Groups satisfy linear surface isoperimetric functions.
problem Isoperimetric functions for surface diagrams in hyperbolic groups.
method Analyzing word-hyperbolic groups and their surface diagrams.
result Linear isoperimetric functions for all surface types in hyperbolic groups.
Explicit polynomial bound found for subgroup Dehn function.
problem Finding explicit bounds on Dehn functions of subgroups of hyperbolic groups.
method Constructing a specific example of a non-hyperbolic subgroup and analyzing its Dehn function.
result Explicit polynomial upper bound n96 on the Dehn function of a non-hyperbolic subgroup. New example of hyperbolic 6-manifold with circle-valued Morse function.
problem Finding Morse functions on hyperbolic manifolds.
method Constructing a circle-valued Morse function on a hyperbolic 6-manifold.
result First example of a hyperbolic 6-manifold with a perfect circle-valued Morse function.
We explain how to construct certain potential functions for the hyperbolic structures of a knot complement, which are closely related to the analytic functions on the deformation space of hyperbolic structures.
Equivalence shown between two mathematical concepts for hyperbolic surfaces.
problem None explicitly stated, but related to mathematical equivalence of concepts.
method Benjamini-Schramm convergence and zeta functions equivalence demonstration.
result Equivalence of Benjamini-Schramm convergence and zeta functions for compact hyperbolic surfaces.
New hyperbolic 4-manifolds found with special functions.
problem Finding hyperbolic 4-manifolds with specific circle-valued Morse functions.
method Constructing hyperbolic 4-manifolds with only index 2 critical points.
result Existence of infinitely many hyperbolic 4-manifolds with bounded Betti numbers.
Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.
problem Establishing inequalities for radial functions on hyperbolic spaces without zero boundary conditions.
method Novel approach considering both bounded and unbounded domains, focusing on weighted Sobolev and Adams-Trudinger-Moser embeddings.
result Theorems 1.2, 1.3, and 1.4 for weighted Sobolev embedding theorems, and Theorems 1.5 and 1.6 for Adams-Trudinger-Moser type embedding theorems.
Study calculates mass of special polyhedra in hyperbolic space.
problem Evaluating mass in hyperbolic geometry.
method Used upper half space model and special polyhedra.
result Evaluated mass functional on polyhedra.
Unified definition of mass aspect function for weakly regular hyperbolic manifolds.
problem Ambiguity in mass definition for asymptotically hyperbolic manifolds.
method Introduced an ADM-style mass aspect function for broad asymptotics and low regularity.
result Unified mass aspect function exhibits favorable covariance properties.
We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce v…
A new learning method using hyperbolic geometry for class labels.
problem Class label representation and learning in machine learning.
method Hyperbolic Prototype Learning with a new loss function based on hyperbolic geometry.
result Hyperbolic Prototype Learning is equivalent to logistic regression in the one-dimensional case.
Study connects hyperbolic geometry to membrane shapes.
problem Understanding the shapes of biological membranes.
method Relates geometry of hyperbolic space to Helfrich model.
result Establishes a connection between membrane shapes and hyperbolic geometry.
Proves globally hyperbolic spacetimes via null distance completeness.
problem No Hopf-Rinow Theorem in Lorentzian Geometry.
method Observation of null distances and their behavior with time functions.
result Proves globally hyperbolic spacetimes via null distance completeness.
Establishes a lower bound for Kähler hyperbolicity modulus in hyperconvex domains and bounded strongly pseudoconvex domains.
problem Kähler hyperbolicity modulus for simply-connected Kähler hyperbolic manifolds
method Computes the Kähler hyperbolicity modulus for bounded symmetric domains
result Establishes a lower bound for the Kähler hyperbolicity modulus in terms of the boundary behavior of the gradient length of a plurisubharmonic function
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.
The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.
problem Bounding the Fridman function for hyperbolic manifolds.
method Analyzing the relationship between the Fridman function and the injectivity radius function.
result The Fridman function is bounded above by the injectivity radius function for certain hyperbolic manifolds.
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.
For hyperbolic 3-manifolds, Ruelle zeta function vanishing order is 4 minus Betti number.
problem Analyzing the Ruelle zeta function at zero for perturbed hyperbolic 3-manifolds.
method Microlocal approach to dynamical zeta functions, first variation, new identity relating pushforwards of resonant and coresonant forms.
result The order of vanishing of the Ruelle zeta function at zero equals 4 minus Betti number for generic perturbations.
Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.
We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric.
Potential theory extended to Gromov hyperbolic spaces.
problem Extending potential theory to a new class of spaces.
method Unified framework for Gromov hyperbolic metric measure spaces.
result Boundary Harnack inequalities and complete classification of positive harmonic functions.
Motivated by the results of Scott and Patel about "untangling" closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back …
Study on hyperbolic manifolds and their boundary data, focusing on volume functions.
problem Determining the hyperbolic metric from boundary data of convex co-compact hyperbolic manifolds.
method Analysis of volume functions and their relation to boundary data, using first variations.
result New connections with physics and probability theory, with open questions remaining.
Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.
problem Spectrum of spin Dirac operator on hyperbolic surfaces with pinched geodesics.
method Trace formula for Dirac operator, Huber's theorem, small-time heat trace asymptotic expansion.
result Convergence of Selberg zeta function for degenerating hyperbolic surfaces.
Study of twisted Ruelle zeta function on hyperbolic manifolds and its relation to analytic torsion.
problem Analyzing the twisted Ruelle zeta function on hyperbolic manifolds.
method Investigating the twisted Ruelle zeta function associated with geodesic flow and acyclic representations.
result The twisted Ruelle zeta function equals the square of the refined analytic torsion multiplied by an exponential involving the eta invariant.
The paper studies Teichmüller TQFT for hyperbolic knots, proving exponential decay of partition functions.
problem Analyzing Teichmüller TQFT for hyperbolic knots with generalized FAMED triangulations.
method Introducing generalized FAMED property, proving exponential decay of partition functions in semi-classical limit.
result Partition functions decay exponentially with the volume of knot complements, and the 1-loop invariant emerges.
Study on a Bahri-Brezis problem on hyperbolic manifolds.
problem Existence of solutions on asymptotically hyperbolic manifolds.
method Algebraic Topological argument of Bahri-Coron.
result Existence of at least one solution under specific conditions.
We establish a correspondence on a Riemann surface between hyperbolic metrics with isolated singularities and bounded projective functions whose Schwarzian derivatives have at most double poles and whose monodromies lie in PSU(1,1). As an application, we construct explicitly a new class of hyperbolic metrics …
These are lectures on discrete groups of isometries of complex hyperbolic spaces, aimed to discuss interactions between the function theory on complex hyperbolic manifolds and the theory of discrete groups.
The paper shows how coarse embeddings affect homological Dehn functions.
problem Characterizing groups with coarse embeddings into hyperbolic groups.
method Demonstrates a coarse embedding theorem for homological filling functions.
result Characterizes groups with coarse embeddings into hyperbolic groups of geometric dimension 2.
This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on hyperbolic groups to the hyperbolike setting, including rationality of generati…
We construct new explicit proper r-harmonic functions on the standard n-dimensional sphere S^n and hyperbolic space H^n for any r\ge 1 and n\ge 2.
The study finds bounds on minimal surfaces in hyperbolic 3-manifolds.
problem Finding bounds on areas of stable minimal surfaces in hyperbolic 3-manifolds.
method Ricci flow with surgery to find lower bounds on minimal surface areas.
result The hyperbolic metric maximizes a functional defined by asymptotic counts of minimal surfaces.
Paper extends tree bijection for hyperbolic surfaces without requiring cusps.
problem Computing volumes and distances on hyperbolic surfaces without cusps.
method Extend tree bijection to half-tight cylinders, using Busemann function.
result Tree bijection can now be applied to surfaces without cusps.
For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the inv…
Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.
problem Investigate functional and geometric inequalities on hyperbolic spaces and Riemannian manifolds.
method Employ symmetrization and semigroup approach based on sharp estimates for heat semigroup.
result Developed robust inequalities and methods relying on geometric and isoperimetric properties.
New spectral Dehn function characterizes word-hyperbolic groups.
problem Characterizing word-hyperbolic groups using spectral properties.
method Introducing spectral Dehn functions and proving inequalities relating them to Dehn functions.
result A spectral Dehn function characterizes word-hyperbolic groups.
Extends Milnor's criterion to biharmonic functions.
problem Deciding surface type for biharmonic functions.
method Generalizes Milnor's criterion to biharmonic functions.
result Characterizes whether a surface is hyperbolic or parabolic for biharmonic functions.
In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both …
Study asymptotics of Selberg zeta function on spin moduli space.
problem Asymptotic behavior of Selberg zeta function for degenerating metrics.
method Analyzes logarithmic derivative of Selberg zeta function for spin Dirac operator on compact surfaces.
result Proves asymptotic expansion up to order t4logt. We prove, in the case of hyperbolic 3-space, a couple of conjectures raised by J. J. Seidel in "On the volume of a hyperbolic simplex", Stud. Sci. Math. Hung. 21, 243-249, 1986. These conjectures concern expressing the volume of an ideal hyperbolic tetrahedron as a monotonic function of algebraic maps. More precisely, …
The paper explores rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
problem Rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
method Characterizations and rigidity investigations for hypersurfaces with constant weighted shifted mean curvatures or ratios.
result Rigidity results for hypersurfaces with constant linear combinations of weighted shifted mean curvatures and radially symmetric shifted mean curvatures.
New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.
problem Acceleration in hyperbolic spaces for strongly geodesically convex functions is impossible.
method Perturbing hard functions with sums of bump functions chosen by a resisting oracle.
result Acceleration is unachievable for any deterministic algorithm in hyperbolic spaces for strongly geodesically convex functions.
It is proved that the Hasse-Weil zeta functions of the canonical components of the SL2 (PSL2)-character varieties of closed orientable complete hyperbolic 3-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed 3-manifol…
Explicit bounds found for shortest orthogeodesics and volumes of hyperbolic manifolds.
problem Finding explicit bounds for shortest orthogeodesics and volumes of hyperbolic manifolds.
method Derived explicit estimates for functions related to volumes and orthospectra, using a new approach.
result Explicit lower bound for the length of the shortest orthogeodesic in terms of volume.
The chapter explores globally hyperbolic spacetimes using topology and functional analysis.
problem Understanding global hyperbolicity in spacetimes.
method Foundational tools from order theory and topology, geometric analysis, and connections to physics.
result A connection between global hyperbolicity and geodesic completeness of space-like surfaces.
Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.
problem Counting minimal Lagrangians in hyperbolic surfaces.
method Uses Mirzakhani functions to show growth rate and proves rigidity of area spectrum.
result Number of minimal Lagrangians grows as A6(k−1). We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.