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.
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.
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…
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.
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 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.
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.
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.
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 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.
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.
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.
We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.
Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.
problem Determining the shortest geodesic loops and injectivity radius on Stiefel manifold.
method Combining bounds on sectional curvature with existing metrics.
result Exact value of the injectivity radius for a wide range of metrics.
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.
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold M: 1) the convexity radius of p, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…
The paper improves bounds on injectivity radius for manifolds with positive scalar curvature.
problem Finding tighter bounds on the injectivity radius for manifolds with positive scalar curvature.
method Utilizing Green's inequality and topological assumptions on manifolds, including specific 3-manifolds and products.
result Stronger upper bounds on injectivity radius for certain manifolds, including products and 3-manifolds with positive scalar curvature.
We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive
Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.
problem Comparing eigenvalues on spherically symmetric manifolds to Euclidean balls.
method Examines Dirichlet Laplace eigenvalues on balls of spherically symmetric manifolds and Euclidean space.
result Eigenvalues on spherically symmetric manifolds are smaller for small radii, but larger for hyperbolic spaces.
The paper establishes lower bounds on injectivity radius and constructs metrics with bounded geometry.
problem Establishing lower bounds on the normal injectivity radius of hypersurfaces and constructing metrics with bounded geometry on manifolds with boundary.
method Pointwise lower estimates and constructions of metrics with bounded geometry.
result The construction of metrics with bounded geometry on arbitrary manifolds with boundary.
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.
We show that there is an upper bound on the injectivity radius of a hyperbolic 3-manifold in terms of the the number of generators of its fundamental group.
In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: ''Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental group of M is virtually free.'' We explain the relevance of this th…
The study shows a 3D manifold's macroscopic dimension is 1 under specific curvature constraints.
problem Understanding the macroscopic dimension of 3D Riemannian manifolds with curvature restrictions.
method Analyzing the volume and homology of balls in the manifold.
result A 3D manifold with the specified curvature constraints has macroscopic dimension 1.
New cell structure on O(3)/O(1)3 derived from injectivity radius computation.
problem Constructing equivariant cell structures on flag manifolds.
method Injectivity radius computation and Dirichlet-Voronoi domains.
result New S3-equivariant cell structure on O(3)/O(1)3. Generalizes a soul-bound for noncompact Alexandrov spaces.
problem Finding a lower bound for injectivity radius in Alexandrov spaces.
method Introduces the soul of Alexandrov spaces and applies a generalized bound.
result Injectivity radius is at least πK⁻¹/² if not equal to the soul's.
Recall that the radius of a compact metric space (X,dist) is given by rad X=minx∈Xmaxy∈Xdist(x,y). In this paper we generalize Berger's 41-pinched rigidity theorem and show that a closed, simply connected, Riemannian manifold with sectional curvature ≥1 and radius $\geq \fracπ{2…
We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M=(0,∞)×Y whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potenti…
Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using Ck,α convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein m…
In this paper, we mainly establish a Cheeger type finiteness theorem for Berwald manifolds. In order to do this, we study the injectivity radius and the convex radius of a Finsler manifold. A Cheeger type estimate on injectivity radii for Finsler manifolds is given and the existence of the center of mass of a Berwald m…
The paper extends radius estimates for stable hypersurfaces in 2, 3, and 4 dimensions.
problem Estimating the radius of nearly stable hypersurfaces in specific dimensions.
method Generalizing existing radius estimates for CMC hypersurfaces in Riemannian manifolds with bounded curvature.
result Radius estimates for nearly stable hypersurfaces in 2, 3, and 4 dimensions are extended.
Small sub-Riemannian balls have diameter close to twice their radius.
problem Understanding the diameter of small sub-Riemannian balls.
method Analyzing C1,1 and C0 sub-Riemannian manifolds. result The diameter of small sub-Riemannian balls equals twice the radius in C1,1 manifolds, and is close to twice the radius in C0 manifolds. Sharp focal radius estimate for hypersurfaces in manifolds with positive curvature.
problem Estimating the focal radius of hypersurfaces in manifolds with positive curvature.
method Proved a sharp Clifford-threshold focal-radius estimate and rigidity under specific curvature conditions.
result Any closed two-sided immersion satisfies a focal radius estimate of π/4, with equality case rigid.
A method to fix radius distortion in generative models on curved spaces.
problem Distortion in geodesic radius measurements across different charts on Riemannian manifolds.
method Radial Compensation (RC) adjusts the tangent-space base distribution to match the geodesic radius law, improving model stability and interpretability.
result RC ensures that the model's geodesic radius matches the intended distribution, improving numerical stability and curvature interpretation.
We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the b…
We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.
In this short note, we study the injectivity radius bound for three dimensional complete and non-compact Riemannian manifold with good leaf foliations and with bounded curvature up to first order. We obtain the injectivity bound by using the minimal surface theory and the Gauss-Bonnet theorem.
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…
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.
In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick applicatio…
This paper considers metric balls B(p,R) in two dimensional Riemannian manifolds when R is less than half the convexity radius. We prove that Area(B(p,R))≥π8R2. This inequality has long been conjectured for R less than half the injectivity radius. This result also yields the upper bound $μ_2(B(p,R)…
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.
Eigenfunction maxima inside high-d nodal domains.
problem Understanding eigenfunction maxima in high-dimensional nodal domains.
method Proving eigenfunction maxima inside nodal domains of high-dimensional manifolds.
result Eigenfunction maxima are within a specific radius of the eigenvalue and dimension.
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.
problem Bounding the Fridman function for hyperbolic manifolds.
method Analyzing the relationship between the Fridman function and the injectivity radius function.
result The Fridman function is bounded above by the injectivity radius function for certain hyperbolic manifolds.
Study on Gehring link problem and width of bands in curved manifolds.
problem Width of bands in positively curved manifolds.
method Same idea applied to focal radius and rigidity of hypersurfaces.
result Sphere theorem for hypersurfaces in Sn involving focal radius and rigidity of Clifford hypersurface. Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.
problem Establishing uniform bounds for eigenvalues of the Hodge Laplacian on manifolds with specific geometric constraints.
method Using uniform bounds on Ricci curvature, injectivity radius, and diameter, we derive eigenvalue estimates for the Hodge Laplacian.
result Uniform upper bounds for eigenvalues of the Hodge Laplacian on differential forms on manifolds with given geometric constraints.