The paper proves that certain spaces have injective balls of any radius.
problem The injectivity radius of certain geometric spaces is infinite.
method Analyzes higher rank simple and semisimple Lie groups with specific properties.
result The locally symmetric spaces have injective balls of any radius.
First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the …
New examples of deformed Hermitian-Yang-Mills connections found.
problem Constructing deformed Hermitian-Yang-Mills connections on manifolds.
method Constructed first higher rank, irreducible deformed Hermitian-Yang-Mills connections in both small and large radius regimes.
result Existence of solutions with any possible angle and ruling out some stability conditions.
In this paper we obtain a simple upper bound for the infimum of the Ricci curvatures of a complete Riemannian manifold with nonzero injectivity radius i(M) depending only on of the i(M). In case of rigidity the Riemannian manifold must be an Euclidean sphere(Euclidean space) conform the injectivity radius be finite(inf…
Improved homological dimension for certain subgroups in Lie groups.
problem Determining homological dimensions of discrete subgroups in Lie groups.
method Using recent results and properties of injectivity radius, the homological dimension gap is calculated.
result Infinite volume torsion-free subgroups of higher rank Lie groups have a homological dimension gap of at least 1/8 of the real rank.
Commentary on Busemann's foundational geometry papers.
problem Foundational axiomatic geometry and Minkowski spaces.
method Translation and analysis of three seminal papers.
result Early ideas and techniques for Minkowski spaces.
We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.
We give an optimal estimate for the norm of any submanifold's second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a C1,α compactness result for submanifolds, …
We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a …
In the presence of model risk, it is well-established to replace classical expected values by worst-case expectations over all models within a fixed radius from a given reference model. This is the "robustness" approach. We show that previous methods for measuring this radius, e.g. relative entropy or polynomial diverg…
In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. With these functions and a classical the…
Let (M,g) be a non-compact and complete Riemannian manifold with minimal horospheres and infinite injectivity radius. We prove that bounded functions on (M,g) satisfying the mean-value property are constant. We extend thus a result of A. Ranjan and H. Shah who proved a similar result for bounded harmonic functions on h…
The paper confirms a conjecture about convex bodies and their properties.
problem The conjecture about the mean width of convex bodies.
method Analyzing λ-convex bodies and their intersections of balls. result The λ-convex lens maximizes the mean width among all λ-convex bodies with a given inradius. Wide neural networks become linear, but adding bottlenecks makes them bilinear or multilinear.
problem Understanding the transition of neural networks from linearity to higher-order functions.
method Analyzing the behavior of randomly initialized wide neural networks with and without bottleneck layers.
result Bottleneck layers transform the network's function from linear to bilinear or multilinear.
Formula for European option pricing under jump diffusion model.
problem Option pricing under complex stochastic processes.
method Infinite series of Black-Scholes terms for Levy-driven processes.
result Series solution converges with a radius of convergence.
The paper proves Sard's theorem for polynomial maps in infinite dimensions.
problem The validity of Sard's theorem for polynomial maps in infinite-dimensional Banach manifolds.
method Sharp quantitative criteria for the validity of Sard's theorem.
result The paper provides criteria for the validity of Sard's theorem in infinite-dimensional Banach manifolds.
The paper establishes estimates for convexity and injectivity radii in Riemannian manifolds.
problem Estimating convexity and injectivity radii in Riemannian manifolds.
method Pointwise and curvature-free estimates on convexity radius, injectivity radius, and local behavior of geodesics.
result Established estimates for convexity and injectivity radii in Riemannian manifolds.
Directly proves positive injectivity radius for Lie manifolds.
problem Injectivity radius positivity for Lie manifolds.
method Direct, geometric proof.
result Injectivity radius is positive for Lie manifolds.
Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…
Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.
problem Bounding curvature of regularized metrics with constraints on Ricci tensor and injectivity radius.
method Mollification of riemannian metrics, uniform W2,p-harmonic radius bounds, Ricci tensor bounds, injectivity radius bounds. result Uniform estimate on the change of sectional curvature for regularized metrics.
Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.
problem Bounding the conjugate radius of open manifolds with specific curvature and spectrum conditions.
method Established an upper bound using scalar curvature and bottom-of-spectrum constraints.
result For certain conditions, the conjugate radius is no more than π.
Proves upper bound on filling radius for manifolds with positive scalar curvature.
problem Bounding the filling radius of manifolds with positive scalar curvature.
method Quantitative operator K-theory and index theory.
result Proves a quantitative upper bound on the filling radius.
Compact theorem for minimal surfaces with lower injectivity radius.
problem Proving compactness of minimal surfaces with lower injectivity radius.
method Variant of Choi--Schoen compactness theorem, focusing on injectivity radius.
result Proved compactness theorem for minimal surfaces.
Injectivity radius on Stiefel manifold is π.
problem Determining the injectivity radius of the compact Stiefel manifold.
method Utilized the property of geodesics being space curves of constant Frenet curvatures.
result The injectivity radius on the Stiefel manifold under the Euclidean metric is π.
Lower bound on boundary injectivity radius for specific tubes.
problem Estimating the boundary injectivity radius of Margulis tubes.
method Using curvature bounds to derive a lower bound.
result A lower bound on the boundary injectivity radius is provided.
Study horocycle and geodesic orbits on hyperbolic surfaces.
problem Understanding the relationship between horocyclic and geodesic orbits on geometrically infinite surfaces.
method Analyzing the topological dynamics of the horocycle flow on hyperbolic surfaces, focusing on specific vectors and their orbits.
result The closure of the horocycle flow of a non-periodic vector intersects geodesic orbits along an unbounded sequence of points.
Study gives bounds on filling radius for Riemannian manifolds.
problem Finding bounds on the filling radius of Riemannian manifolds.
method Curvature-dependent bounds for all closed manifolds and upper bounds for submersion and submetry cases.
result Upper and lower bounds on the filling radius for specific types of manifolds.
Defines John-Nirenberg radius for collapsing conformal metrics and proves a convergence theorem.
problem Analyzing collapsing conformal metrics in a fixed conformal class.
method Defining John-Nirenberg radius and proving convergence using curvature conditions.
result The John-Nirenberg radius is bounded below by a positive constant for collapsing conformal metrics.
The expected covering radius of a translation surface is bounded by a function of log g/g^(1/2).
problem Computing the expected covering radius of translation surfaces.
method Estimating the volume of the thin part of H_1(kappa) and using it to bound the covering radius.
result The expected covering radius is bounded above by a uniform multiple of ((log g)/g)^(1/2).
The paper proves estimates and theorems for Kähler manifolds.
problem Curvature conditions on Kähler manifolds.
method Volume comparison and rigidity theorems.
result Conjugate radius estimates for Kähler manifolds.
Study finds the covering radius of RM(4,8) is 26.
problem Determining the covering radius of RM(4,8).
method Invented a lift by derivation invariant to classify B(5,6,8).
result Covering radius of RM(4,8) is 26.
Hardness result for approximating manifold radius.
problem Approximating the radius of triangulated manifolds.
method Proving NP-hardness for almost-polynomial approximation.
result It is NP-hard to approximate the hyperspherical radius up to an almost-polynomial factor.
Study on optimal ReLU networks with weight decay for interpolation.
problem Interpolating data with radially symmetric distributions using shallow ReLU networks.
method Weight decay regularization in infinite neuron, infinite data limit; analysis of growth rates.
result Existence and growth rates of unique radially symmetric minimizers with weight decay.
Clarifies definition of polarized canonical radius in Kahler Ricci flow.
problem Unclear definition of polarized canonical radius in Kahler Ricci flow.
method Clarification of the definition.
result Clarified definition of polarized canonical radius.
The paper proves an area inequality for metric balls in Riemannian manifolds.
problem Proving an area inequality for metric balls in Riemannian manifolds.
method Analyzing metric balls B(p,R) in two-dimensional Riemannian manifolds. result Proves an area inequality Area(B(p,R))≥π8R2 for R less than half the convexity radius. The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.
problem Bounding bandwidth and focal radius for manifolds with positive isotropic curvature.
method Using spectral properties of a twisted de Rham-Hodge operator.
result Upper bounds on bandwidth and focal radius are derived for hypersurfaces in PIC manifolds.
Upper bound on Stiefel manifold's injectivity radius found.
problem Finding the maximum distance within which the Stiefel manifold remains injective.
method Exhibited conjugate points and calculated the minimum of geodesic lengths.
result Upper bound on Stiefel manifold's injectivity radius is conjectured to be equal to the injectivity radius.
Upper estimate of Dirac eigenvalue linked to hyperspherical radius.
problem Estimating the smallest eigenvalue of the Dirac operator.
method Proved an upper estimate of the smallest eigenvalue in terms of hyperspherical radius.
result Combining with known lower estimates, geometric consequences are derived.
The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.
problem Understanding the properties and behavior of Ricci solitons.
method Analytical proofs and estimates for various types of Ricci solitons.
result Upper bounds and estimates for conjugate radius of Ricci solitons.
Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
Estimates radius and volume of curved spaces with convex boundaries.
problem Estimating the radius and volume of curved spaces with convex boundaries.
method Analyzes Alexandrov spaces with strictly convex boundaries, using Base-Angle and volume estimates.
result Estimates for radius and volume of curved spaces with convex boundaries.
In this paper we study the geometry of metric spheres in the curve complex of a surface, with the goal of determining the "average" distance between points on a given sphere. Averaging is not technically possible because metric spheres in the curve complex are countably infinite and do not support any invariant probabi…
We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.
This paper explores rigid properties of Alexandrov spaces with maximal radius.
problem Rigidity of Alexandrov spaces with maximal radius and specific curvature conditions.
method Analyzes Alexandrov spaces with \curv\geq1, nonempty boundary, and maximal radius \fracπ{2}. Uses rigidity theorems and geometric/topological constraints.
result Shows flexibility and rigidity in Alexandrov spaces with maximal radius under different conditions.
A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with a translation structure with a topology, which we call the immersive topology be…
Uniformly positive scalar curvature implies a lower bound on injectivity radius.
problem Bounding curvature and scalar curvature in three-manifolds.
method Analyzing bounded sectional curvature and uniformly positive scalar curvature properties.
result Uniform lower bound on injectivity radius.
Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.
problem Bounding inscribed radius in asymptotically hyperbolic Einstein manifolds.
method Generalized inscribed radius estimate to AH Einstein manifolds, combining recent work.
result Rigidity result achieved for upper bound of relative volume.