arXiv research

Compact theorem for minimal surfaces with lower injectivity radius.

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 paper proves injectivity and vanishing theorems on compact Kahler manifolds.

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…

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…

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 …

We present a noise-injected version of the Expectation-Maximization (EM) algorithm: the Noisy Expectation Maximization (NEM) algorithm. The NEM algorithm uses noise to speed up the convergence of the EM algorithm. The NEM theorem shows that injected noise speeds up the average convergence of the EM algorithm to a local…

X-ray transform on H-type groups solved, revealing function injectivity.

We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call…

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

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.

Injective X-ray transform on Heisenberg group for regular functions.

The abstract discusses p-harmonic forms and their geometric properties, proving new theorems about Lp-cohomology.

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…

Paper solves injectivity of X-ray transform on surfaces.

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…

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let $π$ be a group and let $M \to N$ be a homomorphism between projective $\Z[π]$-modules such that $\Z_p \otimes_{\Z[π]} M\to \Z_p \otimes_{\Z[π]} N$ is injective; for which other right $\Z[π]…

The paper extends Huber's theorem to higher dimensions using n-Laplace equations.

We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter and lower bounded injectivity radius.

Generalizes a soul-bound for noncompact Alexandrov spaces.

In this work, we show an injectivity result and support theorems for integral moments of a m-tensor field on a simple, real analytic, Riemannian manifold. Integral moments of m-tensor field were first introduced by Sharafutdinov. At first we generalize a Helgason type support theorem proven by Krishnan and Stefanov in …

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 paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.

The paper proves that certain spaces have injective balls of any radius.

Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.

Survey discusses new ideas in geometric group theory and their applications.

Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among o…

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

Study the discontinuity of functions not embeddable in Euclidean space.

Simplified proof of a theorem about 3D shapes.

Sphere theorems for p-Laplacian eigenvalues established.

Maps preserving mass and injective on boundary are isometries.

Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be de…

We study the embedded Calabi-Yau problem for complete embedded constant mean curvature surfaces of finite topology or of positive injectivity radius in a simply-connected three-dimensional Lie group X endowed with a left-invariant Riemannian metric. We first prove a half-space theorem for constant mean curvature surfac…

We prove two injectivity theorems for the geodesic ray transform on two-dimensional, complete, simply connected Riemannian manifolds with non-positive Gaussian curvature, also known as Cartan-Hadamard manifolds. The first theorem is concerned with bounded non-positive curvature and the second with decaying non-positive…

In this paper, we consider the conormal bundle over a submanifold in a Finsler manifold and establish a volume comparison theorem. As an application, we derive a lower estimate for length of closed geodesics in a Finsler manifold. In the reversible case, a lower bound of injective radius is also obtained.

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,α}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein m…

Hyperbolic spaces have many cells in their fibers.

Reduces conjecture for Artin groups to simpler cases.

Let (M,g) be an analytic, compact, Riemannian manifold with boundary, of dimension n >= 2. We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition [23]. Using analytic microlocal analysis, we prove a microlocal regularity theorem for ge…

Injectivity of ReLU networks studied using statistical physics.

We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\…

The study connects polyhedral manifolds to Riemannian ones with geometric bounds.

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…

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answ…

Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in general not determined by its natural weight-two Hodge structure. The text gives …

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…

We prove that some symetric semi-riemannian manifolds do not admit a proper domain which is divisible by the action of a discrete group of isometries. In other words, if a closed semi-riemannian manifold is locally isometric to such a model, and if its developing map is injective, then the manifold is actually geodesic…