Compact method proves Brown-York mass positivity and connects to major conjectures.
problem Proving positivity of Brown-York's mass and its connections to conjectures.
method Compact approach to proving mass positivity and exploring connections.
result Proved the positivity of Brown-York's mass and its relation to conjectures.
Proves positive energy conjecture for a specific metric class.
problem Proving the positive energy conjecture for a class of AHM metrics.
method Analyzes asymptotically Horowitz-Myers (AHM) metrics on R2imesTn−2. result Generalizes previous results on positive energy conjecture.
Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain homological condition is satisfied. We present a counterexample to this conjecture, based…
Study positive braid knots and their taut foliations, proving some L-space conjecture evidence.
problem Understanding positive braid knots and their properties through taut foliations.
method Construct taut foliations in specific 3-manifolds and use them to prove braid positivity and unknot detection.
result Prove some positive evidence towards the L-space conjecture and provide a braid positivity obstruction.
Study crystallographic groups for positive scalar curvature conditions.
problem Examining positive and negative results for Gromov-Lawson-Rosenberg Conjecture.
method Analyzing split extensions of free abelian by cyclic groups.
result Produce infinite counterexamples for the Gromov-Lawson-Rosenberg Conjecture.
Paper finds non-positive Weyl connections on Lie groups, confirming a conjecture.
problem Finding non-positive invariant Weyl connections on Lie groups.
method Investigation of completely solvable Lie groups, focusing on SOL group.
result Only SOL admits non-positive Weyl connections, confirming a conjecture.
Proves a 1930s Hopf conjecture about positive curvature manifolds.
problem Even-dimensional compact Riemannian manifolds with positive sectional curvature and high isometry rank.
method Reduces to a representation theoretic problem involving torus representations.
result Proves the Hopf conjecture under specific conditions.
Study verifies Homogeneity Conjecture for three odd-dimensional spheres in positive curvature.
problem Verifying the Homogeneity Conjecture for three specific odd-dimensional spheres in positive curvature.
method Developed methods to verify the conjecture for three odd-dimensional spheres.
result Completes verification of the Homogeneity Conjecture in positive curvature.
Paper proves Gromov's conjecture on manifolds with certain group properties.
problem Gromov's conjecture on positive scalar curvature and simplicial volume.
method Proves conjecture under a fundamental group decay property.
result Proves Gromov's conjecture for manifolds with a weakened rapid decay property.
Study proves a conjecture for certain spin manifolds.
problem Proving a conjecture about positive scalar curvature metrics.
method Connected sum construction and Gromov-Lawson area enlargement.
result Connected sum of spin manifolds admits no complete metric of positive scalar curvature.
Study shows some contractible complexes can't have certain immersions.
problem Understanding non-positive immersions in contractible complexes.
method Provided counterexamples to a conjecture by Wise.
result Some contractible complexes do not have non-positive immersions.
Simply-connected manifolds of positive sectional curvature M are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipt…
We prove new results about unknotting fibered positive knots and braids.
problem Proving the unknotting number equals genus for fibered positive knots and braids.
method Analyzing positive braid diagrams and fibered positive knots, proving new constraints and conjectures.
result We found fibered positive knots that cannot be unknotted optimally, contradicting Stoimenow's conjecture.
New non-perturbative counterexamples to Min-Oo's Conjecture are created.
problem Min-Oo's Conjecture on positive curvature and mass.
method Quantitative Gromov-Lawson Schoen-Yau surgery.
result Construction of new non-perturbative counterexamples with more complex topology.
This paper proves a conjecture about unique positive harmonic functions in a ball.
problem Proving the uniqueness of positive harmonic functions in a unit ball for specific parameters.
method Analyzing a partial differential equation to show the solution is constant.
result Guo-Wang's conjecture is proven for the specified parameters.
Study positive characteristic Fano 4-folds with nef tangent bundles.
problem Positive characteristic version of the Campana-Peternell conjecture for Fano 4-folds.
method Analyzes Fano 4-folds with nef tangent bundles in positive characteristic.
result Affirmative answer for Fano 4-folds with Picard number > 1 and nef tangent bundle.
Study a relative aspherical conjecture and prove 3-manifold obstruction to positive scalar curvature.
problem Obstructing the existence of positive scalar curvature in higher dimensions.
method Introduced a relative aspherical condition and a new geometric quantity called spherical width.
result Proved results on how 3-manifolds obstruct the existence of positive scalar curvature.
Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.
problem Hopf's conjecture on positive sectional curvature and its failure under relaxed conditions.
method Non-trivial extension of the Four Periodicity Theorem to higher degrees.
result Proves positive Euler characteristic for specific manifolds with positive second intermediate Ricci curvature.
Generalizes positivity conjecture to Roger--Yang skein algebras using polynomials.
problem Positivity conjecture for Roger--Yang skein algebras.
method Used explicit polynomials like Chebyshev polynomials of the first kind to give candidates of positive bases.
result Polynomials form a lower bound in the sense of [Lê18] and [LTY21].
Paper proves positivity of Chern-Weil forms for certain vector bundles.
problem Proving positivity of Chern-Weil forms for Griffiths semipositive vector bundles.
method Analyzing characteristic differential forms and Schur polynomials.
result Positivity of c1(E,h)∧c2(E,h)−c3(E,h) established. Falsehood of Pólya's conjecture for spheres shown.
problem Disproving Pólya's eigenvalue conjecture for spheres.
method Comparison of Laplace spectrum and Weyl function of spheres.
result No analogue of Pólya's conjecture holds for spheres.
Paper proves Fujimoto's conjecture for even m ≥ 4.
problem Proving Fujimoto's conjecture for even dimensions.
method Using a special planar network in the theory of positive matrices.
result Best possible number for even m ≥ 4 is proven.
Positive Quaternion Kaehler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. We prove this conjecture in dimension 20 under additional assumptions and we provide recognition theorems for quaternionic projective spaces …
Positive braids have a signature bound by their Betti number.
problem Bounding the signature of positive braids.
method Using the first Betti number as a lower bound for the signature.
result The signature is bounded from below by one-quarter of the first Betti number.
We study positivity in the conjecture proposed by Lejmi and Székelyhidi on finding effective necessary and sufficient conditions for solvability of the inverse σk equation, or equivalently, for convergence of the inverse σk-flow. In particular, for the inverse σn−1-flow we partially verify their conjecture b…
We give the first examples of rationally inessential but macroscopically large manifolds. Our manifolds are counterexamples to the Dranishnikov rationality conjecture. For some of them we prove that they do not admit a metric of positive scalar curvature, thus satisfy the Gromov positive scalar curvature conjecture. Fu…
In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…
The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new to…
3D manifold with positive curvature has a short geodesic.
problem Finding the shortest closed geodesic in 3-manifolds with positive scalar curvature.
method Analyzing the properties of geodesics in manifolds with positive scalar curvature.
result Existence of a non-trivial closed geodesic of length less than 22500.
The Besse's conjecture was posted on the well-known book Einstein manifolds by Arthur L. Besse, which describes the critical point of Hilbert-Einstein functional with constraint of unit volume and constant scalar curvature. In this article, we show that there is an interesting connection between Besse's conjecture and …
Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem.
problem Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. method Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. result Prove Gromov's Euclidean endpoint C0 rigidity conjecture for positive mass theorem. Proves a conjecture about Riemann surfaces using PDEs.
problem Griffiths' conjecture on holomorphic vector bundles on compact Riemann surfaces.
method Combines techniques from Uhlenbeck-Yau and Pingali's reduction to prove a system of PDEs.
result Analytic proof of Griffiths' conjecture on compact Riemann surfaces.
Paper proves conjecture about Einstein metrics on manifolds with positive isotropic curvature.
problem Proving the Besse conjecture for metrics with positive isotropic curvature.
method Analyzing the critical point equation and using properties of metrics with positive isotropic curvature.
result The Besse conjecture is true for metrics with positive isotropic curvature.
Proves a conjecture about manifolds and scalar curvature.
problem Determining when a manifold admits a positive scalar curvature metric.
method Uses a geometric bound to measure discrepancies between vector fields.
result Proves the conjecture in codimension two.
Positive knots are minimal in a specific knot ordering.
problem Understanding the minimal structure of positive knots in a specific knot ordering.
method Analyzing ribbon concordance and using properties of rational Alexander modules.
result Positive knots are minimal in the ribbon concordance ordering.
The paper confirms a conjecture about manifolds with positive curvature.
problem Estimating the width of manifolds with positive sectional curvature.
method Establishing an optimal Lipschitz lower bound for functions on manifolds.
result Characterization of doubly warped product metrics with positive constant curvature.
We establish metrics of positive 2nd-intermediate Ricci curvature, i.e. Ric2>0, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and Hsiang-Kleiner theorem for positively …
5 minimal tori found in 3-spheres with positive Ricci curvature.
problem Existence of at least 5 embedded minimal tori in 3-spheres with positive Ricci curvature.
method Combination of min-max theory and mean curvature flow heuristics.
result Confirms B. White's conjecture for positive Ricci curvature.
The paper proves a stronger Petersen--Wilhelm conjecture for principal bundles.
problem Conditions for positive sectional curvature submersion metrics on principal bundles.
method Cheeger deformations, good triples, Chaves-Derdzinski-Rigas type condition.
result Any principal bundle over a positively curved base admits a metric of positive sectional curvature if the submersion is fat.
The doubling conjecture for positive scalar curvature is proven under certain conditions.
problem Determining when a manifold with a specific boundary condition admits positive scalar curvature.
method Surgery techniques for positive scalar and mean curvature, and existence of area-minimizing hypersurfaces.
result The doubling conjecture holds true for manifolds with certain split conditions on fundamental groups.
Proves positive mass theorem for non-spin weighted manifolds.
problem Proving the positive mass theorem for non-spin weighted manifolds.
method Establishing density theorem and generalizing Geroch conjecture.
result Proves positive weighted mass theorem for non-spin weighted manifolds.
Proves a quantitative index theorem for positive scalar curvature metrics.
problem Studying conjectures and open questions on positive scalar curvature.
method Quantitative relative index theorem and λ-Lipschitz rigidity theorem. result Positive answers to Gromov's open questions on scalar curvature.
New curvature condition helps characterize Kähler manifolds.
problem Characterize compact Kähler manifolds with specific curvature properties.
method Introduce and utilize 2−positive bisectional curvature condition. result Deduce characterization theorem for manifolds with 2−positive bisectional curvature. The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf product conjecture asks whether a positive curvature metric can exist on product manifolds like S^2 x S^2. By formulating curv…
Study links' arc index and Turaev genus, proving conjectures.
problem Understanding the arc index and Turaev genus of links.
method Computed arc index, established bounds, and conjectured inequalities.
result Proved conjectures linking crossing number, arc index, and Turaev genus.
The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.
problem Proving Liouville-type theorems for positive harmonic functions on specific types of manifolds.
method Employing the P-function method and a closed conformal vector field inherent to such manifolds.
result Confirms some cases of Wang's conjecture and provides a partial verification of Wang's conjecture on warped product manifolds.
New method finds large curved subcomplexes, proving conjectures for specific groups.
problem Proving the K(π,1)-conjecture for Artin groups.
method Finding large non-positively curved subcomplexes in spherical Deligne complexes.
result Proves K(π,1)-conjecture for many Artin groups, except one.
Solved Gromoll-Walschap's conjecture on negative curvature manifolds.
problem Existence of Riemannian foliations on compact manifolds of negative curvature.
method Provided a proof for the conjecture.
result Gromoll-Walschap's conjecture is proven to be true.