Injectivity Radius Research Articles

Finite topology minimal surfaces in homogeneous three-manifolds

We prove that any complete, embedded minimal surface M with finite topology in a homogeneous three-manifold N has positive injectivity radius. When one relaxes the condition that N be homogeneous to that of being locally homogeneous, then we show that the closure of M has the structure of a minimal lamination of N. As an application of this general result we prove that any complete, embedded minimal surface with finite genus and a countable number of ends is compact when the ambient space is S3 equipped with a homogeneous metric of nonnegative scalar curvature.

Symmetry gaps in Riemannian geometry and minimal orbifolds

We study the size of the isometry group $\mathrm{Isom}(M,g)$ of Riemannian manifolds $(M,g)$ as $g$ varies. For $M$ not admitting a circle action, we show that the order of $\mathrm{Isom}(M,g)$ can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of $M$. This generalizes results known for negative Ricci curvature to all manifolds. More generally we establish a similar universal bound on the index of the deck group $\pi_1 (M)$ in the isometry group $\mathrm{Isom}(\widetilde{M},\widetilde{g})$ of the universal cover $\widetilde{M}$ in the absence of suitable actions by connected groups. We apply this to characterize locally symmetric spaces by their symmetry in covers. This proves a conjecture of Farb andWeinberger with the additional assumption of bounds on curvature, diameter, and injectivity radius. Further we generalize results of KazhdannMargulis and Gromov on minimal orbifolds of nonpositively curved manifolds to arbitrary manifolds with only a purely topological assumption.

Some observations concerning multidimensional quasiconformal mappings

For several important objects and quantities in the theory of quasiconformal space mappings, we discuss their dependence on the dimension of the space. In particular, in connection with the global homeomorphism theorem and the theorem on the injectivity radius of quasiconformal immersions, we consider the asymptotic behaviour of the moduli of Grötzsch and Teichmüller rings with respect to the dimension. Bibliography: 23 titles.

The convexity radius of a Riemannian manifold

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.

Renormalized volume and the volume of the convex core

We obtain upper and lower bounds on the difference between the renormalized volume and the volume of the convex core of a convex cocompact hyperbolic 3-manifold which depend on the injectivity radius of the boundary of the universal cover of the convex core and the Euler characteristic of the boundary of the convex core. These results generalize results of Schlenker obtained in the setting of quasifuchsian hyperbolic 3-manifolds.

Building hyperbolic metrics suited to closed curves and applications to lifting simply

Let $\gamma$ be an essential closed curve with at most $k$ self-intersections on a surface $\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $\rho$ for which $\gamma$ has length at most $M \cdot \sqrt{k}$, where $M$ is a constant depending only on the topology of $\mathcal{S}$. Moreover, the injectivity radius of $\rho$ is at least $1/(2\sqrt{k})$. This yields linear upper bounds in terms of self-intersection number on the minimum degree of a cover to which $\gamma$ lifts as a simple closed curve (i.e. lifts simply). We also show that if $\gamma$ is a closed curve with length at most $L$ on a cusped hyperbolic surface $\mathcal{S}$, then there exists a cover of $\mathcal{S}$ of degree at most $N \cdot L \cdot e^{L/2}$ to which $\gamma$ lifts simply, for $N$ depending only on the topology of $\mathcal{S}$.

The local maxima of maximal injectivity radius among hyperbolic surfaces

The function on the Teichmueller space of complete, orientable, finite-area hyperbolic surfaces of a fixed topological type that assigns to a hyperbolic surface its maximal injectivity radius has no local maxima that are not global maxima.

Pucci eigenvalues on geodesic balls

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng’s eigenvalue comparison theorem for the Laplace–Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng’s bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng’s upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principal Pucci eigenvalues of an \({O(n)}\)-invariant hypersurface immersed in \({{\mathbb{R}}^{n+1}}\) with one smooth boundary component are smaller than the eigenvalues of an \({n}\)-dimensional Euclidean ball with the same boundary.

Complete manifolds with bounded curvature and spectral gaps

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 curvature and positive injectivity radius) with an arbitrarily large finite number of gaps in its essential spectrum.

On the Hofer Geometry Injectivity Radius Conjecture

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 $Ham(\Sigma, \omega)$, for $\Sigma$ a closed positive genus surface. In particular we show that any loop in $Ham (S^2)$, respectively $Ham(\Sigma, \omega)$ 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 the loop space level. One dynamical type corollary is that there are no smooth, positive Morse index (Ustilovsky) geodesics, in $Ham (S^2)$, respectively in $Ham(\Sigma, \omega)$ with $L ^{+} $ Hofer length less than $area (S ^{2} )/2$, respectively any length. The above condition on the geodesics can be expanded as an explicit and elementary dynamical condition on the associated Hamiltonian flow. We also give some speculations on connections of this later result with curvature properties of the Hamiltonian diffeomorphism group of surfaces.

Rough metrics on manifolds and quadratic estimates

We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate on sets of measure zero. We also demonstrate how to transmit quadratic estimates between manifolds which are homeomorphic and locally bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato square root problem under Lipschitz transformations of the space and obtain solutions to this problem on functions and forms on compact manifolds with a continuous metric. Furthermore, we show that a lower bound on the injectivity radius is not a necessary condition to solve the Kato square root problem.

Non-divergence of unipotent flows on quotients of rank-one semisimple groups

Let$G$be a semisimple Lie group of rank one and$\unicode[STIX]{x1D6E4}$be a torsion-free discrete subgroup of$G$. We show that in$G/\unicode[STIX]{x1D6E4}$, given$\unicode[STIX]{x1D716}>0$, any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than$\unicode[STIX]{x1D6FF}$for a$1-\unicode[STIX]{x1D716}$proportion of the time, for some$\unicode[STIX]{x1D6FF}>0$. The result also holds for any finitely generated discrete subgroup$\unicode[STIX]{x1D6E4}$and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces.Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed$\unicode[STIX]{x1D716}>0$, there exists an injectivity radius$\unicode[STIX]{x1D6FF}$such that, for any unipotent trajectory$\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$, either it spends at least a$1-\unicode[STIX]{x1D716}$proportion of the time in the set with injectivity radius larger than$\unicode[STIX]{x1D6FF}$, for all large$T>0$, or there exists a$\{u_{t}\}_{t\in \mathbb{R}}$-normalized abelian subgroup$L$of$G$which intersects$g\unicode[STIX]{x1D6E4}g^{-1}$in a small covolume lattice. We also extend these results to when$G$is the product of rank-one semisimple groups and$\unicode[STIX]{x1D6E4}$a discrete subgroup of$G$whose projection onto each non-trivial factor is torsion free.

Volume comparison without curvature-assumption

Let \(\, X^n ,\ M^n\,\) be any pair of compact connected Riemannian manifolds; the only assumption on \(\,M^n\,\) is that the Gromov–Hausdorff distance \(\,\varepsilon \,\) between \(\,M^n\,\) and \(\,X^n\,\) is smaller than the (normalized) injectivity radius of \(\, X^n\,\). Using a transport of measures and a new notion of barycentre, we construct a \(\,C^1\,\) Gromov–Hausdorff \(\,C_0\cdot \varepsilon ^{2/3}\)-approximation \(\,H : M^n \rightarrow X^n\,\), whose energy and Jacobian determinant are sharply bounded from above by \(\, 1 + C_1\cdot \varepsilon ^{1/3}\,\) (the constants \(\,C_i\,\) are explicit), implying a sharp lower bound for the ratio between the volumes of \(\,M^n\,\) and \(\,X^n\,\). This result extends to the non compact case, when \(\,X^n\,\) and \(\,M^n\,\) are only supposed to be quasi-isometric.

Gradient estimates and applications for Neumann semigroup on narrow strip

By using local and global versions of Bismut type derivative formulas, gradient estimates are derived for the Neumann semigroup on a narrow strip. Applications to functional/cost inequalities and heat kernel estimates are presented. Since the narrow strip we consider is non-convex with zero injectivity radius, and does not satisfy the volume doubling condition, existing results in the literature do not apply.

Geometry of shrinking Ricci solitons

The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\text{Rm}$ of $M$ satisfies the estimate $|\text{Rm}|\leqslant cS$ for some constant $c$. Moreover, the curvature operator $\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, where $r$ is the distance function to a fixed point in $M$. As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.

On the Radius of Injectivity for Generalized Quasiisometries in the Spaces of Dimension Higher Than Two

We consider a class of local homeomorphisms more general than the mappings with bounded distortion. Under these homeomorphisms, the growth of the p-module (n−1 < p ≤ n) of the families of curves is controlled by an integral containing an admissible metric and a measurable function Q. It is shown that, under generic conditions imposed on the majorant Q, this class has a positive radius of injectivity (and, hence, a ball in which every mapping is homeomorphic). Moreover, one of the conditions imposed on Q is not only sufficient but also necessary for existence of a radius of injectivity.

Le lemme de Schwarz et la borne supérieure du rayon d’injectivité des surfaces

We study the injectivity radius of complete Riemannian surfaces (S, g) with bounded curvature \({|K(g)|\leq 1}\). We show that if S is orientable with nonabelian fundamental group, then there is a point \({p\in S}\) with injectivity radius R\({_p(g)\geq}\) 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 (Bavard 1984). We establish a similar inequality for surfaces with boundary. The proofs rely on a version due to Yau (J Differ Geom 8:369–381, 1973) of the Schwarz lemma, and on the work of Bavard (1984). This article is the sequel of Gendulphe (2014) where we studied applications of the Schwarz lemma to hyperbolic surfaces.

The centered dual and the maximal injectivity radius of hyperbolic surfaces

We give sharp upper bounds on the maximal injectivity radius of finite-area hyperbolic surfaces and use them, for each g at least 2, to identify a constant r_{g-1,2} with the property that the set of closed genus-g hyperbolic surfaces with maximal injectivity radius at least r is compact if and only if r > r_{g-1,2}. The main tool is a version of the "centered dual complex" that we introduced earlier, a coarsening of the Delaunay complex of a locally finite set. In particular, we bound the area of a compact centered dual two-cell below given lower bounds on its side lengths.

Injectivity radii of hyperbolic integer homology 3–spheres

We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H^3 whose normalized Ray-Singer analytic torsions do not converge to the L^2-analytic torsion of H^3. This contrasts with the work of Abert et. al. who showed that Benjamini-Schramm convergence forces convergence of normalized betti numbers. Our results shed light on a conjecture of Bergeron and Venkatesh on the growth of torsion in the homology of arithmetic hyperbolic 3-manifolds, and we give experimental results which support this and related conjectures.

Embeddings of Riemannian Manifolds with Heat Kernels and Eigenfunctions

AbstractWe show that any closed n‐dimensional Riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points. Both this time and this number can be bounded in terms of the dimension, a lower bound on the Ricci curvature, the injectivity radius, and the volume. It follows that the manifold can be embedded by a finite number of eigenfunctions of the Laplace operator. Again, this number only depends on the geometric bounds and the dimension. In addition, both maps can be made arbitrarily close to an isometry. In the appendix, we derive quantitative estimates of the harmonic radius, so that the estimates on the number of eigenfunctions or heat kernels needed can be made quantitative as well. © 2016 Wiley Periodicals, Inc.