arXiv research

New proof of injectivity for broken non-abelian X-ray transform in Minkowski space.

A labeled oriented tree is called injective if each generator occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof relies on a new relative asphericity test based on a lemma of Stallings.

Summary of tensor tomography proofs on manifolds with boundaries.

Uniform proof of $π_1$-injectivity for certain maps in low dimensions.

Recently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain …

The exponential map fails to be injective near critical points in sub-Riemannian geometry.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

We give lower bounds on the maximal injectivity radius for a closed orientable hyperbolic 3-manifold M with first Betti number 2, under some additional topological hypotheses. A corollary of the main result is that if M has first Betti number 2 and contains no fibroid surface then its maximal injectivity radius exceeds…

Classic braids embed in virtual braids.

We give a new and complete proof of Hamilton's injectivity radius estimate for sequences with bounded and almost nonnegative curvature operators, unbounded diameters, and bump-like origins. Such sequences arise in particular from dilations about a singularity of the Ricci flow on a 3-manifold.

Injective maps in various link homologies induced by ribbon concordances proved.

The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extensi…

In this paper, we study transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs, aiming to establish an analytic theory for log canonical singularities. As a result, in the case of purely log terminal pairs, we give an analytic proof of the injectivity theorem originally proved by the …

Geodesic X-ray transform proves injective for smooth one-forms on gas giant manifolds.

The paper proves that certain spaces are injective and Helly graphs.

We give sharp conditions on a local biholomorphism $F:X \to \mathbb C^{n}$ which ensure global injectivity. For $n \geq 2$, such a map is injective if for each complex line $l \subset \mathbb C^{n}$, the pre-image $F^{-1}(l)$ embeds holomorphically as a connected domain into $\mathbb C \mathbb P^{1}$, the embedding bei…

Proves a general ∂∂̄-lemma and applies it to a Fujino conjecture.

Simplified proof of a theorem about 3D shapes.

This paper gives a proof that the fundamental group of a class of closed orientable 3-manifolds constructed from three injective handlebodies has a solvable word problem. This is done by giving an algorithm to decide if a closed curve in the manifold is null-homotopic. Non-Haken and non-Seifert fibered examples are con…

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…

Injectivity of X-ray transform proven for non-smooth metrics.

Proof of injection from double shuffle to Kashiwara-Vergne Lie algebra.

Reconstructs piecewise constant functions from geodesic integrals.

This paper gives a proof that the universal cover of a closed 3-manifold built from three $π_1$-injective handlebodies is homeomorphic to $\mathbb R^3$. This construction is an extension to handlebodies of the conditions for gluing of three solid tori to produce non-Haken Seifert fibered manifolds with infinite fundame…

The article explores metrics on buildings and symmetric spaces, proving injectivity and proper actions.

The paper proves a cohomological injection for a specific group.

Injectivity of geodesic X-ray transform on low-regularity manifolds.

The paper improves bounds on injectivity radius for manifolds with positive scalar curvature.

Uniform proof for Ricci flows on complete manifolds.

Hyperbolic spaces have many cells in their fibers.

Stability of submanifold cut loci under metric perturbations proved.

A version of a conjecture of McMullen is as follows: Given a hyperbolizable 3-manifold M with incompressible boundary, there exists a uniform constant K such that if N is a hyperbolic 3-manifold homeomorphic to the interior of M, then the injectivity radius based at points in the convex core of N is bounded above by K.…

Study proves solenoidal injectivity for tensor fields on curved manifolds with low regularity.

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…

Maximal spacetimes have unique past/future sets.

We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of Bökstedt, Hsiang, and Madsen, …

The Blaschke conjecture claims that every compact Riemannian manifold whose injectivity radius equals its diameter is, up to constant rescaling, a compact rank one symmetric space. We summarize the intuition behind this problem, the proof that such manifolds have the cohomology of compact rank one symmetric spaces, and…

Study filling links in 3-manifolds to understand their topological properties.

The paper bounds eigenvalues of specific operators on certain manifolds.

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $Δ_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than $δ$, absolute values of their sectional curvatures $|K_σ|\leq C$, and their injectivity radii…

The survey is devoted to Toponogov's conjecture, that {\it if a complete simply connected Riemannian manifold with sectional curvature $\le 4$ and injectivity radius $\ge π/2$ has extremal diameter $π/2$, then it is isometric to CROSS}. In Section 1 the relations of problem with geodesic foliations of a round sphere ar…

Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S^2 x S^2's. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a certain map from the second homology of the fundamental group with coefficients in Z/2 to…

We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact Kähler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of the uniform estimate does not rely on the exsitence of Kähler-Einstein metrics on $M…

New tests detect asphericity in complex pairs, simplifying previous proofs.

Direct proof of cohomology isomorphism for pullback Lie algebroids.

We study the injectivity radius of complete Riemannian surfaces (S,g) with curvature |K(g)| bounded by 1. We show that if S is orientable with nonabelian fundamental group, then there is a point p in S with injectivity radius at least arcsinh(2/\sqrt{3}). This lower bound is sharp independently of the topology of S. Th…

Paper proposes using logic networks to inject prior knowledge for better reinforcement learning.

Affirmative proof that rank 3 3-manifolds have filling links.