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.
Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.
problem Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
method Show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal and that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature.
result Partial affirmative answers to Chen's conjecture, generalized Chen's conjecture in hyperbolic spaces, and Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.
Researchers confirm conjecture for complex nilmanifolds in higher dimensions.
problem Confirming the conjecture for compact Hermitian manifolds with constant holomorphic sectional curvature.
method Focused on complex nilmanifolds, proving the conjecture for these specific manifolds.
result The conjecture is confirmed for complex nilmanifolds in higher dimensions.
The Homogeneity Conjecture explores if constant displacement isometries imply homogeneous spaces.
problem Whether constant displacement isometries on a manifold imply its homogeneity.
method Survey and verification of cases, including new results.
result New results and open problems suggested.
The article confirms a complex geometry conjecture for a specific type of manifold.
problem Compact Hermitian manifolds with constant holomorphic sectional curvature.
method Restricting to pluriclosed manifolds and confirming the conjecture for Strominger Kähler-like manifolds.
result The conjecture is confirmed for a specific type of Hermitian manifold.
New findings on Chern's conjecture for Dupin hypersurfaces.
problem Chern's conjecture for hypersurfaces with constant scalar curvature.
method Combining topology and geometry to reduce assumptions in algebraic arguments.
result Closed proper Dupin hypersurfaces with specific conditions are isoparametric.
The study proves constant-curvature analogues of hot spots conjecture for triangles.
problem Proving the hot spots conjecture in constant curvature domains.
method Analyzing geodesic triangles of constant negative curvature and using Killing fields.
result First mixed Dirichlet-Neumann Laplace eigenfunctions have no non-vertex critical points in constant curvature triangles.
The paper confirms a conjecture for Bismut torsion parallel metrics.
problem The existence of metrics with constant holomorphic sectional curvature on non-Kähler manifolds.
method Investigation of Bismut torsion parallel metrics.
result The conjecture is confirmed for all non-balanced BTP manifolds.
The paper confirms a conjecture about Hermitian manifolds with constant mixed curvature.
problem Compact Hermitian manifolds with non-zero constant mixed curvature must be Kähler.
method Verification for specific types of Hermitian manifolds including complex nilmanifolds, solvmanifolds, and Lie algebras.
result Partial evidence supporting Kai Tang's conjecture.
Proves ropelength conjecture for alternating knots.
problem Determining the minimum ropelength of alternating knots.
method Proved the conjecture using a constant b0 and knot properties. result Ropelength of any alternating knot is at least proportional to its crossing number.
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
We prove that proper biharmonic hypersurfaces with constant scalar curvature in Euclidean sphere S5 must have constant mean curvature. Moreover, we also show that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space E5 or hyperbolic space H5, …
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…
This paper generalizes biharmonic Riemannian submersions to higher dimensions.
problem Classifying biharmonic Riemannian submersions from manifolds with constant sectional curvature.
method Constructing an adapted orthonormal frame to simplify the biharmonic equation and analyzing curvature properties.
result A Riemannian submersion is biharmonic if and only if it is harmonic from an (n+1)-dimensional manifold with constant sectional curvature to an n-dimensional manifold. Paper disproves conjecture about log-Sobolev constants.
problem Log-Sobolev constants and curvature bounds.
method Counterexample on birth-death chains.
result Conjecture about Ollivier curvature is incorrect.
Paper proves biharmonic hypersurfaces in nonzero space form have constant mean curvature.
problem Proving constant mean curvature for biharmonic hypersurfaces.
method Careful analysis of Gauss and Codazzi equations.
result Positive answer to Balmus-Montaldo-Oniciuc's conjecture for four dimensional hypersurfaces.
Paper proves a conjecture about minimal hypersurfaces in spheres.
problem Proving a conjecture about the second gap of minimal hypersurfaces with constant scalar curvature.
method Analyzing the squared norm of the second fundamental form of minimal hypersurfaces in spheres.
result Proves the Chern conjecture about the second gap of minimal hypersurfaces in spheres.
Two-layer neural networks need more neurons to be robust.
problem Understanding the robustness of two-layer neural networks and the role of overparametrization.
method Investigation of the tradeoffs between network size and robustness, using Lipschitz constant as a measure.
result A conjecture that robustness requires overparametrization, with precise bounds for different cases.
We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and…
Minimal surfaces help prove a conjecture about special metrics.
problem Proving Arthur L. Besse's conjecture about CPE metrics.
method Using the theory of minimal surfaces.
result The conjecture is proven for 3D manifolds.
We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity.
We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.
The Han-Li conjecture states that: Let (M,g0) be an n-dimensional (n≥3) smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and c be any real number, then there exists a conformal metric of g0 with scalar curvature 1 and boundary mean curvature c. Combining…
For a closed hypersurface Mn⊂Sn+1(1) with constant mean curvature and constant non-negative scalar curvature, the present paper shows that if tr(Ak) are constants for k=3,…,n−1 for shape operator A, then M is isoparametric. The result generalizes the theorem of d…
Let Mn be a biharmonic hypersurface with constant scalar curvature in a space form Mn+1(c). We show that Mn has constant mean curvature if c>0 and Mn is minimal if c≤0, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and…
Lu conjecture proven for minimal 2-spheres and surfaces under certain conditions.
problem Discreteness of constant scalar curvatures of compact minimal submanifolds in unit spheres.
method Refined Simons' first gap theorem and Yau's theorems for high-codimensional submanifolds.
result Lu's conjecture for minimal 2-spheres and surfaces proved under inequality conditions.
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…
Upper bound for total mean curvature of spin fill-ins is proven.
problem Bounding the total mean curvature of spin Riemannian manifolds.
method Proving Gromov's conjecture for spin manifolds with specific conditions.
result Explicit upper bounds for total mean curvature are derived under various conditions.
Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.
problem Negative curvature manifolds and their volume classes.
method Integration of volume forms and isoperimetric constants.
result Vanishing of bounded volume class implies positivity of Cheeger constant and vice versa.
Quantum modularity proved for a knot manifold.
problem Proving quantum modularity for a specific closed hyperbolic 3-manifold.
method Using factorization of state integrals and proving quantum modularity for functions and q-series. result Quantum modularity for the closed manifold provides a unification of volume conjecture and Witten's asymptotic expansion conjecture.
Totally geodesic minimal hypersurfaces in H5 with specific curvature properties.
problem Characterizing minimal hypersurfaces in hyperbolic space with certain curvature conditions.
method Analyzing properties of minimal hypersurfaces in H5 with constant scalar curvature and zero Gauss-Kronecker curvature. result Any complete minimal hypersurface in H5 with constant scalar curvature and zero Gauss-Kronecker curvature is totally geodesic. The simplest non-collision solutions of the N-body problem are the "relative equilibria", in which each body follows a circular orbit around the centre of mass and the shape formed by the N bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured tha…
New stability concept for Poisson structures leads to constant curvature metrics.
problem Finding constant scalar curvature metrics in generalized Kähler geometry.
method Introducing Poisson K-stability and using infinite-dimensional momentum map techniques.
result Existence of constant scalar curvature symplectic generalized Kähler structures on Kähler-Einstein Fano manifolds.
Authors construct hypertori with constant negative mean curvature in a sphere.
problem Constructing constant mean curvature hypertori in a sphere.
method Constructing two different constant mean curvature (2n−1)-dimensional hypertori in a 2n-dimensional sphere. result Two different constant mean curvature (2n−1)-dimensional hypertori with negative mean curvature in a 2n-dimensional sphere. For biharmonic maps, there is a famous conjecture named Chen's conjecture. In later paper, Wang and Ou gave an affirmative partial answer to submersion version of Chen's conjecture. In this paper, we give an affirmative partial answer to submersion version of generalized Chen's conjecture, that is, triharmonic Riemanni…
The article confirms a conjecture for solvmanifolds with complex commutator.
problem Confirming a conjecture about compact Hermitian manifolds with constant holomorphic sectional curvature.
method Analyzing solvmanifolds with complex commutator, extending results on nilmanifolds.
result The conjecture is confirmed for all solvmanifolds with complex commutator.
The paper tackles Kakeya and Nikodym sets on curved manifolds, reducing problems to Euclidean space.
problem Analyzing Kakeya and Nikodym sets on curved manifolds.
method Reduction of problems on curved manifolds to Euclidean space, using Bourgain's condition and recent breakthroughs.
result Establishes the Nikodym conjecture for three-dimensional manifolds with constant sectional curvature.
Classifies invariant hypersurfaces with singularities.
problem Classifying invariant hypersurfaces with singularities.
method Analyzing O(p)imesO(q)-invariant constant mean curvature hypersurfaces. result Solved Wu-yi Hsiang's conjecture.
The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.
problem Understanding the rotational symmetry of specific shrinking gradient Yamabe solitons.
method Analyzing nontrivial complete shrinking gradient Yamabe solitons with bounded scalar curvature.
result The assumption of bounded scalar curvature and strict inequality at some point is necessary and sufficient for rotational symmetry.
We address the issue of strong cosmic censorship for T^2-symmetric spacetimes with positive cosmological constant. In the case of collisionless matter, we complete the proof of the C^2 formulation of the conjecture for this class of spacetimes. In the vacuum case, we prove that the conjecture holds for the special case…
Minimal hypersurfaces are the only H-tensional in 4D space forms.
problem Classifying H-tensional hypersurfaces in 4D space forms. method Investigation of H-tensional hypersurfaces in 4-dimensional space forms of constant sectional curvature. result Minimal hypersurfaces are the only H-tensional hypersurfaces in 4D space forms. Effective Yau-Tian-Donaldson conjecture for spherical varieties.
problem Finding effective K-stability criteria for spherical varieties.
method Formulated an effective variant of the Yau-Tian-Donaldson conjecture and reviewed effective K-stability criteria for spherical varieties.
result Effective K-stability criteria can be computed given combinatorial data.
We solve two classical conjectures by showing that if an action of a connected Lie group on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations.
Proves log-concavity of cluster algebra coefficients for type An.
problem Log-concavity of cluster algebra coefficients.
method Introduced atomic theta basis and proved log-concavity for type An. result Proved log-concavity of coefficients for cluster algebra variables of type An. Introduces Poisson K-stability for Kähler manifolds and proves existence of constant scalar curvature structures.
problem Stability conditions for Poisson structures on Kähler manifolds.
method Infinite-dimensional momentum map techniques.
result Existence of constant scalar curvature symplectic generalized Kähler structures on Kähler-Einstein Fano manifolds.
Verify conjecture for special Hermitian manifolds.
problem Conjecture about space forms for canonical metric connections.
method Verify conjecture for complex nilmanifolds and Bismut torsion-parallel manifolds.
result Verify conjecture for two special types of Hermitian manifolds.
We prove the classical Yano-Obata conjecture by showing that the connected component of the group of holomorph-projective transformations of a closed, connected Riemannian Kähler manifold consists of isometries unless the metric has constant positive holomorphic curvature.
We show that one-sided Alexandrov embedded constant mean curvature cylinders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are rotational.