New condition prevents hyperbolic spaces from matching curve complexes.
problem Identifying when hyperbolic spaces cannot match curve complexes.
method Analyzing specific hyperbolic complexes and identifying a condition.
result Identified a condition preventing quasi-isometry between hyperbolic spaces and curve complexes.
We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic surfaces and their compactifications.
Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.
problem Characterizing and constructing minimal Lagrangian surfaces in complex hyperbolic quadric.
method Loop of flat connections, isometric deformations, DPW-type representation.
result Explicit examples of minimal Lagrangian surfaces, including catenoid-type examples.
We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved Kähler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, ou…
Study minimal surfaces in complex hyperbolic space, linking entropy and volume.
problem Characterize minimal submanifolds in complex hyperbolic space.
method Analyze asymptotic regularity and introduce Colding-Minicozzi entropy and CR-volume.
result Establish a connection between Colding-Minicozzi entropy and CR-volume.
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
The paper studies Lagrangian surfaces in a specific Riemannian product space.
problem Characterizing and classifying Lagrangian surfaces in a particular geometric space.
method Analyzes various types of Lagrangian surfaces and their properties.
result Classification of different types of Lagrangian surfaces.
The abstract proves the existence of CMC-1 surfaces with any complex structure in hyperbolic space.
problem Proving the existence of CMC-1 surfaces with arbitrary complex structures in hyperbolic space.
method Using a jet interpolation theorem and a uniform approximation theorem for holomorphic null curves.
result Existence of complete densely immersed CMC-1 surfaces in hyperbolic space with arbitrary complex structure.
Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.
problem Investigating relationships between three invariants of complex hyperbolic disc orbibundles.
method Analyzing Euler characteristic, Euler number, and Toledo invariant of disc orbibundles over 2-orbifolds.
result Proved that -3|τ| = 2e + 2χ holds for certain complex hyperbolic disc orbibundles.
This paper studies deformations of hyperbolic surfaces with special structures.
problem Infinitesimal deformations of hyperbolic surfaces with boundary and ideal vertices.
method Description of the admissible cone of deformations in terms of the arc complex.
result Realization of the admissible cone and its faces as arc complexes for specific surface families.
The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an…
The paper proves a linear diameter bound for hyperbolic knot complexes.
problem Understanding the diameter of Kakimizu complexes for hyperbolic knots.
method Defined a complex ISℓ(K) to study incompressible Seifert surfaces and proved its diameter has a linear upper bound. result The diameter of the Kakimizu complex for hyperbolic knots grows linearly with genus, confirming a conjecture.
New rigidity result for hyperbolic surfaces based on curve lengths.
problem Determining hyperbolic metrics on surfaces from curve lengths.
method Investigating oriented graphs on curve complexes and Dehn quasi-homothetic functions.
result Knowing which curve is longer suffices to determine the hyperbolic metric on a surface.
We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our c…
The paper constructs metrics with negative curvature on complex manifolds.
problem Constructing complete Kähler metrics with negative bisectional curvature on hyperbolic complex manifolds.
method Introducing a mechanism for constructing complete Kähler metrics with negative bisectional curvature.
result Realized Chern slopes c12/c2 for surfaces with negative holomorphic sectional curvature. A proof that the separating curve complex of the closed genus two surface has a quasi-distance formula and is delta hyperbolic using tools of Masur and Schleimer. This answers in the affirmative a Conjecture of Schleimer.
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
The pants graph has proved to be influential in understanding 3-manifolds concretely. This stems from a quasi-isometry between the pants graph and the Teichmüller space with the Weil-Petersson metric. Currently, all estimates on the quasi-isometry constants are dependent on the surface in an undiscovered way. This pape…
Compact hyperbolic complex manifolds are rigid under deformation.
problem Studying the deformation behavior of compact hyperbolic complex manifolds.
method Analyzing smooth families of compact complex manifolds over the unit disk and compact Riemann surfaces.
result The H-locus is either at most a discrete subset or the whole domain, depending on the family structure. The study proves a tube theorem for complex hyperbolic manifolds.
problem Understanding the geometry of complex hyperbolic manifolds.
method Tubular neighborhood theorem and geometric combination theorem.
result Explicit estimates and bounds for tube widths in complex hyperbolic manifolds.
Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.
problem Quantifying the complexity of non-simple closed geodesics on hyperbolic surfaces.
method Analyzing the geometry of shortest figure eight curves and constructing geodesic representatives.
result Explicit upper bounds for the length of shortest geodesics with k self-intersections improved from 512 to 128. We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated b…
Study circle patterns and polyhedral surfaces in hyperbolic ends, proving manifold properties.
problem Understanding the space of complex projective structures on surfaces with circle patterns.
method Analyzing ideal polyhedral surfaces in hyperbolic ends, proving manifold properties and Lagrangian immersions.
result The space of complex projective structures on surfaces with circle patterns is a manifold of dimension 6g-6.
The paper determines the structure of Kakimizu complexes for genus one hyperbolic knots.
problem Understanding the structure of Kakimizu complexes for genus one hyperbolic knots.
method Analyzing the simplicial complex of minimal genus Seifert surfaces in the exterior of the knots.
result The Kakimizu complex for genus one hyperbolic knots consists of a single d-simplex for d=0,4 and otherwise of at most two d-simplices which intersect in a common (d−1)-face. New minimal surfaces found in spheres and hyperbolic spaces.
problem Constructing minimal submanifolds in even-dimensional spheres and hyperbolic spaces.
method Using complex-valued harmonic morphisms.
result Explicit examples of minimal submanifolds in S4 and H4. Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's spher…
We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that…
New 2D complex hyperbolic structures found on sphere orbibundles.
problem Locally rigid complex hyperbolic structures on sphere orbibundles.
method Constructing families of complex hyperbolic structures on disc orbibundles.
result Examples of non-locally rigid complex hyperbolic structures.
We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hami…
In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.
We show that for each aspherical compact complex surface X whose fundamental group π fits into a short exact sequence 1→K→π→π1(S)→1 where S is a compact hyperbolic Riemann surface and the group K is finitely-presentable, there is a complex structure on S and a nonsingular holomorphic fibr…
This study finds a special class of representations that dominate others in a complex hyperbolic group.
problem Domination of surface-group representations in complex hyperbolic groups.
method Analysis of T-bent representations and their domination by discrete and faithful representations. result A discrete and faithful representation exists that dominates a given T-bent representation in the Bergman translation length spectrum. Paper studies complex Lagrangian surfaces and their relation to SL(3,C)-representations.
problem Minimal Lagrangian surfaces in bi-complex hyperbolic space and their representations.
method Introduces bi-complex Higgs bundles and parameterizes SL(3,C)-quasi-Fuchsian representations. result Parameterization of SL(3,C)-quasi-Fuchsian representations by an open set in Teichmüller space. We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyp…
New simplicial complex for infinite-type surfaces shows graph properties.
problem Characterizing infinite-type surfaces using graph theory.
method Constructing grand arc graph and analyzing its properties.
result Grand arc graph is infinite-diameter and δ-hyperbolic under certain conditions.
Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.
problem Bounding the L2-norm of harmonic forms in hyperbolic 3-manifolds. method Using Thurston norm and interaction with minimal surfaces.
result Generalizes inequalities of Brock-Dunfield and studies sharpness in closed and cusped cases.
Study strip deformations of hyperbolic polygons with decorated vertices.
problem Understanding deformations of hyperbolic polygons with decorated vertices.
method Analyzing strip deformations of ideal hyperbolic polygons with horoballs.
result Arc complexes parameterize uniformly lengthening deformations.
We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each g at least 2, to identify a constant r_{g-1,2} with the property that the set of closed genus-g hyperbolic surfaces with maximal injectivity radius at least r is compact if and only if r > r_{g-1,2}. T…
We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichmüller space of the torus, inspired by Thurston's Lipschitz metric for the case of hyperbolic surfaces. Based on extremal Lipschitz maps, the Teichmüll…
Let p be a branched covering of a Riemann surface to the Riemann sphere P1, with branching set B⊂P1. We define the complexity of p as infinity, if P1∖B does not admit a hyperbolic structure, or the product of its degree and the hyperbolic area of $\mathbb{P}^1 \…
The study shows that surface groups are the only non-free infinite index subgroups of certain hyperbolic groups.
problem Identifying the only non-free infinite index subgroups of specific hyperbolic and one-relator groups.
method Careful analysis of free and cyclic splittings of cubulated groups.
result Proves that surface groups are the only non-free infinite index subgroups of certain hyperbolic and one-relator groups.
In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…
Rafi and Schleimer recently proved that the natural relation between curve complexes induced by a covering map between two surfaces is a quasi-isometric embedding. We offer another proof of this result using a distance estimate via hyperbolic 3-manifolds.
Abstract: Study of geometric structures on surfaces using various tools.
problem Understanding geometric structures on surfaces.
method Use of volume, contact, symplectic, complex, and almost complex structures; local rigidity results; higher-dimensional analogues; constructions with Riemann surfaces; definitions using surjective homomorphisms; models of hyperbolic plane and 3-space; conformal structures.
result Introduction of new models and constructions for hyperbolic plane and 3-space.
We prove that the curve graph $\calC^{(1)}(S)$ is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface S. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.
Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.
problem Localization of Bergman kernels on Kähler manifolds with complex hyperbolic cusps.
method Revisiting Tian's peak section method, applying to Kähler-Einstein metrics and quotients of complex balls.
result Partial localization result for Poincaré type cusps.
Novel defects in hyperbolic sheets explain complex wrinkling patterns in nature.
problem Understanding complex wrinkling patterns in thin elastic hyperbolic surfaces.
method Non-Euclidean plate theory and investigation of branch points.
result Branch points are natural defects in hyperbolic sheets, influencing their morphology robustly.