This paper classifies Blaschke para-umbilical hypersurfaces in conformal space.
problem Classifying Blaschke para-umbilical hypersurfaces in conformal space.
method Classification through mathematical analysis and extension of previous classifications.
result Classification of Blaschke para-umbilical hypersurfaces in conformal space.
Study classifies space-like hypersurfaces with parallel Blaschke tensor in conformal space.
problem Classifying hypersurfaces with parallel Blaschke tensor.
method Classifies hypersurfaces up to conformal equivalence.
result All such hypersurfaces are classified.
The paper classifies a specific type of hypersurfaces in Lorentzian space forms.
problem Classifying para-Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms.
method Analyzing the conformal transformation group and properties of para-Blaschke tensor.
result Classification of para-Blaschke isoparametric spacelike hypersurfaces.
Proves conjectured capillary Blaschke-Santaló inequality for certain convex hypersurfaces.
problem Proving a conjectured inequality for specific convex hypersurfaces.
method Analyzing unconditional, strictly convex capillary hypersurfaces.
result The inequality holds for θ∈(0,2π) and has no finite upper bound for θ∈(2π,π). Study classifies space-like hypersurfaces in de Sitter space with parallel tensors.
problem Characterizing space-like hypersurfaces with parallel para-Blaschke tensors.
method Conformal classification of hypersurfaces.
result Complete classification of space-like hypersurfaces in de Sitter space with parallel para-Blaschke tensors.
The paper classifies space-like hypersurfaces with parallel Blaschke tensors in de Sitter space.
problem Classifying hypersurfaces with specific geometric properties.
method Using conformal non-homogeneous coordinate systems to study hypersurfaces in de Sitter space.
result A complete classification of regular space-like hypersurfaces with parallel Blaschke tensors.
We solve Blaschke's problem for hypersurfaces of dimension n≥3. Namely, we determine all pairs of Euclidean hypersurfaces f,f~:Mn→Rn+1 that induce conformal metrics on Mn and envelope a common sphere congruence in Rn+1.
The paper proves inequalities for convex hypersurfaces in spheres and hyperbolic spaces.
problem Understanding geometric properties of convex hypersurfaces in curved spaces.
method Proving identities and inequalities for closed, strictly convex hypersurfaces in spheres and hyperbolic/de Sitter space.
result Generalized Blaschke-Santaló type inequalities and quermassintegral inequalities in hyperbolic/de Sitter space.
This paper classifies hypersurfaces in n+1 with parallel Fubini-Pick form.
problem Classifying hypersurfaces with parallel Fubini-Pick form in \(\mathbb{R}^{n+1}\).
method Defining a generalized Calabi product and proving decomposition theorems.
result Complete classification of Calabi hypersurfaces in \(\mathbb{R}^{n+1}\) with parallel Fubini-Pick form.
The volume distance from a point p to a convex hypersurface M of the (N+1)-dimensional space is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane H(p) we ob…
For a convex domain D that is enclosed by the hypersurface ∂D of bounded normal curvature, we prove an angle comparison theorem for angles between ∂D and geodesic rays starting from some fixed point in D, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtai…
Researchers develop Q-curvature for convex hypersurfaces using ambient metrics.
problem Calculating Q-curvature for convex hypersurfaces in projective manifolds.
method Using the ambient metric, they construct GJMS operators and relate Q-curvature to the logarithmic coefficient in volume expansion.
result Derived first and second variation formulas for strictly convex domains.
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.
In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a pointwise symmetry if at every point there exists a linear transformation preserving the affine metric, the affin…
The paper proves ellipsoids are the only centroaffine Tchebychev hyperovaloids.
problem Generalizing Blaschke and Deicke's theorem to centroaffine differential geometry.
method Characterized hypersurfaces by a closed conformal vector field and used properties of Riemannian manifolds.
result Ellipsoids are the only centroaffine Tchebychev hyperovaloids.
This paper classifies Blaschke parallel submanifolds in the unit sphere with three distinct eigenvalues.
problem Classifying submanifolds with parallel Blaschke tensors in the unit sphere.
method Analyzing the Blaschke tensor and its eigenvalues.
result Classification of Blaschke parallel submanifolds with exactly three distinct eigenvalues.
New method proves mateability of triangle groups with Blaschke products.
problem Proving mateability of triangle groups with Blaschke products.
method Associating two piecewise analytic circle maps to the triangle group, mating these with Blaschke products, and constructing a common lift.
result Proves mateability of all cusped triangle groups with suitable Blaschke products.
Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.
problem Finding optimal bounds between Riemannian and Hilbert metrics in convex projective domains.
method Optimal control techniques applied to Riemannian metrics induced by centro-affine hypersurface immersions.
result Optimal inequalities between Riemannian and Hilbert metrics for a class of convex projective domains.
The paper studies Blaschke products, proving uniformization and non-degeneracy of pressure metrics.
problem Analytic aspects of Blaschke products and their moduli space.
method Definition of complex structure and proof of uniformization theorem.
result Pressure semi-norms are non-degenerate outside the super-attracting locus.
Study on Blaschke locus with covariance metric properties.
problem Riemannian structure of Blaschke locus.
method Analysis of Blaschke locus as a manifold, study of covariance metric properties.
result Geodesics in Blaschke locus have infinite length with respect to the covariance metric.
The paper proves Bonnet-type theorems for hypersurfaces in 4D space.
problem Characterizing hypersurfaces in 4D space with constant mean curvatures.
method Using relative differential geometry and properties of relative mean curvatures.
result Proves Bonnet-type theorems for hypersurfaces with constant relative mean curvatures.
Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
Affine spheres can be realized with a specific Blaschke metric condition.
problem Realizing affine spheres with a prescribed Blaschke metric.
method Proving the necessary and sufficient condition for local realizability.
result The equality Δln∣κ−λ∣=6κ is a condition for local realizability of the metric as an affine sphere's Blaschke metric. Study Blaschke's asymptotic lines on surfaces in 3D space.
problem Characterize Blaschke's asymptotic lines on surfaces in 3D.
method Analyze binary differential equations near cusp and umbilic points.
result Describe Blaschke's asymptotic lines near Euclidean parabolic set.
The paper classifies submanifolds in a sphere with a special tensor.
problem Classifying submanifolds with a parallel Blaschke tensor.
method Defining new examples and proving a classification theorem.
result Classification of immersed umbilic-free submanifolds with a parallel Blaschke tensor.
Study Blaschke metrics to compactify Hitchin component.
problem Compactify the SL(3,R)-Hitchin component.
method Study degeneration of Blaschke metrics on equivariant affine spheres.
result Establish closure in projectivized geodesic currents space.
The study shows how many diameter directions in Besse manifolds relate to Blaschke manifolds.
problem Understanding the relationship between diameter directions and Blaschke manifolds in Besse manifolds.
method Analyzing the properties of Besse manifolds and pinched curvature metrics.
result Besse manifolds with many diameter directions are Blaschke manifolds.
The paper extends equiaffine structure to frontals and defines Blaschke vector fields.
problem Defining equiaffine structure on frontals.
method Defining Blaschke vector fields and providing necessary conditions for frontals to have such fields.
result A fundamental theorem for frontals, akin to the fundamental theorem of affine differential geometry.
We give a stability version of of the Blaschke-Santaló inequality in the plane.
Study volume expansion on convex domains using Blaschke metric.
problem Volume expansion of Blaschke metric on strictly convex domains.
method Expressed logarithmic coefficient L as integrals of affine invariants over the boundary and formulated intrinsic geometry as conformal Codazzi structure.
result L is a global conformal invariant of the boundary.
Established a new inequality for convex bodies in high dimensions.
problem Bounding the product of dual quermassintegrals of convex bodies and their polar sets.
method Induction on dimensions, focusing on convex bodies in \(\mathbb{R}^{n+1}\).
result Proved a generalised Blaschke-Santalò inequality with upper bounds.
Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An applicat…
In this paper, using functional Steiner symmetrizations, we show that Meyer and Pajor's proof of the Blaschke-Santalo inequality can be extended to the functional setting.
The study proves a discrete Blaschke theorem for convex polygons in 2-dimensional space forms.
problem Investigating curvature and circumradius constraints for convex polygons in 2-space forms.
method Defining curvature at each vertex and proving a Blaschke-type theorem.
result The circumradius of a convex polygon satisfies a specific inequality related to its vertex curvatures.
The aim of this paper is to give a local description of affine surfaces, whose induced Blaschke structure is projectively flat. We show that such affine surfaces with constant Gauss affine curvature and indefinite induced Blaschke metric are described by soliton equations.
Constructs correspondences on hyperelliptic surfaces combining orbifold groups and Blaschke products.
problem Combining geometric and dynamical properties on hyperelliptic surfaces.
method Analytic combinations on Riemann sphere, algebraic characterization, and Teichmüller spaces.
result Explicit description of correspondences and injection into Hurwitz spaces.
The Blaschke rolling disk theorem is extended to non-convex domains.
problem Classical inclusion principle for non-convex domains.
method Geometric conditions based on curvature, algorithm for decomposition.
result Necessary and sufficient conditions for rolling disks in non-convex domains.
The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n > 2. In this paper we describe a necessary condition that th…
The study classifies geometrically finite polynomials on the boundary of Blaschke products.
problem Understanding the boundaries of hyperbolic components of Blaschke products.
method Combinatorial classification and construction of self-bumps.
result The closure of the main hyperbolic component is not a topological manifold with boundary for d≥4. We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.
The paper surveys pressure metrics in geometry and dynamics.
problem Understanding pressure metrics in various deformation spaces.
method Survey and discussion of pressure semi-norms and their degeneracy loci.
result Discussion of pressure semi-norms and their degeneracy loci in quasi-Blaschke products.
We show that isothermic surfaces and S-Willmore surfaces are also the solutions to the corresponding Blaschke's problem for both spacelike and timelike surfaces in pseudo-Riemannian space forms. For timelike surfaces both Willmore and isothermic, we obtain a description by minimal surfaces similar to the classical resu…
Novel algorithm compresses ECG signals with preserved R peaks.
problem Efficiently compressing ECG signals while preserving R peak information.
method Blaschke unwinding AFD for faster convergence and higher fidelity.
result The proposed algorithm outperforms state-of-the-art approaches in ECG signal compression.
The paper derives a formula for Riemannian manifolds involving Ricci curvatures.
problem Computing Jacobian determinants and circumscribed spheres in Riemannian manifolds.
method Local Blaschke-Petkantschin formula derivation for Riemannian manifolds.
result Explicit two-term expansion for small radii with Ricci curvature corrections.
Laguerre geometry of surfaces in R3 is given in the book of Blaschke [1], and have been studied by E.Musso and L.Nicolodi [5], [6], [7], B. Palmer [8] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in Rn. For any umbilical free hypersurface x:M→Rn with non-zero …
We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. As a side result, we show that the number of linearizations in the Gronwall conject…
A homogeneous Gibbons-Hawking ansatz is described, leading to 4-dimensional hyperkahler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkahler metrics that are, in a suitable sense, comp…
Study of K-surfaces with free boundaries hitting a manifold at a fixed angle.
problem Determine a K-surface with a free boundary hitting a manifold at a fixed angle. method Analytic formulation reduces to a Bernoulli type free boundary problem for the Monge-Ampère equation. Introduced a notion of Blaschke extension.
result Solved a model case of the problem for both 0-curvature and positive curvature.