The paper proves that certain spaces are injective and Helly graphs.
problem Understanding the structure of certain geometric and algebraic spaces.
method Building Helly graphs and injective metric spaces from lattices.
result The natural piecewise ℓ∞ metric on Euclidean buildings and Deligne complexes is injective. A quasi-geodesic is Morse if and only if it is strongly contracting in injective spaces.
problem Characterizing Morse quasi-geodesics in injective spaces.
method Proving equivalence between Morse and strongly contracting quasi-geodesics.
result Injective metric spaces have the Morse local-to-global property and acylindrically hyperbolic groups with Morse elements.
Lecture notes on group actions on injective spaces and Helly graphs.
problem Understanding group actions on specific metric spaces.
method Review of injective metric spaces and Helly graphs, elementary properties, constructions, and exercises.
result Presentation of various constructions of injective metric spaces and Helly graphs with interesting group actions.
The article explores metrics on buildings and symmetric spaces, proving injectivity and proper actions.
problem Injectivity of metrics on buildings and symmetric spaces.
method Analyzing norms, Helly property, and group actions.
result Most classical buildings and symmetric spaces can be endowed with injective metrics.
Word metrics from large balls in injective spaces are exponentially generic and growth tight.
problem Understanding genericity and growth in word metrics from injective spaces.
method Geometric arguments and large ball considerations.
result Exponential genericity and growth tightness for word metrics.
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.
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.
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.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. 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.
Injective X-ray transform on Heisenberg group for regular functions.
problem Injectivity of X-ray transform on sub-Riemannian manifolds.
method Group Fourier Transform and analysis of taming metrics.
result Sufficiently regular functions on Heisenberg group are determined by their line integrals.
The paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.
problem Understanding the geometry of convex sums of Riemannian metrics.
method Quantitative inverse function theorem and Riemannian geometry techniques.
result Injectivity radii of convex sums have uniform lower bounds.
Stability of submanifold cut loci under metric perturbations proved.
problem Stability of submanifold cut loci under metric perturbations.
method Continuity of injectivity radius and Whitney C2 perturbation of submanifolds. result Hausdorff stability of submanifold cut loci under C2 metric perturbations. 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.
We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive
Local X-ray transform works well near boundaries in hyperbolic spaces.
problem Injectivity of X-ray transform near boundaries for hyperbolic metrics.
method Local injectivity proof for geodesic X-ray transform on asymptotically hyperbolic manifolds.
result Local injectivity near a boundary point for X-ray transform in dimensions 3 and higher, up to O(ρ5). 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…
Survey discusses new ideas in geometric group theory and their applications.
problem Understanding geodesic metric spaces and their equivariant wall structures.
method Introduces and highlights the impact of injective metric spaces and cubical approximation theorem.
result Rich equivariant wall structures in various geodesic metric spaces.
New infinite K(π,1) arrangements found in higher dimensions.
problem Finding new infinite K(π,1) arrangements in higher dimensions. method Endowing Falk complexes with an injective metric.
result New examples of infinite K(π,1) arrangements in dimension n>2. Introduces injective category number for continuous maps, linking classical and contemporary research.
problem Understanding conditions for a continuous map to be injective.
method Defines injective category number and examines its behavior under various operations.
result Provides a cohomological lower bound and expressions for injective category numbers in specific cases.
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
Estimates for harmonic forms on a 3-Torus, proving their existence.
problem Existence of nowhere vanishing harmonic 1-forms on a 3-Torus.
method Explicit computation of injectivity estimates using the Laplace operator on the 3-Torus and its perturbations.
result Existence of a nowhere vanishing harmonic 1-form on a perturbed metric on the 3-Torus.
The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.
problem Uniform Lipschitz bounds on geometric functions in Teichmüller space.
method Injectivity radius analysis and Lipschitz bounds on systole function.
result Uniform Lipschitz constant for the square root of the systole function on Teichmüller space.
The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the h…
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.
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.
Proves X-ray transform injectivity on specific manifolds.
problem Injectivity of X-ray transform on closed Anosov manifolds and spherical boundary.
method Perturbative argument of the 0-eigenvalue of elliptic operators via microlocal analysis.
result Generically injective X-ray transform on specified manifolds.
Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then t…
We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equati…
In this paper we prove that if a point p in a complete Riemannian manifold is not a cut point of any point whose distance to p is r, then the injectivity radius of p is strictly large than r. As a corollary we give a positive answer to a problem raised by Z. Sun and J. Wan.
We study ray transforms on spherically symmetric manifolds with a piecewise C1,1 metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of L2 functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds …
We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge…
The paper studies harmonic maps between surfaces homotopic to a covering map, proving uniqueness and injectivity of Hopf differential.
problem Analyzing harmonic maps between surfaces in the homotopy class of a covering map.
method Proving the uniqueness of critical points and injectivity of Hopf differential for harmonic maps.
result The uniqueness of critical points of energy function and injectivity of Hopf differential are proven under specific conditions.
We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the W2,p-harmonic radius…
We show that on every manifold, every conformal class of semi-Riemannian metrics contains a metric g such that each k-th-order covariant derivative of the Riemann tensor of g has bounded absolute value ak. This result is new also in the Riemannian case, where one can arrange in addition that g is complete wi…
New method to bound Laplacian eigenvalues of geodesic balls.
problem Computing upper bounds for the first eigenvalue of Laplacian on geodesic balls.
method Transforming metric tensor into rotationally symmetric form preserving geodesic sphere areas.
result Upper bound for Laplacian eigenvalues is sharp and computable using geodesic sphere areas.
Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric,…
In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem.
Paper proves rigidity of metrics near hyperbolic ones in 3D.
problem Proving rigidity of metrics near hyperbolic ones in 3D.
method Introducing marked Poincaré determinant and proving local rigidity.
result Lichnerowicz Laplacian is injective in negative curvature.
Equivalence of norms on manifolds with curvature bounds established.
problem Establishing equivalence of norms on manifolds with bounded sectional curvature.
method Using spectral projector and thickness condition for subsets.
result Constant in equivalence depends only on manifold dimension, curvature bounds, and frequency threshold.
Reduces conjecture for Artin groups to simpler cases.
problem Proving K(π,1) for Artin groups with specific spherical parabolics. method Reduces to simpler cases, uses injective metric spaces, combinatorial convexity, and Bestvina-type inequalities.
result Deduces K(π,1) conjecture for specific Artin groups. 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)…
The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generali…
New Einstein metrics found in curved spaces.
problem Finding Einstein metrics in curved spaces.
method Analyzing almost-Einstein metrics to find genuine Einstein metrics.
result Negative curvature preserved in Einstein metrics.
Study coning totally geodesic boundaries of hyperbolic manifolds.
problem Understanding metrics on coned-off spaces of hyperbolic manifolds.
method Analyzing the geometric and group-theoretic properties of coned-off spaces.
result Explicit conditions for negatively curved metrics and locally convex subsets.
The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a rela…
We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements for the geodesic flow.
We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat metrics asymptotic to the product of a real line with a flat 3-manifold.