Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
Unique hyperbolic manifolds identified by boundary pleating.
problem Identifying hyperbolic manifolds from their boundary pleating.
method Used pleating measured lamination on the boundary of convex cores.
result Convex co-compact hyperbolic manifolds are uniquely determined by their pleating lamination.
Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial result…
Plumbing of surfaces embeds in hyperbolic 4-manifolds.
problem Embedding specific geometric structures in hyperbolic manifolds.
method Using plumbings of disc bundles over surfaces.
result Plumbed structures can be embedded in closed hyperbolic 4-manifolds.
Gradient flow converges to a minimal convex structure.
problem Finding the minimal convex structure in hyperbolic manifolds.
method Weil-Petersson gradient vector field of renormalized volume.
result The flow converges to the structure with minimum convex core volume.
We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling w…
Constructs examples of domains divided by groups in dimensions 3 and above.
problem Dividing convex sets with properly embedded cones.
method Uses Zariski dense relatively hyperbolic groups and properly embedded cones.
result Answers a question of Benoist and provides a topological criterion for convex projective structures.
Study finds optimal loops in hyperbolic space with Finsler structure.
problem Optimal loops in Finsler hyperbolic plane.
method Left-invariant Finsler structure, convex trigonometry functions.
result Optimal isoperimetric loops found in terms of trigonometry functions.
In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-triv…
Paper constructs geometric transitions between hyperbolic and Anti-de Sitter geometries.
problem Transitioning between hyperbolic and Anti-de Sitter geometries.
method Construction via Half-pipe geometry on ΣimesS1 with cone singularities. result Deformation of convex core structure as bending laminations collapse.
The paper characterizes groups acting on real projective spaces.
problem Understanding groups acting on convex domains in real projective geometry.
method Proves structure theorem for relatively hyperbolic groups in real projective spaces.
result Characterizes groups in terms of invariant convex subsets.
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
New method for computing hyperbolic structures on 3-manifolds with torus boundaries.
problem Computing a complete hyperbolic structure on 3-manifolds with torus boundaries.
method Convex optimization and combinatorial modifications to find a triangulation that admits a solution to the gluing equations.
result Experimental results support the new method for modifying triangulations and updating their geometry.
The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.
problem Characterizing and understanding geometric properties of 4D projective manifolds.
method Analyzing geometric decompositions and using properties of locally symmetric spaces.
result Closed, indecomposable 4D projective manifolds are either real hyperbolic or have real hyperbolic pieces.
For d=4,5,6, we exhibit the first examples of complete finite volume hyperbolic d-manifolds M with cusps such that infinitely many d-orbifolds Mm obtained from M by generalized Dehn filling admit properly convex real projective structures. The orbifold fundamental groups of Mm are Gromov-hyperbolic …
In this paper we show that bending a finite volume hyperbolic d-manifold M along a totally geodesic hypersurface Σ results in a properly convex projective structure on M with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We th…
The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.
problem Determining a closed convex set in hyperbolic 3-space by its boundary metric.
method Pogorelov's rigidity theorem, Hausdorff measure, and complex analysis techniques.
result The intrinsic path metric on the boundary determines a closed convex set up to isometry under certain conditions.
We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture…
We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex cocompactness in mapping class groups and Out(Fn). Corrigendum: there is an u…
Suppose that N is a geometrically finite orientable hyperbolic 3-manifold. Let P(N,C) be the space of all geometrically finite hyperbolic structures on N whose convex core is bent along a set C of simple closed curves. We prove that the map which associates to each structure in P(N,C) the lengths of the curves in the b…
Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
Geometric data uniquely determines convex subsets in hyperbolic manifolds.
problem Determining convex subsets in hyperbolic manifolds based on boundary data.
method Using conformal structure, induced metric, and third fundamental form on boundary components.
result Convex subsets are uniquely determined by boundary data.
A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its…
A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to …
The paper studies convex cocompact structures using Weil-Petersson flow.
problem Understanding the space of convex cocompact hyperbolic structures.
method Weil-Petersson gradient flow for renormalized volume.
result The renormalized volume difference is bounded by Weil-Petersson distance.
We consider globally hyperbolic maximal anti de Sitter 3-manifolds M with a closed Cauchy surface S of genus greater than one and prove that any pair of hyperbolic metrics on S can be realized as the boundary metrics of the convex core of a maximal globally hyperbolic anti de Sitter 3-manifold structure on M. T…
Paper finds inequalities for convex domains in hyperbolic space.
problem Finding inequalities for convex domains in hyperbolic space.
method Introducing hyperbolic ellipsoids and using orthogonal projection to establish inequalities.
result Affine isoperimetric inequalities for static convex domains in hyperbolic space characterized by hyperbolic ellipsoids.
Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
Let S be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural correspondence between convex RP^2 structures on S and pairs (Σ,U) consisting of a conformal structure Σon S and a holomorphic cubic differential U over Σ. …
The paper characterizes sets with infinite hyperbolic convex hull volume.
problem Characterizing sets with infinite hyperbolic convex hull volume.
method Geometric conditions and self-similar sets.
result Characterizes continua and planar self-similar sets with infinite hyperbolic convex hull volume.
Study projective deformations of hyperbolic 3-orbifolds with turnover ends.
problem Deformations of hyperbolic 3-orbifolds with turnover ends in projective geometry.
method Projective deformations of hyperbolic 3-orbifolds with turnover ends, focusing on totally geodesic generalized cusps.
result Turnover funnels remain totally geodesic and the deformed projective 3-orbifold remains properly convex.
Study of infinite type surfaces' mapping class groups via hyperbolic structures.
problem Understanding mapping class groups of infinite type surfaces.
method Definition of a topology on a moduli space of marked hyperbolic structures.
result Continuous action of mapping class groups on the marked moduli space.
We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian …
For an n-dimensional real hyperbolic manifold M, we calculate the Zariski tangent space of a character variety χ(π1(M),SL(n+1,R)),n>2 at Fuchisan loci to show that the tangent space consists of cubic forms. Furthermore we prove the Weil's local rigidity theorem for uniforml hyperbolic lattices using rea…
We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This…
We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particle…
The paper examines hyperbolicity in bounded strongly minimally convex domains in R^d.
problem Investigating hyperbolicity in bounded strongly minimally convex domains.
method Analyzing the minimal metric and Hilbert metric in convex domains.
result Every bounded strongly minimally convex domain is Gromov hyperbolic.
Study bounds changes in hyperbolic 3-manifold structures after drilling short geodesics.
problem Bounding changes in complex projective structures after drilling short geodesics.
method Analyzes L2-bounds on changes in conformally compact hyperbolic 3-manifolds. result Change is bounded by a universal constant times the square root of the length of the drilled geodesics.
New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.
problem Analyzing convex co-compact hyperbolic 3-manifolds with compressible boundaries.
method Defining and analyzing a new version of the renormalized volume.
result The adapted renormalized volume is bounded and has properties analogous to the classical renormalized volume.
The paper proves the existence of convex hyperbolic metrics on 3-manifolds with specific properties.
problem Proving the existence of convex hyperbolic metrics on 3-manifolds with given properties.
method Analyzing the induced metrics and bending lamination on the boundary of convex hyperbolic 3-manifolds.
result Existence of convex hyperbolic metrics that induce specific metrics and lamination structures.
The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
problem Proving new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
method Locally constrained inverse curvature flows in hyperbolic and spherical spaces.
result Established new Alexandrov-Fenchel and Minkowski inequalities involving general convex weight functions.
The paper defines equations for constructing projective structures on 3-manifolds.
problem Constructing projective structures on 3-manifolds.
method Defining a system of real valued equations and inequalities.
result These equations can be used to detect properly convex structures on 3-manifolds.
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.
problem Investigating Gromov hyperbolicity of specific metrics.
method Using isoperimetric inequalities to characterize Gromov hyperbolicity.
result Characterization of domains where these metrics are Gromov hyperbolic.
Study convex hyperbolic cone-metrics on 3-manifold boundaries, proving unique bent realizations.
problem Convex hyperbolic cone-metrics on 3-manifold boundaries and their bent realizations.
method Alexandrov-Weyl-type problem, bent metrics, controllably polyhedral, Lipschitz topology.
result Unique bent realizations for convex hyperbolic cone-metrics on 3-manifold boundaries.
To a complex projective structure Σ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms ∥φΣ∥∞ and ∥φΣ∥2 of the quadratic differential φΣ of Σ given by the Schwarzian derivative of the associated locally univalent map.…
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.