New finding links hyperbolic manifold systolic volume to triangulation complexity.
problem Understanding the relationship between systolic volume and triangulation complexity in hyperbolic manifolds.
method Proof based on Jørgensen and Thurston's theorem of hyperbolic volume.
result Systolic volume of hyperbolic manifolds is related to triangulation complexity.
Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
Proves volume conjecture for twist knots using complex analysis.
problem Volume conjecture for twist knots.
method Equivalence relation, complex analysis, analytic continuation, function of several complex variables.
result Proves volume conjecture for twist knots.
Minimal volume entropy vanishes or is positive under certain fiber growth conditions.
problem Conditions for minimal volume entropy to be zero or positive.
method Analyzes topological conditions related to fiber growth of maps.
result Examples of finite simplicial complexes with zero simplicial volume and large minimal volume entropy.
The study examines conditions for minimal volume entropy of simplicial complexes.
problem Conditions for minimal volume entropy of simplicial complexes.
method Topological conditions and growth of fundamental groups.
result Examples of simplicial complexes with zero simplicial volume and large minimal volume entropy.
The paper proves an equality linking torsion, volume, and a product for hyperbolic 3-manifolds.
problem Understanding the volume of hyperbolic 3-manifolds.
method Using Reidemeister torsion and Zograf's infinite product.
result Proves an equality involving torsion, volume, and Zograf's product.
Estimates Kähler metrics with noncollapsing volume under complex Monge-Ampère constraints.
problem Volume noncollapsing for Kähler metrics induced by complex Monge-Ampère equations.
method Proves local volume noncollapsing estimate with Ricci curvature lower bound.
result Establishes diameter and gradient estimates for Kähler metrics.
In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depen…
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…
The paper proves an equality involving torsion, volume, and a product for hyperbolic 3-manifolds.
problem Understanding the geometry and topology of hyperbolic 3-manifolds with cusps.
method Using Reidemeister torsion, complex volume, and Zograf's infinite product.
result Proves an equality linking these mathematical concepts.
We compare the volume of a hyperbolic 3-manifold M of finite volume and the complexity of its fundamental group.
Study intrinsic volume forms on complex hypersurfaces.
problem Computing volume functionals on pseudoconvex hypersurfaces.
method Compute first and second variation formulae, explore infinite dimensional aspects.
result Discuss possible analogues of the affine isoperimetric inequality.
New inequality linking geodesic length and volume in complex projective plane.
problem Understanding geometric properties of complex projective plane.
method Combining recent results on area minimizers and geodesics with Kronheimer-Mrowka's proof.
result Proved a new inequality relating volume and length of geodesics.
The paper shows that certain bundles have unique volumes.
problem Volume rigidity of specific circle bundles.
method Proving volume rigidity for principal circle bundles over complex projective spaces.
result Principal circle bundles are volume rigid among K-contact manifolds. The paper introduces a new complex analytic invariant called the pointed harmonic volume and its relation to the Johnson homomorphism.
problem Exploring new complex analytic invariants related to the complex structure of Riemann surfaces.
method Defining and computing the pointed harmonic volume as a natural extension of Chen's iterated integrals.
result Established a relationship between the harmonic volume and the first extended Johnson homomorphism.
We try to give a cluster algebraic interpretation of complex volume of knots. We construct the R-operator from the cluster mutations, and we show that it is regarded as a hyperbolic octahedron. The cluster variables are interpreted as edge parameters used by Zickert in computing complex volume.
Estimates simplicial volume for complex hyperbolic surfaces.
problem Bounding the simplicial volume of complex hyperbolic surfaces.
method Estimates Gromov norm and uses top dimensional class in Hc4. result Explicit upper bound for simplicial volume.
In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, N…
The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We e…
We construct an explicit lower bound for the volume of a complex hyperbolic orbifold that depends only on dimension.
The study identifies Hermitian metrics preserving the total Monge-Ampere volume.
problem Understanding Hermitian metrics preserving volume invariance.
method Characterizations and comparison principles for complex Monge-Ampere operator.
result Several characterizations of Hermitian metrics satisfying the comparison principle.
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.
The study links Ricci curvature and convexity in complex tori.
problem Characterizing Ricci curvature signs in toric manifolds.
method Characterization through convexity of volume functional.
result Relationships between Ricci curvature, volume, submanifolds, and pluri-subharmonic functions.
Essential minimal volume bounds for Einstein 4-manifolds.
problem Bounding the essential minimal volume of Einstein 4-manifolds.
method Introduced a new volume concept, essential minimal volume, and showed bounds for closed Einstein 4-manifolds.
result Closed Einstein 4-manifolds satisfy linear bounds on essential minimal volume.
Estimates Schwarzian derivative on long complex projective tubes.
problem Behaviour of Schwarzian derivative on complex projective structures.
method Analyzes Schwarzian derivative on long complex projective tubes, estimating its pairing with infinitesimal earthquakes and graftings.
result Obtains bounds for the variation of renormalized volume under complex earthquake paths and its asymptotic behavior under pinching.
For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern--Gauss--Bonnet formula. Relations between…
Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.
Study volumes of Bott-Chern classes on complex manifolds.
problem Understanding volumes of transcendental Bott-Chern classes.
method Extending non-pluripolar products to quasi-positive currents, establishing quasi-monotonicity of Monge-Ampère masses, and solving degenerate complex Monge-Ampère equations.
result Positive answer to Demailly-Păun-Boucksom conjecture regarding bounded mass property.
The paper proves complex Monge-Ampère equations with new singularity types and confirms log-concavity of volume.
problem Existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularities.
method Proves existence and uniqueness of solutions to complex Monge-Ampère equations with general model type singularities in big cohomology classes.
result Log-concavity of volume of closed positive (1,1)-currents is confirmed.
Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he re…
Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.
problem Characterize minimal volume entropy for aspherical simplicial complexes with these groups as fundamental groups.
method Algebraic and geometric characterization, using fiber π1-growth collapse and non-collapsing assumptions. result Provide bounds and criteria for minimal volume entropy in aspherical simplicial complexes.
Complex b-6j symbols relate to hyperbolic tetrahedron volumes and determinants.
problem Analyzing asymptotics of complex b-6j symbols. method Relating asymptotics to hyperbolic tetrahedron volumes and determinants.
result Complex b-6j symbols' asymptotics linked to tetrahedron volumes and determinants. The volume conjecture is extended to all orders for hyperbolic 3-manifolds using complex Chern-Simons theory.
problem Extending the volume conjecture to all orders for hyperbolic 3-manifolds.
method Deriving formulas for the perturbative expansion of the partition function of complex Chern-Simons theory and comparing it to Witten-Reshetikhin-Turaev invariants.
result The conjecture that the perturbative expansion of the partition function of complex Chern-Simons theory matches the Witten-Reshetikhin-Turaev invariants at roots of unity in the limit of infinitely many invariants.
We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.
We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a co…
The paper proves bounded domains are biholomorphic to the unit ball.
problem Covering manifolds with finite volume.
method Analyzes C2 and C1,ε boundary conditions for bounded domains. result Bounded domains are biholomorphic to the unit ball under certain conditions.
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group G⊂PU(n,1) acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…
The paper defines a volume form on moduli spaces of meromorphic differentials.
problem Defining a volume form on moduli spaces of meromorphic differentials.
method Shows the existence of a canonical volume form on these spaces that is parallel.
result The total volume of the projective space of these spaces is finite.
Efficiently samples arbitrary compact bodies with polynomial complexity.
problem Uniform sampling from arbitrary compact bodies efficiently.
method Warm start algorithm under isoperimetry and volume growth condition.
result Substantial generalization of known results for convex and star-shaped bodies.
Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the …
Finite simplicial complexes dominate certain manifolds with a bounded number of simplices.
problem Understanding the finite domination of manifolds by simplicial complexes.
method Proving that a manifold can be dominated by the n-skeleton of a finite simplicial complex with a bounded number of simplices. result The total number of simplices in the n-skeleton is bounded above by a constant depending only on n and the embolic volume of the manifold. Study bounds on Monge-Ampère volumes for degenerate complex equations.
problem Bounds on volumes of Monge-Ampère measures for degenerate complex equations.
method Fine use of quasi-plurisubharmonic envelopes.
result Established a transcendental version of the Grauert-Riemenschneider conjecture.
The study solves open problems in complex geometry by analyzing bounded domains with finite-volume quotients.
problem Analyzing bounded pseudoconvex domains with finite-volume quotients in complex geometry.
method Using semi-simplicity of automorphism groups and applying results to specific settings.
result The automorphism group of certain domains is discrete, and domains with specific properties are biholomorphic to the unit ball.
The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another complex analytic invariant called the harmonic volume. It is a natural extension o…
Improved lower bounds on 2-bridge link complexity.
problem Finding minimal triangulations of 2-bridge link complements.
method Explicit angle structures and volume estimates.
result Explicit lower bounds on link complexity.
Constructs simplified or complexified simplicial complexes.
problem Efficiently simplifying or complexifying complex spaces.
method Embeddings of simplicial complexes into a simplicial ball with bounded degrees and low volume.
result Realizes complicated spaces as parts of a ball/sphere or gives spheres specific metrics.
We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to assume furthermore that the Kronheimer-Mrowka invariant is non vanishing.
This paper tackles target-dependent label complexity gap in active learning.
problem Target-dependent label complexity gap in Agnostic Active Learning.
method Introduces a novel distribution-splitting strategy based on number density to reduce label complexity and error rate.
result Provides theoretical guarantees and practical advantages for reducing label complexity and error rate.