Paper solves a conjecture about minimal surfaces using sphere intersections and Weierstrass data.
problem Solving the Fraser-Li conjecture for minimal surfaces.
method Using the Weierstrass representation formula and sphere intersections.
result The conjecture can be translated into problems about the Gauss map.
The paper generalizes a Steklov eigenvalue inequality for substatic triples under non-negative Ricci curvature.
problem Estimating Steklov eigenvalues for substatic triples under non-negative Ricci curvature.
method Generalization of Fraser-Li type inequality for substatic triples under non-negative Ricci curvature associated with an affine connection.
result The paper provides a new inequality for Steklov eigenvalues of substatic triples.
Compactness proven for CMC surfaces with bounded topology and boundary length.
problem Proving compactness of CMC surfaces with specific constraints.
method Graphical Ck compactness proof for surfaces with bounded topology, area, and boundary length. result Space of free boundary CMC surfaces is compact in the Ck graphical sense away from a finite set of points. Strict convexity is essential for compact minimal surfaces in curved spaces.
problem Conditions for compact minimal surfaces in curved spaces.
method Analysis of minimal surfaces in curved manifolds with free boundaries.
result Strict convexity of the boundary is necessary for compact minimal surfaces.
Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of…
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.
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…
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…
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.
We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum-Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which tog…
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.
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 \(n=4\).
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.
In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open Question 1 in 2016. In this paper we discuss further these conjectures and put forward several new conjectures which will be shown equivalent to Conjecture 2. In particula…
In this paper, we generalize the Cosmetic Surgery Conjecture to an n-cusped hyperbolic 3-manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For n=1 and 2, we show them without the assumption.
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. 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.
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.
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.
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.
We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…
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.
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.
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…
There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Propert…
Paper proves contractible fake surfaces up to complexity 6 are deformable.
problem Stable Andrews-Curtis conjecture and contractible fake surfaces.
method Induction scheme proving contractibility up to complexity 6.
result Contractible fake surfaces up to complexity 6 are 3-deformable.
Paper analyzes Bartnik's quasi-local mass conjectures and their validity.
problem Understanding the validity of Bartnik's quasi-local mass conjectures.
method Developed a framework to analyze Bartnik's static vacuum extension conjecture.
result Proved the static vacuum extension conjecture is not true in general.
This memoire consists of two main results. In the first one we describe Ricci flow theory and we give an educative way for proving Elliptization Conjecture and then we prove Poincare conjecture which is the second proof of Perelman for Poincare conjecture. In the second one which is the main propose of our memoire, we …
Proves Stolz conjecture for specific types of manifolds.
problem String manifolds with positive Ricci curvature and Witten genus.
method Analyzes toric string Fano manifolds and string torus manifolds with invariant metrics.
result Proves conjecture for specified types of manifolds.
In this paper we propose counterexamples to the Geometrization Conjecture and the Elliptization Conjecture.
Established a version of the Atiyah-Floer conjecture for SO(3)-bundles.
problem Atiyah-Floer conjecture for admissible SO(3)-bundles
method Adapted to admissible SO(3)-bundles
result A version of the Atiyah-Floer conjecture established
The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope conjecture to links. We prove the conjecture for all alternating and more generally adequ…
Upper bound conjecture for Yokota invariant proved for polyhedral graphs.
problem Growth of Yokota invariant of polyhedral graphs
method Barrett's Fourier transform
result Proved upper bound conjecture for large family of examples
Bianchi proves Mumford conjecture using branched covers.
problem Mumford conjecture in algebraic topology.
method Moduli spaces of branched covers.
result Proof of Mumford conjecture.
Paper proves Horowitz-Myers conjecture in 3-7 dimensions.
problem Proving Horowitz-Myers conjecture in specific dimensions.
method Alternative proof and local isometry demonstration.
result Metrics achieving conjecture equality locally match Horowitz-Myers metrics.
The paper proves the Singer conjecture for aspherical complex surfaces and refines Gromov's inequality.
problem Proving the Singer conjecture and refining Gromov's inequality for aspherical complex surfaces.
method Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group, and refinement of Gromov's inequality for non-residually finite surfaces.
result Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group.
Analytic proof for minimal rank Sard conjecture.
problem Proving the minimal rank Sard conjecture in the analytic category.
method Using subanalytic abnormal distribution from [4], we establish a proof.
result The set of points accessible through singular horizontal curves of minimal rank has Lebesgue measure zero.
In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.