Study on a weighted Suita conjecture for higher derivatives and their geometric properties.
problem Analyzing the Suita conjecture for higher derivatives with weights.
method Examining the set of points for equality in a weighted Suita conjecture and relating it to harmonic functions and Dirichlet problems.
result Relations between the set of points and integer-valued points of harmonic functions and Dirichlet problems for planar domains.
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
New proof of Suita's conjecture without L2 extension theorem.
problem Equality in Suita's conjecture for open Riemann surfaces.
method Exploration of Maitani and Yamaguchi's variation formula for the Bergman kernel and harmonicity.
result Characterization of surfaces by constant Gaussian curvature properties of the Bergman kernel or metric.
In this paper, we solve the optimal constant problem in the setting of Ohsawa's generalized L2 extension theorem. As applications, we prove a conjecture of Ohsawa and the extended Suita conjecture, we also establish some relations between Bergman kernel and logarithmic capacity on compact and open Riemann surfaces…
The Bergman kernel's minimal point determines domain properties.
problem Characterizing domains based on the Bergman kernel's minimal point.
method Using Suita conjecture and properties of Bergman and Szegö kernels.
result The Bergman kernel's minimal point identifies specific domain types.
Polyhedra volume conjecture supports Stoker conjecture weakly.
problem Proving the Stoker conjecture for polyhedra.
method Using an extension of Montcouquiol and Weiss' result on polyhedra angles as local coordinates.
result Volume Conjecture for polyhedra implies a weak version of the Stoker conjecture.
Survey on two non-Kähler geometry conjectures.
problem Constant holomorphic sectional curvature and Fino-Vezzoni conjectures in non-Kähler geometry.
method Survey and discussion of historical and recent developments.
result Discussion of conjectures without new results.
The paper generalizes a surgery conjecture and proves it under the Zilber-Pink conjecture.
problem Generalizing the Cosmetic Surgery Conjecture to n-cusped hyperbolic 3-manifolds. method Proves the generalized conjecture under the assumption of the Zilber-Pink conjecture, and without assuming it for n=1 and 2. result Proves the generalized Cosmetic Surgery Conjecture for n=1 and 2 without assuming the Zilber-Pink conjecture. Numerical study confirms Brennan's conjecture for a counterexample to Thurston's K=2 conjecture.
problem Thurston's K=2 conjecture and Brennan's conjecture in planar domains. method Numerical analysis of a specific counterexample to Thurston's conjecture.
result The counterexample does not contradict Brennan's conjecture.
The non-vanishing conjecture implies the abundance conjecture in certain cases.
problem Abundance conjecture in algebraic geometry.
method Proof of the abundance conjecture under specific conditions.
result The abundance conjecture holds in dimensions ≤ 5 when κ ≥ 0 and ν ≤ 1.
Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
problem Gromov's flat corner domination conjecture and Stoker's conjecture for convex polyhedra.
method Same techniques applied to prove conjectures.
result Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…
The Burghelea conjecture is proven for many groups, but not all, with counter-examples provided.
problem Computing the periodic cyclic homology of complex group rings.
method Analyzing groups of finite asymptotic dimension and constructing counter-examples.
result The Burghelea conjecture holds for many classes of groups but not for all, with specific counter-examples provided.
Paper discusses conjectures and proves some related inequalities.
problem Unified generalization of BW and DDVV inequalities.
method Unified discussion and proofs of conjectures.
result Proves Conjecture 2 and obtains a new upper bound for Conjecture 3.
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…
Survey on topological rigidity problems and related conjectures.
problem Topological rigidity of closed aspherical manifolds.
method Review of recent results and open problems.
result Status and implications of Farrell-Jones Conjectures.
Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
problem Integrality structure of framed knots' quantum invariants.
method Explicit formulas of colored HOMFLY-PT invariants of torus knots, verified in a limit form for any framed knots.
result Proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
New proof shows most thin knots satisfy Cabling Conjecture.
problem Cabling Conjecture for thin knots using Heegaard Floer homology.
method Heegaard Floer homology and immersed curves techniques.
result Almost all thin knots satisfy the Cabling Conjecture.
The study confirms a conjecture for a specific type of knot.
problem Determining the topology of essential surfaces in knot theory.
method Analyzing twisted, generalized Whitehead doubles of knots.
result Twisted, generalized Whitehead doubles satisfy the Slope and Strong Slope Conjectures.
Equivalent conjectures proved for group presentations.
problem Balanced presentations of the trivial group.
method Equivalence of cyclic and satellite versions, restrictive cancellative version, stabilizations consideration.
result Original conjecture equivalent to cyclic version.
Metric SYZ conjecture proved using non-archimedean geometry.
problem Proving the metric SYZ conjecture in large generality.
method Using non-archimedean geometry to support the conjecture.
result Metric SYZ conjecture can be proved in large generality.
Proved Burghelea Conjecture for specific groups.
problem Proving Burghelea Conjecture for certain groups.
method Used additional cohomological properties to prove the conjecture.
result Proved Burghelea Conjecture for specified groups.
Proves Simon's conjecture for 3-manifolds.
problem Simon's conjecture for 3-manifolds.
method Proof for 3-manifolds.
result Proved Simon's conjecture for 3-manifolds.
Paper confirms Chen's biharmonic conjecture for hypersurfaces in 5D.
problem Chen's conjecture on biharmonic submanifolds in Euclidean spaces.
method Analyzes hypersurfaces in \(\mathbb{R}^5\) for \(n=4\).
result Chen's conjecture is confirmed for hypersurfaces in \(\mathbb{R}^5\) when \(n=4\).
Proved Farrell-Jones conjecture for free-by-cyclic groups.
problem Proving the Farrell-Jones conjecture for a specific class of groups.
method Geometric methods for establishing the Farrell-Jones Conjecture.
result Proved the Farrell-Jones conjecture for free-by-cyclic groups.
Counterexample disproves recent Penrose conjecture variant.
problem A variant of the Penrose conjecture.
method Provided a counterexample.
result The conjectured variant of the Penrose inequality is false.
In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.
Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
Proves isomorphism conjecture for braid groups.
problem Proving isomorphism conjecture for braid groups.
method Fibered Isomorphism Conjecture of F. T. Farrell and L. E. Jones.
result Proved K and L theoretic versions of the conjecture. Survey on Novikov conjecture and its applications.
problem Novikov conjecture in topology.
method Survey and recent developments.
result Applications to topological rigidity and non-rigidity.
Paper connects AJ conjecture and colored Jones polynomial potential function.
problem Relationship between A-polynomial and colored Jones polynomial. method Connects AJ conjecture and colored Jones polynomial potential function.
result Establishes connection between A-polynomial and colored Jones polynomial potential function. Paper disproves conjectures about hyperbolic knots.
problem Conjectures about hyperbolic knots and their properties.
method Analyzes and refutes several conjectures about hyperbolic knots.
result Proves the conjecture about hyperbolic knots contradicts other plausible conjectures.
Counterexample disproves conjectures about log canonical thresholds.
problem Conjectures about log canonical thresholds were disproved.
method Provided a counterexample to both conjectures.
result Conjectures about log canonical thresholds are false.
Study confirms conjecture on Hermitian manifolds with bounded mass.
problem Morse-type integrals in nef (1,1) classes on compact Hermitian manifolds with bounded mass.
method Analyzes conjecture using bounded mass property on compact Hermitian manifolds.
result Confirms Demailly-Păun and Tosatti-Weinkove's conjectures.
Akbulut and Kirby conjectured that two knots with the same 0-surgery are concordant. In this paper, we prove that if the slice-ribbon conjecture is true, then the modified Akbulut-Kirby's conjecture is false. We also give a fibered potential counterexample to the slice-ribbon conjecture.
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
Reformulated Markov's conjecture in combinatorial terms.
problem Markov's uniqueness conjecture in integral necklaces.
method Geometric reformulation and combinatorial description.
result Explicitly described set of lengths on modular torus.
Proves Vol-Det Conjecture for many alternating links using Mahler measures.
problem Vol-Det Conjecture relating hyperbolic link volumes and determinants.
method Exact computations of Mahler measures of two-variable polynomials.
result Proves Vol-Det Conjecture for many infinite families of alternating links.
Exploring conjectures in constrained Willmore problem.
problem Understanding the Willmore functional over compact surfaces.
method Analyzing conjectures from partial results and numerical experiments.
result Ramifications for deeper understanding of the Willmore functional.
Verifies a conjecture for the figure eight knot.
problem Relates A-ideal and recurrence ideal of knots.
method Uses quantum A-ideals, q-holonomicity, and AJ conjecture.
result Strong AJ conjecture verified for figure eight knot.
Review of Yau's conjecture on zero sets of Laplace eigenfunctions.
problem Yau's conjecture on zero sets of Laplace eigenfunctions.
method Discussion of old and new results and methods related to the conjecture, including solutions and new results in smooth settings.
result Discussion of Donnelly and Fefferman's solution of the conjecture in the real-analytic Riemannian manifold case and new results in the smooth setting.
Proves Gromov's conjecture on metric operators.
problem Non-free isometric immersions and metric operators.
method Proof of a conjecture.
result Infinitesimal invertibility of metric inducing operators proven.
Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.
problem Fundamental group of nonnegative curvature manifolds.
method Observation in dimension 4.
result Fukaya-Yamaguchi conjecture holds in 4D.
Proved a combinatorial conjecture in machine learning.
problem None explicitly stated in the abstract.
method Binomial and multinomial sums identities.
result Proved a combinatorial conjecture.
The Strong Slope Conjecture is proven for specific knot types.
problem Proving the Strong Slope Conjecture for various knot types.
method Using connect sums and cabling, the conjecture is shown to be closed under these operations.
result The Strong Slope Conjecture is established for graph knots.
Verify knot polynomial conjectures for specific knots.
problem Relate knot polynomial degree to surfaces in knot complements.
method Analyze 3-string Montesinos knots under specified conditions.
result Verify Slope and Strong Slope Conjectures for these knots.
In their study of the Yamabe problem in the presence of isometry group, Hebey and Vaugon announced a conjecture. This conjecture generalizes Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey--Vaugon conjec…