New invariant found in spaces with curvature below, related to volume.
problem Geometric invariant of spaces with curvature below.
method Relations between new invariant and normalized volume, rigidity for maximal case.
result New invariant related to volume and rigidity for maximal case.
Paper extends circle pattern theory to obtuse angles.
problem Circle patterns with obtuse angles not previously covered.
method Using topological degree theory, extends Koebe-Andreev-Thurston Theorem.
result Generalized Andreev's Theorem for obtuse dihedral angles.
Paper generalizes Andreev's theorem with obtuse angles.
problem Characterizing hyperbolic polyhedra with obtuse angles.
method Established discrete analog of weak solution/regularity theory.
result Generalized Andreev's Theorem to include obtuse angles.
Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the ca…
The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.
problem Understanding the combinatorial types of 3D polytopes in specific Riemannian manifolds.
method Analysis of mean curvature convex Riemannian polyhedra with non-obtuse dihedral angles in positive scalar curvature 3-manifolds.
result Determination of combinatorial types of 3D simple convex polytopes.
Given a combinatorial description C of a polyhedron having E edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize C is generally not a convex subset of RE \cite{DIAZ}. If C has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angle…
We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume b…
The paper studies circle patterns on surfaces with specific angles and curvature maps.
problem Investigating circle patterns with obtuse angles on surfaces of finite type.
method Characterizing curvature maps and establishing combinatorial Ricci flow conditions.
result Generalizations of circle pattern theorem and a computational method to find patterns.
The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.
problem Proving the Discrete Schwarz-Pick Lemma for circle packings with various inversive distances.
method Using a variational principle for circle packings with inversive distances, the paper extends the lemma to a broader range of packings.
result The Discrete Schwarz-Pick Lemma holds for circle packings with inversive distances in (−1,1], provided an additional condition on triangle weights. Trapezoids uniquely identified by their Dirichlet Laplace spectrum.
problem Identifying trapezoids based on their spectral properties.
method Analyzing the Dirichlet Laplace spectrum of non-obtuse trapezoids.
result Non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum.
We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's Method. Under the restriction that the dihedral angles are non-obtuse, Andreev's Theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyh…
Introduces Coxeter polyhedra in various geometries.
problem None explicitly stated, focuses on introducing concepts.
method Introduction of Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries.
result Fundamental theorems and descriptions of polyhedra and tessellations.
We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real n-space proposed by Allen Knutson and Jean-Claude Hausmann. This cor…
Study on least symmetric triangles in spherical geometry.
problem Finding the least symmetric triangle, both in planar and molecular contexts.
method Using Grassmannian correspondence and hyperoctahedral group action, compute the furthest point from the boundary in the Grassmannian.
result Exact computation of least symmetric triangles, including obtuse and acute types.
The paper studies how spaces collapse to Alexandrov spaces with mild singularities.
problem Understanding how Riemannian manifolds collapse to Alexandrov spaces with isolated singularities.
method Analyzes the structure of locally trivial fibrations over compact Alexandrov spaces.
result Proves that collapsing sequences of Riemannian manifolds admit locally trivial fibrations over the limit space.
From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these d…
Study on folded ribbon knots and their minimum length.
problem Finding the minimum length of folded ribbon knots.
method Using Kauffman's model of folded ribbon knots and analyzing their properties.
result Proved bounds on the minimum folded ribbonlength for various types of knots.
In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, C, Andreev's Theorem provides five classes of linear inequalities, depending on C, for the dihedral angles, which are necessar…
The study examines how shallow neural nets converge to training samples or manifold points during diffusion.
problem Understanding when and how shallow neural nets converge to training samples or manifold points during diffusion.
method Analysis of shallow ReLU neural network denoisers trained with minimal ℓ2 norm, comparing score flow and diffusion flow. result Probability flow converges to training points, sums of training points, or manifold points, depending on the diffusion time scheduler.
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than 2π is the metric of the Gauss image of som…
Fermat constants fail to fully identify Clairaut constants for certain geodesics on a surface of revolution.
problem Identifying Clairaut constants from Fermat constants for specific geodesics.
method Analytical proof for a specific class of geodesics on a surface of revolution.
result Fermat constants do not fully determine Clairaut constants for some geodesics, except for a standard sphere.
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
The paper classifies hypersurfaces in H2imesH2 with constant curvature.
problem Classifying hypersurfaces in H2imesH2 with constant sectional curvature. method Analyzing the geometry of H2imesH2 and constructing specific examples. result Examples of hypersurfaces in H2imesH2 with non-constant product angle function. Study on biconservative hypersurfaces with constant scalar curvature in space forms.
problem Characterize biconservative hypersurfaces with constant scalar curvature in space forms.
method Analyzing biconservative hypersurfaces in space forms Nn+1(c), proving properties and finding specific examples. result Proves that biconservative hypersurfaces with constant scalar curvature in N4(c) have constant mean curvature, and in N5(c), they are either rotational or constant mean curvature. Study constant angle surfaces in 4D Minkowski space, proving their properties.
problem Characterize surfaces in 4D Minkowski space with constant angle between tangent planes.
method Define complex angle, prove curvature properties, use PDE methods, analyze special cases.
result Constant angle surfaces have vanishing Gauss and normal curvatures; not complete for ψeq0 [π/2]. The study classifies surfaces with constant slope in 4D space.
problem Classifying surfaces with constant slope in higher dimensions.
method Analyzing generalized constant ratio surfaces in Euclidean 4-space.
result A classification of constant slope surfaces.
We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…
The paper studies curves of constant-ratio in pseudo-Galilean space.
problem Characterizing curves of constant-ratio in pseudo-Galilean space.
method Analyzing spacelike curves with constant-ratio in terms of curvature functions.
result Characterization of special curves of constant-ratio in pseudo-Galilean space.
Study CR Yamabe constant and CR structures on manifolds.
problem Understanding CR Yamabe constant and its role in CR geometry.
method Developed integral formulae and constructed families of CR structures.
result Found an infinite family of CR structures with varying CR Yamabe constants.
The Cheeger constant increases under Ricci flow on spheres.
problem Behavior of the Cheeger constant under Ricci flow.
method Evolution identities for parallel curves and viscosity formulation of logh. result The Cheeger constant is non-decreasing under Ricci flow on surfaces diffeomorphic to S2. Study on higher-order Escobar constants for planar domains.
problem Understanding Escobar constants for planar domains of higher order.
method Investigation of higher-order Escobar constants Ik(M) on bounded planar domains M. result Escobar constants Ik for the unit disk and a family of polygons are provided. The paper defines new constants for p-Laplacian on manifolds.
problem Bounding eigenvalues of the p-Laplacian on compact manifolds. method Introducing Steklov and Neumann isocapacitary constants.
result Two-sided bounds for (p,α)-Sobolev constants and eigenvalues. Study classifies 3D self-shrinkers with constant second form norm.
problem Classifying self-shrinkers with specific geometric properties.
method Analyzes 3D self-shrinkers in Euclidean space with constant second form norm.
result Classifies complete self-shrinkers with constant norm of the second fundamental form.
In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in R3 with constant width, constant brightness, and boundary of class C2 is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.
Study refines Siegel-Veech constants for abelian differentials.
problem Computing Siegel-Veech constants for abelian differentials.
method Intersection theory and quasimodular forms.
result New identity for Siegel-Veech constants of cylinders.
Curves with constant torsion can be deformed arbitrarily.
problem Deforming curves of constant torsion in Euclidean space.
method Convex integration and degree theory.
result Existence of knots with constant torsion in each isotopy class.
New upper bound for Cheeger constant of hyperbolic surfaces.
problem Bounding the Cheeger constant of hyperbolic surfaces.
method Random construction based on Poisson--Voronoi tessellation.
result The Cheeger constant of closed hyperbolic surfaces is less than that of the hyperbolic plane.
Paper finds conditions for generalized Kropina spaces to have constant curvature.
problem Classifying Finsler spaces of constant curvature.
method Obtained necessary and sufficient conditions for generalized Kropina spaces to be of constant flag curvature.
result Conditions for generalized Kropina spaces to have constant curvature.
Simplified proof for Cheeger's isoperimetric constant.
problem Cheeger's isoperimetric constant
method Simplified proof of Buser's result
result Simplified proof for Cheeger's isoperimetric constant
Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.
problem Characterizing and constructing hypersurfaces with specific curvature properties in Finsler geometry.
method Analyzing isoparametric and constant-curvature hypersurfaces on Randers manifolds.
result Construction of a conformally flat Randers manifold with nonisoparametric hyperplanes of constant curvatures.
This paper mixes constant sum and constant product market makers to improve their features.
problem Improving the balance between stable exchange rates and liquidity in automated market makers.
method Mixing and designing new methods for AMMs with specific features.
result Demonstrates new tools for creating markets with desired characteristics.
Ruled surfaces with Ricci metrics use curves of constant torsion.
problem Characterizing ruled surfaces with Ricci metrics.
method Using curves of constant torsion to construct ruled surfaces.
result Helicoid is the only surface with constant mean curvature.
Paper proves a Liouville theorem for solitons with constant curvature.
problem Understanding harmonic functions on specific geometric structures.
method Proved a Liouville theorem without gradient estimates.
result Finite dimensionality of harmonic functions with polynomial growth.
Study connects geodesic sums to area constant.
problem Understanding geodesic sums in Teichmüller space.
method Relates trimmed sums of twists to area Siegel-Veech constant.
result Established connection between geodesic sums and area constant.
The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.
problem Classifying surfaces with constant astigmatism in space forms.
method Classification and construction of surfaces using variational problems and binormal evolution.
result Locally constructed all rotational surfaces of constant astigmatism.
The paper classifies biharmonic hypersurfaces with constant scalar curvature.
problem Classifying biharmonic hypersurfaces with constant scalar curvature.
method Analyzing biharmonic hypersurfaces in space forms and spheres.
result Supports conjectures on biharmonic submanifolds and hypersurfaces.
Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.
problem Estimating eigenvalues and spectrum for graph substructures.
method Introducing Cheeger type constants via isocapacitary constants to estimate eigenvalues and spectrum.
result Estimates for first Dirichlet, Neumann, and Steklov eigenvalues, as well as the bottom of the spectrum of the Laplace operator and Dirichlet-to-Neumann operator.
Paper builds singular metrics with constant Q-curvature.
problem Constructing metrics with constant Q-curvature on manifolds.
method Utilizes tools from recent years to build weak solutions.
result First construction of singular metrics with positive Q-curvature.