Edson S. Figueiredo, Jaime Ripoll
It is given a topological pinching for the injectivity radius of a compact embedded surface either in the sphere or in the hyperbolic space
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that under a suitable pinching condition, the hypersurface is diffeomorphic and almost isometric to a geodesic hypersphere.
We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. We show that if the pinching condion is strong enough with a dependance on the norm of the second foundamental form, then the hypersurface is diffeomorphic and almost isometric to a geodesic hypersphere.
We determine the cut locus of arbitrary non-simply connected, compact and irreducible Riemannian symmetric space explicitly, and compute injectivity radius and diameter for every type of them.
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.
We explore relationship between the cut locus of an arbitrary simply connected and compact Riemannian symmetric space and the Cartan polyhedron of corresponding restricted root system, and compute injectivity radius and diameter for every type of irreducible ones.
Anisotropic extrinsic radius pinching for hypersurfaces and the stability of the Wulff shape
本文证明了涉及各向异性平均曲率的Hasanis--Koutroufiotis型外半径不等式，并研究了等号情形，证明几乎极值超曲面在Hausdorff距离意义下接近Wulff形状。
We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive
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.
In this paper, we study the injectivity radius bound for 3-d Ricci flow. As applications we show the long time existence of the Ricci flow with positive Ricci curvature. We also partially settle a question in page 302 of the book of Chow-Lu-Ni (2006).
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.
Compactness for minimal surfaces with injectivity radius bounded from below
证明了闭黎曼3-流形中面积有上界且单射半径有下界的闭嵌入极小曲面空间的紧致性定理，用单射半径下界替代了Choi-Schoen紧致性定理中的亏格界。
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(infinite). Furthermore with the additional assumption that the second derivative of the Ricci tensor is null we prove that the same upper bound for the infimum of the Ricci curvatures holds for the supremum of the Ricci curvatures and $M^n$ has, in fact, parallel Ricci tensor.
A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic volume ratio of gradient Ricci solitons.
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 $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname{inj}(p)}(p))\}$, where $\operatorname{inj}(p)$ is the injectivity radius of $p$ and $\operatorname{foc}(B_r(p))$ is the focal radius of open ball centered at $p$ with radius $r$; 2) for any two points $p,q$ in $M$, $\operatorname{inj}(q)\ge \min\{\operatorname{inj}(p), \operator…
In this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3-manifolds. We require the metric have the form g_{ij}=δ_{ij}+h_{ij} with h_{ij}=O_{4}(r^{-1}) and R=O(r^{-3-τ}),τ>0. We do not require the metric to be close to Schwarzschild metric in any sense or to satisfy RT conditions. We prove that, when the mass is not 0, stable CMC spheres that separate a certain compact part from infinity satisfy the radius pinching estimate r_{1}\leq Cr_{0} , which in many cases is critical to prove the uniqueness of the CMC spheres. As applications of thi…
Bi-Lipschitz Smoothing under Ricci and Injectivity Bounds
该研究证明了在正的里奇曲率和注入半径下界条件下，流形可被双利普希茨光滑度量近似，回答了开放问题。
Using Lie groupoids, we prove that the injectivity radius of a manifold with a Lie structure at infinity is positive.
Lower bounds on the normal injectivity radius of hypersurfaces and bounded geometries on manifolds with boundary
本文首次对超曲面法向注入半径进行点wise下界估计，并在任意带有边界的流形上构造具有有界几何的度量，同时证明边界可变为凸。
Strengthened injectivity radius bounds for manifolds with positive scalar curvature
本文通过额外拓扑假设，加强了具有正标量曲率流形的注入半径界，并证明了对于大直径3流形的加强不等式。
Stability of the Injectivity Radius and the Cut Locus of Submanifolds under Perturbations
本文证明了子流形的注入半径关于度量连续依赖，并在度量C²扰动下得到子流形割迹的Hausdorff稳定性。
We give a lower bound on the boundary injectivity radius of the Margulis tubes with smooth boundary constructed by Buser, Colbois, and Dodziuk. This estimate depends on the dimension and a curvature bound only.
The Injectivity Radius of Souls of Alexandrov Spaces
本文将Šarafutdinov关于非紧非负曲率黎曼流形注入半径的下界推广到Alexandrov空间，证明当空间注入半径不等于其灵魂的注入半径时，其至少为$πK^{-1/2}$，并系统引入灵魂概念及比较两种注入半径定义。
The paper is concerned with regularity properties of boundaries of causal pasts of points in a 3+1-dimensional Einstein-vacuum spacetime. In a Lorentzian manifold such boundaries play crucial role in propagation of linear and nonlinear waves. We prove a uniform lower bound on the radius of injectivity of these null boundaries in terms of the Riemann curvature flux along them and some additional quantities arising specifically in a problem of a large data breakdown criterion in General Relativity
The ultimate upper bound on the injectivity radius of the Stiefel manifold
该研究针对Hüper等人2021年提出的含典范、欧几里得度量的黎曼度量族，通过分析Stiefel流形的共轭点与测地环得到单射半径上界，并用数值实验验证该上界等于单射半径的猜想。
Scalar Curvature, Injectivity Radius and Immersions with Small Second Fundamental Forms
本文在 spin 条件及 n≤8 限制下，研究正标量曲率流形的单射半径、光滑浸入的焦半径以及区域平均曲率的拓扑下界。
We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates. In contrast with earlier results based on a global bound for derivatives of the curvature, our method requires only a sup-norm bound on the curvature near the given observer.
The injectivity radius of the compact Stiefel manifold under the Euclidean metric
本文研究欧几里得度量下紧斯蒂弗尔流形的单射半径，通过证明其测地线为常弗雷内曲率空间曲线，最终确定该流形的单射半径为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 characterization of the convexity radius that resembles a classical result of Klingenberg about the injectivity radius.
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 there is a subsequence which converges to an n-dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov-Hausdorff sense to a space which i…
Shortest Geodesic Loops, Sectional Curvature, and Injectivity Radius of the Stiefel Manifold
本文确定了Stiefel流形在Hüper等人引入的单参数黎曼度量族下非平凡测地线环的最短长度，并结合截面曲率界限精确计算了该流形在广泛度量参数下的单射半径。
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 $Area(B(p,R)) \geq \frac{8}πR^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $μ_2(B(p,R)) \leq 2(\fracπ{2 R})^2$ on the first nonzero Neumann eigenvalue $μ_2$ of the Laplacian in terms only of the radius. This has also been conjectured for $R$ up to half the injectivity radius.
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 manifold is proved.
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 $|Rm|\leq Ct^{-1}$ for all $t\in (0,T]$, where $C$ depends only on the dimension $n$ and $T$ depends on $K$ and the injectivity radius $inj_{g(t)}$. In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly…
Very recently, the Fridman function of a complex manifold $X$ has been identified as a dual of the squeezing function of $X$. In this paper, we prove that the Fridman function for certain hyperbolic complex manifold $X$ is bounded above by the injectivity radius function of $X$. This result also suggests us to use the Fridman function to extend the definition of uniform thickness to higher-dimensional hyperbolic complex manifolds. We also establish an expression for the Fridman function (with respect to the Kobayashi metric) when $X = \mathbb{D} \diagup Γ$ and $Γ$ is a torsion-free discrete su…
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 a minimal parking garage structure on $\mathbb{R}^3$, whose theory we also develop.
A counterexample to Yau's bounded-mean-curvature embedding question
针对Yau关于有界Ricci曲率与正单射半径流形是否存在有界平均曲率等距嵌入的问题，构造了四维及以上维度的反例，证明此类嵌入不存在。
In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $ϕ$ into $§^{n+1}$. Let $q>n$ and $A>0$ be some real numbers satisfying $|M|^\frac{1}{n}(1+\|B\|_q)\leq A$. Suppose that $ϕ(M)\subset B(p_0,R)$, where $p_0$ is a center of gravity of $M$ and radius $R<\fracπ{2}$. We prove that there exists a positive constant $\e$ depending on $q$, $n$, $R$ and $A$ such that if $n(1+\|H\|_\infty^2)-\e\leq ł_1$, then $M$ is diffeomo…
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, and are formulated via a reference Riemannian metric that we canonically associate with a given observer $(p,T)$ --where $p$ is a point of the manifold and $T$ is a future-oriented time-like unit vector prescribed at $p$. The proofs are based on a generalization of arguments fro…
Dirichlet-Voronoi domain and injectivity radius of flag manifolds -- equivariant cell structure on $O(3)/O(1)^3$
第一部分研究黎曼流形离散等距群的Dirichlet-Voronoi区域，用于构造齐次旗流形的Weyl群等变胞腔结构；第二部分计算实和复旗流形的内射半径，并具体构造$O(3)/O(1)^3$的$\mathfrak{S}_3$-等变胞腔结构。
We verify here some variants of topological and dynamical flavor of the injectivity radius conjecture in Hofer geometry, Lalonde-Savelyev \cite{citeLalondeSavelyevOntheinjectivityradiusinHofergeometry} in the case of $Ham (S^2)$ and $Ham(Σ, ω)$, for $Σ$ a closed positive genus surface. In particular we show that any loop in $Ham (S^2)$, respectively $Ham(Σ, ω)$ with $L ^{+}$ Hofer length less than $area(S ^{2} )/2$, respectively any $L ^{+} $ length is contractible through ($L ^{+} $) Hofer shorter loops, in the $C ^{\infty} $ topology. We also prove some stronger variants of this statement on…
Injectivity radius lower bound of convex sum of tame Riemannian metrics and applications to symplectic topology
该研究证明了一对曲率有界、单射半径有正下界的完备驯黎曼度量的凸和仍保持曲率有界且单射半径有一致下界，进而证明驯度量拟等距类的凸性与可缩性，以及对应$C^3$-驯殆复结构集合的可缩性，为辛拓扑研究提供了黎曼几何工具。
We prove a completely new integral criterion for the existence and completeness of the wave operators $W_{\pm}(-Δ_h,-Δ_g, I_{g,h})$ corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators $-Δ_j$, $j=1,2$, that are induced by two quasi-isometric complete Riemannian metrics $g$ and $h$ on an open manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra of $-Δ_g$ and $-Δ_h$ to coincide. Our proof relies on estimates that are obtained using a probabilistic Bismut type formula for the gradient of a heat semigroup. Unlike a…
Infinite Volume and Infinite Injectivity Radius
该研究证明了Margulis关于高秩单李群中无穷协体积离散子群对应局部对称空间有任意半径嵌入球的猜想，还拓展到带Kazhdan性质(T)的高秩半单群，得到相关刚性与平稳极限结果。
Regularity and structure of spaces with synthetic Ricci bounds and positive injectivity radius
本文为具合成Ricci界的RCD空间建立结构理论，在单射半径为正的假设下重构光滑微分结构并证正则性结果，还建立调和半径下界等，证实了V. Kapovitch的相关猜想。
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 base so that the lifted metric has an arbitrarily large finite number of gaps in its essential spectrum. Also, for any complete noncompact manifold with bounded curvature and positive injectivity radius we construct a metric uniformly equivalent to the given one (also of bounded c…
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. This result was conjectured by Bavard who has already proved the genus zero cases. We establish a similar inequality for surfaces with boundary. The proofs rely on a version due to Yau of the Schwarz lemma, and on the work of Bavard. This article is the sequel of a previous one whe…
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 surfaces. This half-space theorem applies to certain properly immersed constant mean curvature surfaces of X contained in the complements of normal R^2 subgroups F of X. In the case X is a unimodular Lie group, our results imply that every minimal surface in X-F that is properly immers…
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 $a_k$. This result is new also in the Riemannian case, where one can arrange in addition that $g$ is complete with injectivity and convexity radius greater than 1. One can even make the radii rapidly increasing and the functions $a_k$ rapidly decreasing at infinity. We prove generalizations to foliated manifolds, where curvature, second fundamental form and injectivity radius of the leaves…
In this paper we study the injectivity radius based at a fixed point along Weil-Petersson geodesics. We show that the square root of the injectivity radius based at a fixed point is $ 0.3884$-Lipschitz on Teichmüller space endowed with the Weil-Petersson metric. As an application we reprove that the square root of the systole function is uniformly Lipschitz on Teichmüller space endowed with the Weil-Petersson metric, where the Lipschitz constant can be chosen to be $0.5492$. Applications to asymptotic geometry of moduli space of Riemann surfaces for large genus will also be discussed.
We study the relationship between two norms on the first cohomology of a hyperbolic 3-manifold: the purely topological Thurston norm and the more geometric harmonic norm. Refining recent results of Bergeron, Şengün, and Venkatesh as well as older work of Kronheimer and Mrowka, we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3-manifold itself. Moreover, we give families of examples showing that some (but not all) qualitative aspects of our estimates are sharp. Finally, we exhibit closed hyperbolic 3-…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。