Researchers determine the Thurston unit ball for a family of n-chained links and find conditions for fibered faces.
problem Determining the Thurston unit ball and conditions for fibered faces in a family of n-chained links. method Analyzing the family of n-chained links C(n,p), proving the Thurston unit ball is an n-dimensional cocube for p>0, and finding conditions for fibered faces. result The Thurston unit ball for C(n,p) is an n-dimensional cocube for p>0 and provides at least one fibered face for any p. Article provides polytopes as dual unit balls of Thurston norms on 3-manifolds.
problem Understanding the dual unit ball shape of Thurston norms.
method Introduced a family of polytopes in Z^2g that can be dual unit balls of Thurston norms on 3-manifolds.
result Polytopes with mod 2 congruent vertices can be realized as dual unit balls of Thurston norms.
Volume of unit balls defined by quadratic differentials is not proper and has integrable volume.
problem Characterizing the behavior of volume functions associated with quadratic differentials.
method Analyzing the Thurston volume of unit balls in measured lamination spaces.
result The volume function is not proper and is p-integrable for any 0<p<1. The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.
problem Determining if integral points on the Thurston norm dual ball correspond to geometric structures.
method Examining various geometric, topological, and dynamical structures on 3-manifolds.
result Integral points on the Thurston norm dual ball correspond to the Euler class of taut foliations and other structures.
The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds M with b1(M)=2, and whose fundamental groups admit presentations with two generators and one relator. We show that even among this…
We simplify Thurston norm computation for 2-bridge link complements.
problem Understanding the complexity of Thurston norm unit balls in 3-manifolds.
method Utilized Floyd and Hatcher's surface description and integral class minimization.
result Thurston norm unit balls of 2-bridge link complements have at most 8 faces.
Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.
problem Understanding Thurston norms of graph manifolds and their realizability.
method Analyzing the structure of Thurston norms as sums of linear functionals and showing realizability.
result Every Thurston norm of a graph manifold can be expressed as a sum of absolute values of linear functionals with rational coefficients.
For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3-manifold Y, there is a bijection between vertices of the HF^+(Y) polytope carrying the group Z and the faces of the Thurston norm unit ball that corresp…
For sutured 3-manifolds M, there is a sutured Thurston norm due to Scharlemann. We show how depth one foliations of M and corresponding fibrations and the usual Thurston norm on the double of M are useful tools for computing this norm. In many examples, the faces of the unit ball of the sutured norm are related to cone…
We present a new proof of Thurston's theorem that the unit ball of a seminorm on Rd taking integer values on Zd is a polyhedra defined by finitely many inequalities with integer coefficients.
Algorithm computes Thurston norm for hyperbolic 3-manifolds.
problem Computing the Thurston norm for hyperbolic 3-manifolds.
method Developed a theory of spun-normal immersed surfaces and implemented an algorithm.
result Computed the unit ball of the Thurston norm for cusped hyperbolic 3-manifolds.
Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.
Analyzes convex structures in Teichmüller space unit tangent spheres.
problem Characterize faces and extreme points of unit tangent spheres in Teichmüller space.
method Analyzes Finsler infinitesimal balls of Thurston metric, characterizes faces, exposed faces, and extreme points.
result Characterizes faces and extreme points of unit tangent spheres in Teichmüller space.
The paper disproves a conjecture about 3D manifolds using even lattice points.
problem Thurston's Euler class one conjecture for fillable contact structures.
method Analyzing finite covers of hyperbolic 3-manifolds and properties of their dual Thurston norm unit balls.
result Found counter-examples to the conjecture using even lattice points on boundary.
Let M be a closed hyperbolic 3-manifold with a fibered face σ of the unit ball of the Thurston norm on H2(M). If M satisfies a certain condition related to Agol's veering triangulations, we construct a taut branched surface in M spanning σ. This partially answers a 1985 question of Oertel, and extends an e…
New flows represent Thurston norm ball faces, differing by veering mutations.
problem Dynamic representation of Thurston norm ball faces by distinct flows.
method Combining veering triangulations and mutations to represent faces by multiple flows.
result Non-fibered faces can be represented by two distinct flows differing by veering mutations.
The Whitehead link exterior lacks most Euler class taut foliations.
problem Existence of co-orientable taut foliations with specific Euler classes.
method Combining foliation theory techniques and topological obstructions.
result Most lattice points in the dual Thurston ball cannot be Euler classes of co-orientable taut foliations.
We highlight several analogies between the Finsler (infinitesimal) properties of Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space. Thurston defined his asymmetric metric in analogy with Teichmüllers' metric, as a solution to an extremal problem, which consists, in the case of the asymmetric me…
Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by non-negative integer weights, 14 for each 3-simplex, that describe how many copies o…
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
Study constructs disks with curved boundaries in a 3D ball.
problem Constructing non-planar free boundary disks in a unit ball.
method Infinite family of non-planar disks with non-positive Gaussian curvature.
result Constructs disks with curved boundaries in a unit ball.
We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic manifolds having 3 cusps, so called "the magic manifold". We compute the entropy function on the fiber face of the unit ball with respect to the Thurston norm, determine homology classes whose representatives are gen…
Study of fibred faces of Thurston polyhedra for 2-component 2-bridge links
problem Study of fibred faces of Thurston polyhedra for 2-component 2-bridge links
method Novikov homology associated with the universal covering of the exterior of the link
result Prove that a cohomology class can be represented by a fibration over a circle if and only if its 2-variable Alexander polynomial is ξ-monic Veering triangulations link Thurston norm and isotopy of surfaces.
problem Understanding the relationship between Thurston norm and isotopy of surfaces.
method Analyzing veering triangulations and their relation to Thurston norm and isotopy.
result Veering triangulations specify faces of Thurston norm balls and link isotopy of surfaces.
Let Σbe a k-dimensional minimal surface in the unit ball B^n which meets the unit sphere orthogonally. We show that the area of Σis bounded from below by the volume of the unit ball in R^k. This answers a question posed by R. Schoen.
Study inverse curvature flows for capillary hypersurfaces in a unit ball.
problem Understanding the behavior of capillary hypersurfaces under inverse curvature flows.
method Investigate inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball.
result Establish existence and convergence results for inverse curvature flows.
Two minimal hypersurfaces in a ball intersect in any half-ball.
problem Intersection properties of minimal hypersurfaces in a ball.
method Analyzing the intersection of two minimal hypersurfaces in a unit Euclidean ball.
result Intersection point in any half-ball, strong Frankel property.
The unit ball is characterized by a Kähler-Einstein potential.
problem Characterizing the unit ball in complex geometry.
method Using a global potential function of the Kähler-Einstein metric.
result A compact Kähler manifold with an ample canonical bundle is the unit ball if it has a specific potential function.
The paper proves new inequalities on the unit ball in higher dimensions.
problem Establishing new weighted inequalities on the unit ball.
method Limiting approach to prove Carleman and Huber inequalities.
result Sharp weighted Carleman and Huber inequalities on the unit ball.
Sharp upper bound for minimal graph area in unit ball established.
problem Determining the exact upper limit for the area of minimal graphs intersecting a unit ball.
method Constructing a sequence of minimal graphs via solutions to a Dirichlet problem.
result The areas of constructed minimal graphs tend to the upper bound of 2π. The paper describes geometric properties of Teichmüller space metrics.
problem Analyzing the geometry of Teichmüller space with weak Finsler metrics.
method Geometric description of unit spheres in weak Finsler metrics.
result Introduced a family of weak Finsler metrics interpolating between Thurston's metric and Teichmüller metric.
New combinatorial structures for Teichmüller spaces with Thurston's metric are explored.
problem Understanding the combinatorial structures of Teichmüller spaces with Thurston's metric.
method Analyzing the unit tangent and cotangent spheres of Teichmüller space, proving formulas for dimensions and codimensions of faces.
result The combinatorial structure of unit spheres in Teichmüller spaces is independent of the underlying point and is isomorphic to the extended mapping class group.
The paper extends Thurston's method to new variants of Mather-Thurston theorem.
problem Proving new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
method Generalizing Thurston's technique to prove new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
result The paper answers questions posed by Gelfand-Fuks and Greenberg on PL foliations and Rybicki on contactomorphisms.
Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is f, when f possesses certain reflection or rotation symmetry.
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.
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
Minimal normal curvature immersions in the unit ball studied.
problem Minimal normal curvature immersions in the unit ball.
method Gromov's problem, differentiable sphere theorem, existence result.
result Determined the minimal possible value of the normal curvature of SnimesS1. 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.
Proves rigidity of maps between balls with Hölder boundary continuity.
problem Rigidity of proper holomorphic maps between unit balls with Hölder boundary continuity.
method Proves rigidity for maps with symmetries and Hölder boundary continuity.
result Proves rigidity for maps with Hölder exponent > 1/2 on the boundary.
The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.
problem Characterizing unit balls among Stein spaces with specific groups.
method Study of Bergman metric on finite ball quotients and its Kähler-Einstein property.
result The Bergman-Einstein metric exists only on the unit ball itself for finite ball quotients with trivial groups.
In this paper, we construct an asymptotically hyperbolic metric with scalar curvature -6 on unit ball D3, which contains multiple horizons.
Geometric structures on quaternionic unit ball for slice regular Möbius transformations.
problem No new problem introduced.
method Introducing Hermitian, Riemannian, and Kähler-like structures on quaternionic unit ball using regular Möbius transformations.
result Geometric structures are natural generalizations of complex setup and solve problems not achieved by other geometries.
This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as their logarithmic limit sets are computed. The projective solution space of spun-nor…
We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection with a set D×{c} grows monotonically with c.…
The purpose of the present paper is to show that the components of the unit normal of any minimal surface with free boundary in the unit ball, are eigenfunctions associated with the eigenvalue −2, for some (new) natural eigenvalue problem for the Jacobi operator; this fact has analytic (spectral) consequences for fre…
In this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our…
Sharp inequalities in unit ball with constraints on moments.
problem Establishing Sobolev trace inequalities with constraints.
method Constructing smooth test functions for higher order moments.
result Almost optimal Sobolev trace inequalities for 2nd and 4th orders.
Paper calculates ball number of links using Lorentz geometry and circle packing.
problem Calculating the minimum number of balls needed to represent a link.
method Lorentz geometry and circle packing theorem applied to ball packings.
result Shows ball(L)≤5cr(L) for any link L.