Estimates spacelike surfaces' curvature in de Sitter space.
problem Estimating maximal curvatures of spacelike hypersurfaces.
method Obtained local estimates for k-symmetric curvature functions.
result Curvatures depend on interior and boundary data.
The paper confirms conjectures about ancient ovals and provides counterexamples.
problem Understanding the uniqueness and nonuniqueness of ancient ovals under different symmetries.
method Analyzing mean curvature flow solutions and constructing symmetric ancient ovals.
result Confirms conjectures about ancient ovals and provides counterexamples.
Study proves existence of special hypersurfaces in de Sitter space.
problem Existence of compact spacelike hypersurfaces with prescribed curvature.
method Proves existence using prescribed k-curvature in de Sitter space.
result Compact spacelike hypersurfaces with prescribed k-curvature exist in de Sitter space.
The study constructs equivariant harmonic maps into symmetric spaces with applications to Willmore surfaces.
problem Constructing harmonic maps into symmetric spaces.
method Equivariant primitive harmonic maps construction.
result Examples of S1-equivariant Willmore Moebius strips in S3. We define a large class of integrable nonlinear PDE's, \emph{k-symmetric AKS systems}, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the soluti…
The study confirms that certain symmetric spaces are formal.
problem Understanding the equivariant cohomology of symmetric spaces.
method Analyzing the isotropy action and using Rational Homotopy Theory.
result Formality of Z2×Zk-symmetric spaces. Flow deforms locally convex curves to curves of constant k-order width.
problem Evolve locally convex curves to curves of constant k-order width.
method Introduced a nonlocal curvature flow to evolve locally convex curves in the plane.
result The flow converges to a smooth, locally convex curve of constant k-order width as time goes to infinity.
The paper classifies ancient ovals in higher dimensions and proves their symmetry and uniqueness.
problem Classifying compact ancient noncollapsed mean curvature flows in arbitrary dimensions.
method Analyzing k-ovals and using spectral ratio parameters to prove symmetry and uniqueness. result Ancient k-ovals are uniquely determined by (k−1)-dimensional spectral ratio parameters and are Z2kimesO(n+1−k)-symmetric. We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…
The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on Z2-grading of Lie algebras. In our case, we consider homogeneous spaces G/H such that the Lie algebra $\g$ of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Rieman…
We derive a class of variational functionals which arise naturally in conformal geometry. In the special case when the Riemannian manifold is locally conformal flat, the functional coincides with the well studied functional which is the integration over the manifold of the k-symmetric function of the Schouten tensor of…
Flag manifolds are in general not symmetric spaces. But they are provided with a structure of Z2k-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for the flag manifold SO(5)/SO(2)×SO(2)×SO(1) what are the conditions…
New forms of symmetric shift-invariant subspaces found for harmonic maps.
problem Understanding harmonic maps into symmetric and k-symmetric spaces. method Imposing a symmetry condition on shift-invariant subspaces of a Hilbert space.
result Obtained new general forms for symmetric shift-invariant subspaces and extended solutions.
We collect the recent results on invariant f-structures in the generalized Hermitian geometry. Here the canonical f-structures on homogeneous k-symmetric spaces play a remarkable role. Specifically, these structures provide a wealth of invariant examples for the classes of nearly Kaehler f-structures, Hermitian f-struc…
We prove that all immersions of a genus one surface into G/T possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and the k-symmetric space structure on G/T i…
We give a geometric interpretation of all the m-th elliptic integrable systems associated to a k′-symmetric space N=G/G0 (in the sense of C.L. Terng). It turns out that we have to introduce the integer mk′ defined by m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three cases : the prim…
The notion of a Γ-symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group Γ replaces the group Z2. The case Γ=Zk has also been studied, from the algebraic point of view by V.Kac \cite{VK} and from the point of view of the differential geometry by…
This paper is concerned with the structure of Gromov-Hausdorff limit spaces (Min,gi,pi)⟶dGH(Xn,d,p) of Riemannian manifolds satisfying a uniform lower Ricci curvature bound RcMin≥−(n−1) as well as the noncollapsing assumption Vol(B1(pi))>v>0. In such cases, there is …
We study geodesics of the form γ(t)=π(exp(tX)exp(tY)), $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces G/K, where π:G→G/K is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of G (i.e. γ(t)=π(exp(tX)), $X\in …
Survey on real forms of a complex equation and their connection to surface theory.
problem Describing real forms of the complex A2(2)-Toda equation and their geometric implications. method Analyzing the integrability of Maurer-Cartan forms for different real forms of loop groups.
result Each real form of A2(2) corresponds to a specific surface class with integrable frames. The study quantifies topological expansion properties of complexes and their embeddings.
problem Understanding topological expansion properties of simplicial complexes.
method Quantifying topological expansion through sublinear functions and proving monotonicity under regular maps.
result Proves topological expanders contain graphical expanders and gives lower bounds for specific embeddings.
We study different notions of Riemannian curvatures: The p-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the (p,q)-curvatures, which incorporate …
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with co…
Study on Hermitian metrics and curvature properties of complex manifolds.
problem Analyzing curvature properties of Hermitian metrics on complex manifolds.
method Derivation of formulae and proofs for Chern-Ricci curvatures and holomorphic sectional curvatures.
result Examples of metrics with specific curvature properties.
New metric with negative curvature found near positive-curvature spaces.
problem Understanding spaces with positive curvature bounds.
method Analyzing metrics in relation to curvature bounds.
result Near any point in a positive-curvature space, there exists a metric with negative upper curvature.
Paper explores entropic curvature in Markov chains, comparing it to other curvatures.
problem Comparing entropic curvature to other curvatures in Markov chains.
method Adapted Γ-calculus for θ-curvatures, explicit lower bounds, curvature perturbation.
result Entropic curvature differs significantly from other curvature notions.
Study examines preservation of curvature-adaptedness during mean curvature flow.
problem Preservation of curvature-adaptedness during mean curvature flow.
method Investigates curvature-adaptedness in locally symmetric spaces.
result Curvature-adaptedness is preserved along mean curvature flow.
The paper studies Berwald scalar curvature properties in Finsler geometry.
problem Characterizing Finsler manifolds based on Berwald scalar curvature.
method Analyzes properties of Berwald scalar curvature and its implications for Finsler manifolds.
result Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.
The paper studies Finsler manifolds with a new curvature concept.
problem Understanding Finsler manifolds with positive weighted flag curvature.
method Introducing a new curvature concept based on the flag curvature and a non-Riemannian quantity, T-curvature.
result Positive weighted flag curvature implies the manifold is diffeomorphic to Euclidean space.
New proof shows holomorphic sectional curvature fully determines curvature tensor.
problem Determining the curvature tensor from holomorphic sectional curvature.
method Representation-theoretic means to calculate L2-norm of holomorphic sectional curvature. result Holomorphic sectional curvature fully determines the curvature tensor.
Given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero and the evolution equation by T-curvature at the boundary with the condition that the Q-curvature …
The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n-volumic scalar curvature. method Using the curvature-dimension condition mCD(κ,n) and smGH-convergence. result The stability of n-volumic scalar curvature ≥κ under smGH-convergence. Compact shrinkers with curvature pinching conditions proven.
problem Ensuring shrinkers are compact under curvature pinching conditions.
method Various curvature pinching conditions applied to shrinkers with positive Ricci curvature and asymptotically nonnegative sectional curvature.
result Shrinkers with curvature pinching conditions are proven to be compact.
Study geodesic curvature of logarithmic spirals on curved surfaces.
problem Understanding geodesic curvature on curved surfaces.
method Computed geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature.
result Asymptotic behavior of geodesic curvature is independent of the ambient surface's curvature.
New curvature definitions for networks simplify complex computations.
problem Complex curvature calculations for networks.
method Introducing new curvature definitions based on Menger and Haantjes curvatures.
result Simplified and faster computation of network curvatures.
Study on singularity behavior of mean curvature flow with bounded curvature and index.
problem Understanding singularity formation in mean curvature flow with constraints.
method Analyzing flow with bounded mean curvature and Morse index.
result Either mean curvature or Morse index blows up at first singular time.
Study on curvature in finitely generated groups, showing positive curvature in specific cases.
problem Understanding curvature in finitely generated groups.
method Analyzing dead-end elements and related elements to find curvature, studying effect of radius.
result Examples of positive curvature for arbitrary radius in lamplighter and Houghton's group.
Introduces new curvature concept for Kähler manifolds.
problem Optimizing curvature constraints for projective Kähler manifolds.
method Introduces weighted orthogonal Ricci curvature and proves vanishing theorems.
result Proves optimal curvature constraints for projective Kähler manifolds.
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
Proves curvature estimates for 3D surfaces solving scalar curvature equations.
problem Estimating curvature of 3D surfaces solving scalar curvature equations.
method Integral method and new Lagrangian submanifold observation.
result Interior curvature estimates for 3D hypersurfaces are proven.
We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p,q)-curvatures. They are a generalization of the p-curvat…
The paper examines geometric properties of a unique spacetime model.
problem Investigating the geometric properties of a point-like global monopole spacetime.
method Analyzing the spacetime's pseudosymmetry structures, energy-momentum tensor, and curvature properties.
result The point-like global monopole spacetime exhibits various pseudosymmetry structures and properties.
Quantitative estimate for curvature in mean curvature flow.
problem Estimating curvature in mean curvature flow.
method Proving a curvature estimate for smooth convex ancient flows.
result Curvature grows at most quadratically in terms of rescaled extrinsic distance.
Study on hypersurfaces with specific curvature conditions.
problem Characterizing hypersurfaces with certain curvature properties.
method Defined and analyzed the Opozda-Verstraelen affine curvature tensor for hypersurfaces.
result Conditions for pseudosymmetry types of hypersurfaces with specific curvature properties.
Equi-affine curvature of curves in 2-manifolds linked to Frenet curvature.
problem Understanding equi-affine curvature in pseudo-Riemannian 2-manifolds.
method Expressing equi-affine curvature using Frenet curvature.
result Equi-affine curvature can be derived from Frenet curvature.
New insights into SGD and generalization via shift-curvature and bias-curvature mechanisms.
problem Understanding the role of curvature in generalization and how SGD affects it.
method Derivation of new SGD steady-state distribution and analysis of shift-curvature and bias-curvature mechanisms.
result Shift-curvature is a significant factor in test performance, especially for small SGD noise.