nonseparating curve graph of covers.
Study curves evolving on hypersurfaces with free boundaries, preserving length.
problem Evolution of curves on hypersurfaces with free boundaries.
method Nonlocal evolution equation with nonlinear boundary conditions, short-time existence, uniqueness, and parabolic energy estimates.
result Global existence and convergence to critical points proved.
Extended de Rham theorem for manifolds with boundaries.
problem Applying de Rham decomposition to manifolds with boundaries.
method Using development of curves to extend classical theorem.
result Extended de Rham theorem for manifolds with boundaries.
Study of a beam sliding freely along two curves.
problem Finding the shortest curve with free sliding endpoints.
method Geometric formulation on smooth manifolds with boundary.
result Rigorous geometric approach to free boundary problems.
The study shows that certain curve graphs are hierarchically hyperbolic but not Gromov hyperbolic.
problem Characterizing the hyperbolicity of curve graphs and their boundaries.
method Using hierarchical hyperbolicity and framed curves, the study examines the properties of curve graphs and their boundaries.
result The curve graphs and their boundaries are hierarchically hyperbolic but not Gromov hyperbolic.
Study the Gromov boundary of fine curve graph for surface homeomorphisms.
problem Understanding the boundary of fine curve graph for surface homeomorphisms.
method Examined the Gromov boundary and local topology near specific foliations and laminations.
result Found elements with positive stable commutator length and proved a Tits alternative.
Spheres in curve graphs are connected, proving Gromov boundary linearity.
problem Understanding connectivity in curve graphs and their boundaries.
method Defining spheres and analyzing their connectivity for different complexities.
result Spheres in high complexity curve graphs are always connected, with weaker results for low complexity.
Let S be the boundary of a handlebody M. We prove that the set of curves in S that are boundaries of disks in M, considered as a subset of the complex of curves of S, is quasi-convex.
Proves nonnegatively curved hypersurfaces on a sphere are convex disks.
problem Characterizing nonnegatively curved hypersurfaces with free boundary on a sphere.
method Analyzes hypersurfaces in Euclidean space with constant mth mean curvature. result Compact hypersurfaces are embedded convex disks.
Study the boundary of curve complex and Teichmüller space for surfaces.
problem Characterize the boundary at infinity of curve complex and Teichmüller space.
method Quasi-isometric study and hyperbolicity analysis.
result Boundary at infinity of curve complex and Teichmüller space are equivalent.
We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two boundary curves are either parallel or sufficiently close to parallel, and when th…
The paper proves compactness for holomorphic curves with boundary on nearby Lagrangians.
problem Compactness of holomorphic curves with boundary on nearby Lagrangians.
method Generalizes earlier work on compactness, proving a limit configuration of holomorphic curves joined by gradient flow lines.
result Exponential estimate analyzing the interface between holomorphic parts and gradient flow lines.
Defines timelike ideal boundary for non-positively curved Lorentzian spaces.
problem Understanding the geometry of non-positively curved Lorentzian spaces.
method Introduces timelike ideal boundary as asymptotic classes of geodesic rays, endows with topology and metric, and studies upper curvature bounds.
result Established upper curvature bounds for the resulting metric space.
In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent--Leininger--Schleimer and Mitra, we construct a universal Cannon--Thurston map from a subset of the circle at infinity for the closed surfac…
Study of limits of Cantor and \sier sets, showing homeomorphic spaces.
problem Understanding topological properties of certain geometric structures.
method Analyzing direct limits of embedded Cantor sets and \sier curves, showing homeomorphism.
result Morse boundaries of specific groups are homeomorphic to limits of Cantor and \sier sets.
Solves curve migration problem with elastic flows.
problem Curve migration problem with natural boundary conditions.
method Constructing migrating elastic flows.
result Extends previous work to purely local flow.
The Riemannian submersion π:SO0(1,n)→Hn is a principal bundle and its fiber at π(e) is the imbedding of SO(n) into SO0(1,n), where e is the identity of both SO0(1,n) and SO(n). In this study, we associate a curve, starting from the identity, in $\…
New method proves rigidity of minimal hypersurfaces in curved 4-manifolds.
problem Proving rigidity of minimal hypersurfaces in curved 4-manifolds.
method Combining nonnegative 2-intermediate Ricci curvature and strict positivity of scalar curvature, extending Chodosh-Li-Stryker method.
result Rigidity of two-sided free boundary stable minimal hypersurfaces in 4-manifolds with bounded geometry and weakly convex boundary.
The Poincaré series for surfaces with boundary extends to the complex plane.
problem Counting geodesics on surfaces with boundaries.
method Analytic continuation of Poincaré series.
result Poincaré series extend meromorphically to the whole complex plane.
Study shows how a curve shortens to a half-circle under specific flow.
problem Stability of a semi-circle under curve shortening flow.
method Sharp rate of convergence for a free-boundary curve shortening flow in a convex domain.
result Established a sharp rate of convergence to a round half-point.
Mathematical study connects curve counting to quantum topology.
problem Counting holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds.
method Organized 1-parameter families of holomorphic curves and matched them to HOMFLYPT skein relations.
result Holomorphic curve counts match HOMFLYPT polynomial coefficients of links in the 3-sphere.
Classifies special quartic curves up to equivalence.
problem Classifying maximal quartic curves up to equivalence.
method Analyzing intersections of quartic polynomials and their level sets.
result Quartic generalised projective special real manifolds have non-regular boundary behavior.
Geometric interpretation of 3-manifold invariants using immersed curves.
problem Obstructing smooth equivalences between 4-manifolds and surfaces with boundary.
method Relating morphisms between bordered Floer invariants to cobordism maps via immersed curves in the punctured torus.
result Morphisms between immersed curve invariants compute certain cobordism maps.
In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…