Proves the validity of Bartnik's mass minimization conjecture for some specific 3-balls.
problem Bartnik's conjecture on mass minimization for asymptotically flat extensions of 3-balls.
method Analyzes Riemannian 3-balls with non-negative scalar curvature and proves the existence or non-existence of mass-minimizing extensions.
result Validates the second part of Bartnik's conjecture for some specific cases but disproves the first part.
Paper proves rigidity of minimal disks in 3-balls with non-negative Ricci curvature.
problem Rigidity of free boundary minimal disks in 3-balls with non-negative Ricci curvature.
method Min-max methods and rigidity statements for half-balls with non-negative Ricci curvature.
result Existence and properties of minimal disks with least area in 3-balls.
The study calculates the volumes of 3-ball quotients and their orbifold Euler characteristics.
problem Calculating the volumes and orbifold Euler characteristics of 3-ball quotients.
method Explicit description and presentation of 3-ball quotients, deduction of orbifold Euler characteristics.
result Explicit values for the orbifold Euler characteristics of 3-ball quotients.
3-balls in 4-sphere become isotopic in 5-ball.
problem Whether 3-balls in 4-sphere become isotopic in 5-ball.
method Analyzing the embedding of 3-balls in 4-sphere and 5-ball.
result Affirmative answer to Gay, Hughes, Kim, and Miller's question.
Constructs minimal surfaces in a 3-ball using PDE gluing.
problem Finding minimal surfaces in a 3-ball with boundary constraints.
method PDE gluing construction of discrete free boundary minimal annuli.
result Discrete family of non-rotational free boundary minimal annuli in a unit 3-ball.
Researchers create infinite Brunnian links of 3-balls in 4-sphere.
problem Constructing infinite Brunnian links of 3-balls in 4-sphere.
method Using the third author's result on splitting spheres for trivial 2-spheres link in 4-sphere, and providing a new proof.
result Infinitely many n-component Brunnian links of 3-balls in 4-sphere constructed.
New findings show many 3-balls are not Mogami.
problem Determining if all 3-balls are Mogami.
method Characterized Mogami property in terms of nuclei.
result Many 3-balls are not Mogami.
Unique 3-balls in S^4 can be non-isotopic.
problem Existence and uniqueness of spanning 3-balls in S^4.
method Introducing barbell diffeomorphisms, implantations, and twistings; 2-parameter calculus of embeddings; framed cobordism method.
result Non-uniqueness of spanning 3-balls in S^4.
In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth's theorem by proving that all c…
The paper proves the stability of a 3-ball under curvature constraints.
problem Stability of a 3-ball under L2-curvature pinching.
method Elementary computations based on Bochner formula and 2-spheres effective uniformisation result.
result The manifold is diffeomorphic to the Euclidean ball and metrically close to it.
Metric of a 3-ball determined by minimal curve areas.
problem Determining a Riemannian metric from minimal areas.
method Continuous sweep-out of M by area-minimizing surfaces, combining ideas from inverse problems, minimal surface theory, and pseudo-differential equations. result Least areas circumscribed by curves uniquely determine the metric under specific conditions.
Based on the work of Durhuus-J{ó}nsson and Benedetti-Ziegler, we revisit the question of the number of triangulations of the 3-ball. We introduce a notion of nucleus (a triangulation of the 3-ball without internal nodes, and with each internal face having at most 1 external edge). We show that every triangulation can b…
Extends pillowcase homology for immersed curves in a 3-ball.
problem Constructing algebraic versions of Lagrangian Floer homology.
method Associate algebra A and A-infinity modules M(L) to immersed curves in the pillowcase, proving isomorphism with algebraic module pairings.
result Lagrangian Floer homology is isomorphic to a suitable algebraic pairing of modules.
A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked if and only if it is not outer-planar. A graph is outer-flat if it can be embed…
We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional directi…
Study finds a minimal surface in a ball with specific properties.
problem Finding minimal surfaces in bounded domains.
method 6-sweepout technique to prove existence and properties of minimal surfaces.
result Existence of a free boundary minimal surface with specified topological and geometric constraints.
Minimal surfaces in a ball have limited area.
problem Bounding the area of genus zero minimal surfaces in a unit ball.
method Proving an area inequality and showing convergence of saturating sequences.
result The area of each nonflat surface is less than its radial projection, with sharp asymptotic bounds.
New mappings solve long-standing problems in 3-space.
problem Longstanding problems for bounded locally homeomorphic quasiregular mappings in 3-space.
method Constructed bounded locally homeomorphic quasiregular mappings in the unit 3-ball using non-trivial hyperbolic cobordisms.
result Solved long-standing problems for quasiregular mappings in 3-space.
Paper proves surfaces with specific symmetries have the first Steklov eigenvalue.
problem Proving surfaces with certain symmetries have the first Steklov eigenvalue.
method Analyzing surfaces with reflection planes and genus zero.
result Surfaces with n distinct reflection planes have the first Steklov eigenvalue. The paper studies Pansu spheres in a sub-Riemannian 3-sphere and their area-minimizing properties.
problem The study of Pansu spheres and their area-minimizing properties in a sub-Riemannian 3-sphere.
method Calibration arguments.
result The closed half-spheres of S0 with boundary C0 minimize sub-Riemannian area among compact C1 surfaces with the same boundary. We construct the first explicit example of a simplicial 3-ball B_{15,66} that is not collapsible. It has only 15 vertices. We exhibit a second 3-ball B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable. Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125} (with only 1…
If a tangle, K, in the 3-ball has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.
The study finds at least 2 free-boundary minimal disks in convex 3-balls.
problem Finding minimal disks in convex 3-balls.
method Combining mean curvature flow, min-max theory, and degree theory.
result Existence of at least 2 free-boundary minimal disks in convex 3-balls for generic metrics.
The paper divides minimal surfaces in a ball into two parts.
problem Characterizing minimal surfaces in a ball with free boundaries.
method Analyzing planes and mean convex regions to divide surfaces and proving their properties.
result Every plane through the origin divides a minimal surface into exactly two parts.
We give a simple combinatoric proof of an exponential upper bound on the number of distinct 3-manifolds that can be constructed by successively identifying nearest neighbour pairs of triangles in the boundary of a simplicial 3-ball and show that all closed simplicial manifolds that can be constructed in this manner are…
New minimal surfaces in a ball created by merging catenoids and disks.
problem Creating minimal surfaces in a bounded space.
method Desingularizing a critical catenoid and equatorial disk using singular perturbation methods.
result Constructed high genus free boundary minimal surfaces with three boundary components.
Nonpositive towers property in 3-manifolds spines.
problem Properties of 3-manifold spines.
method Analysis of 2-dimensional spines in aspherical 3-manifolds and 3-ball.
result Nonpositive towers property in 2-dimensional spines of aspherical 3-manifolds.
Constructs minimal surfaces with unbounded genus in a ball.
problem Finding minimal surfaces with arbitrary genus in a bounded domain.
method Variational methods, min-max procedure, equivariant approach.
result Constructs surfaces of genus g for large g. For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms of it to the mapping class group of it has no section when g is at least 6.
The study explores conditions for a manifold to be a leaf of a complete foliation on the open unit ball.
problem Conditions for a manifold to be a leaf of a complete foliation on the open unit ball.
method Analyzes conditions for manifolds to be leaves of complete foliations.
result Provides answers to the question of when a manifold can be a leaf of a complete foliation on the open unit ball.
New invariant detects more elements in 4D diffeomorphism group.
problem Detecting more elements in the mapping class group of 4D handlebodies.
method Defined and computed a new invariant (W3)m for π0Diff(aturalmS1imesD3,∂). result Infinitely many non-isotopic separating 3-balls found.
We study the Kakimizu complex of a split link. As part of this, we also study Seifert surfaces and the Kakimizu complex for a non-split link in a 3-ball. In addition, we show that a simplex of the Kakimizu complex of a non-split link can be realised in an essentially unique way.
Generalizes Seifert algorithm to integral homology spheres.
problem Finding Seifert circles for integral homology spheres.
method Planar diagram representation of Heegaard splitting, tangle projection.
result Natural construction of Seifert circles and bands for integral homology spheres.
Solves a triangulation problem by showing minimum tetrahedra equals minimum integral 3-chain.
problem Finding the minimum number of tetrahedra to extend a triangulation of a 2-sphere to a 3-ball.
method Relates the minimum number of tetrahedra to the minimum integral 3-chain norm, proving them equal and showing how to achieve the minimum.
result The minimum number of tetrahedra needed to extend a triangulation of a 2-sphere to a 3-ball equals the minimum integral 3-chain norm.
A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We show that irreducibility is preserved under certain deformations of embedded graphs…
New functionals defined for free boundary minimal submanifolds in higher dimensions.
problem Characterizing metrics for free boundary minimal submanifolds in geodesic balls.
method Introducing and studying new functionals Θr,i and Ωr,i for higher-dimensional free boundary minimal submanifolds. result Critical metrics for these new functionals are the metrics induced by free boundary minimal immersions.
Sharp bounds on minimal surfaces in compact 3-manifolds with convex boundary.
problem Finding bounds on the length of minimal surfaces in compact 3-manifolds.
method Using Toponogov's theorem and geometric inequalities, the authors derive sharp upper bounds for the boundary length of minimal surfaces.
result If the boundary length of a minimal surface saturates the upper bound, the manifold must be a Euclidean 3-ball and the surface a Euclidean disk.
The paper calculates homology and intersection pairing of branched covers using disoriented homology.
problem Computing homology and intersection pairing of branched covers of 4-ball.
method Associate disoriented homology groups to projections of links and surfaces, show isomorphism to branched cover homology, define pairing on first disoriented homology of surfaces.
result Disoriented homology is isomorphic to the homology of branched cover and pairing is equal to the intersection pairing.
Let M be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric ≥0. We suppose that M is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let Σ be a compact connected and orientable surface immersed in M which is a stable constan…
The generalized Cartan-Hadamard conjecture says that if Ω is a domain with fixed volume in a complete, simply connected Riemannian n-manifold M with sectional curvature K≤κ≤0, then the boundary of Ω has the least possible boundary volume when Ω is a round n-ball with constant curvature K=κ. The c…
We give a complete enumeration of all combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product S2×S1 and 615 triangulations of the twisted sphere product $S^2_\times_S^1$. All the 3-spheres…
New invariant for links in handlebodies defined using braid groups and Soergel bimodules.
problem Defining invariants for links in handlebodies.
method Using braid groups and complexes of Soergel bimodules.
result Generalized HOMFLYPT homology for links in handlebodies.
4-ball can be tiled with knotted surfaces.
problem Tiling the 4-ball with knotted surfaces.
method Using congruent knotted surfaces isotopic to the original surface.
result Tiling of the 4-ball with knotted surfaces.
This paper contains a construction of a finite set X in the boundary of the unit 3-ball in R^3 whose minimal tree is knotted. The example answers Problem 5.17 in ''Problems in Low-dimensional Topology'' by Rob Kirby posed by Michael Freedman: ''Given a finite set of points X in the boundary of B^3, let T be a tree in B…
New minimal surfaces found in a ball, connected and with high genus.
problem Finding minimal surfaces with connected boundaries in a ball.
method Gluing construction tripling the equatorial disc.
result First examples of compact free boundary minimal surfaces with connected boundary in a ball.
We consider a generalised complex Monge-Ampère equation on a compact Kähler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also prove that (for three-folds and a related real PDE in a ball), as long as the Hessian is bounded below by a pre-determined co…
Develops a method for random manifolds and submanifolds, focusing on 3-ball knots.
problem Understanding random submanifolds in triangulated manifolds.
method Coloring vertices of triangulated manifolds to generate random submanifolds.
result Probability of generating an unknot decays exponentially in 3-ball with 3 colors.
For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skei…