Injectivity Radius Research Articles

L2-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds

We give L2-bounds on the change in the complex projective structure on the boundary of conformally compact hyperbolic 3-manifold with incompressible boundary after drilling short geodesics. We show that the change is bounded by a universal constant times the square root of the length of the drilled geodesics. While L∞-bounds of this type where obtained in [7], our bounds here do not depend on the injectivity radius of the boundary.

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Frøyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in 3- and 4-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal L -spaces. In particular, we study relations between Frøyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive explicit upper bounds for the Frøyshov invariants of minimal hyperbolic L -spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Frøyshov invariants of minimal L -spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Frøyshov invariants for all spin ^{c} structures on the Seifert–Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.

$L^{p}$ gradient estimates and Calderón–Zygmund inequalities under Ricci lower bounds

In this paper, we investigate the validity of first and second order L^{p} estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present L^{p} estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain L^{p} estimates for the second order Riesz transform (or, equivalently, the validity of L^{p} Calderón–Zygmund inequalities) on the whole scale 1<p<+\infty by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When 1<p \leq 2 , analogous L^{p} bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order.

The Injectivity Radius and Shortest Arcs of the Oblate Ellipsoid of Revolution

The Injectivity Radius and Shortest Arcs of the Oblate Ellipsoid of Revolution

Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds

In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds M with Ricci curvature bounded from below, positive injectivity radius and spectral gap.Our first main result states that, if L is the positive Laplace–Beltrami operator on M, then the Riesz–Hardy space HR1(M) is the isomorphic image of the Goldberg type space h1(M) via the map L1/2(I+L)−1/2, a fact that is false in Rn. Specifically, HR1(M) agrees with the Hardy type space X1/2(M) recently introduced by the first three authors; as a consequence, we prove that HR1(M) does not admit an atomic characterisation.Noninclusions are mostly proved in the special case where the manifold is a Damek–Ricci space S. Our second main result states that HR1(S), the heat Hardy space HH1(S) and the Poisson–Hardy space HP1(S) are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.

On foliations of bounded mean curvature on closed three-dimensional Riemannian manifolds

The notion of systole of a foliation sys(ℱ) on an arbitrary foliated closed Riemannian manifold (M,ℱ) is introduced. A lower bound on sys(ℱ) of a bounded mean curvature foliation is given. As a corollary we prove that the number of Reeb components of a bounded mean curvature foliation on a closed oriented Riemannian 3-manifold M is bounded above by a constant depending on the volume, the radius of injectivity, and the maximum value of the sectional curvature of the manifold M.

Fitting a manifold of large reach to noisy data

Let [Formula: see text] be a [Formula: see text]-smooth compact submanifold of dimension [Formula: see text]. Assume that the volume of [Formula: see text] is at most [Formula: see text] and the reach (i.e. the normal injectivity radius) of [Formula: see text] is greater than [Formula: see text]. Moreover, let [Formula: see text] be a probability measure on [Formula: see text] whose density on [Formula: see text] is a strictly positive Lipschitz-smooth function. Let [Formula: see text], [Formula: see text] be [Formula: see text] independent random samples from distribution [Formula: see text]. Also, let [Formula: see text], [Formula: see text] be independent random samples from a Gaussian random variable in [Formula: see text] having covariance [Formula: see text], where [Formula: see text] is less than a certain specified function of [Formula: see text] and [Formula: see text]. We assume that we are given the data points [Formula: see text] [Formula: see text], modeling random points of [Formula: see text] with measurement noise. We develop an algorithm which produces from these data, with high probability, a [Formula: see text] dimensional submanifold [Formula: see text] whose Hausdorff distance to [Formula: see text] is less than [Formula: see text] for [Formula: see text] and whose reach is greater than [Formula: see text] with universal constants [Formula: see text]. The number [Formula: see text] of random samples required depends almost linearly on [Formula: see text], polynomially on [Formula: see text] and exponentially on [Formula: see text].

Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus

We give a quantitative estimate for the quantum mean absolute deviation on hyperbolic surfaces of finite area in terms of geometric parameters such as the genus, number of cusps and injectivity radius. It implies a delocalisation result of quantum ergodicity type for eigenfunctions of the Laplacian on hyperbolic surfaces of finite area that Benjamini-Schramm converge to the hyperbolic plane. We show that this is generic for Mirzakhani’s model of random surfaces chosen uniformly with respect to the Weil-Petersson volume. Depending on the particular sequence of surfaces considered this gives a result of delocalisation of most cusp forms or of Eisenstein series.

Stable isoperimetric ratios and the Hodge Laplacian of hyperbolic manifolds

AbstractWe show that for a closed hyperbolic 3‐manifold, the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1‐forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3‐manifolds with injectivity radius bounded below and volume going to infinity for which the 1‐form Laplacian has spectral gap vanishing exponentially fast in the volume.

Injectivity radius of manifolds with a Lie structure at infinity

Using Lie groupoids, we prove that the injectivity radius of a manifold with a Lie structure at infinity is positive.

Manifold Embeddings by Heat Kernels of Connection Laplacian

We show that any closed n-dimensional manifold (M, g) can be embedded by a map constructed using the heat kernels of the connection Laplacian as well as maps constructed using truncated heat kernel at a certain time t from a $$\delta $$ -net $$\{q_i\}_{i=1}^{N_0}$$ via a rescaling technique. Both the time t and $$N_0$$ are bounded in terms of the dimension, bounds on the Ricci curvature and its derivative, the injectivity radius, and the volume. Moreover, both maps can be made arbitrarily close to an isometry.

Effective Discreteness Radius of Stabilizers for Stationary Actions

We prove an effective variant of the Kazhdan–Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a nontrivial intersection with a small r-neighborhood of the identity is at most βrδ for some explicit constants β,δ>0 depending only on the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a novel proof of the fact that all lattices in semisimple groups are weakly cocompact.

A new uniform lower bound on Weil–Petersson distance

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 Teichm\"uller space endowed with the Weil-Petersson metric. As an application we reprove that the square root of the systole function is uniformly Lipschitz on Teichm\"uller 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.

Reconstruction and stability in Gelfand’s inverse interior spectral problem

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $\Delta_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit $\log\log$-type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gel'fand's inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric Boundary Control method.

SO(2)×SO(3)$SO(2)\times SO(3)$‐invariant Ricci solitons and ancient flows on S4$\mathbb {S}^4$

Consider the standard action of S O ( 2 ) × S O ( 3 ) $SO(2)\times SO(3)$ on R 5 = R 2 ⊕ R 3 $\mathbb {R}^5=\mathbb {R}^2\oplus \mathbb {R}^3$ . We establish the existence of a uniform constant C > 0 $\mathcal {C}>0$ so that any S O ( 2 ) × S O ( 3 ) $SO(2)\times SO(3)$ -invariant Ricci soliton on S 4 ⊂ R 5 $\mathbb {S}^4\subset \mathbb {R}^5$ with Einstein constant 1 must have Riemann curvature and volume bounded by C $\mathcal {C}$ , and injectivity radius bounded below by 1 C $\frac{1}{\mathcal {C}}$ . This observation, coupled with basic numerics, gives strong evidence to suggest that the only S O ( 2 ) × S O ( 3 ) $SO(2)\times SO(3)$ -invariant Ricci solitons on S 4 $\mathbb {S}^4$ are round. We also prove existence of the so-called ‘pancake’ ancient solution of the Ricci flow.

Diameter of homogeneous spaces: an effective account

In this paper we prove explicit estimates for the size of small lifts of points in homogeneous spaces. Our estimates are polynomially effective in the volume of the space and the injectivity radius.

Tubular Excision and Steklov Eigenvalues

Given a closed manifold M and a closed connected submanifold \(N\subset M\) of positive codimension, we study the Steklov spectrum of the domain \(\varOmega _\varepsilon \subset M\) obtained by removing the tubular neighborhood of size \(\varepsilon \) around N. All nonzero eigenvalues in the mid-frequency range tend to infinity at a rate which depends only on the codimension of N in M. Eigenvalues above the mid-frequency range are also described: they tend to infinity following an unbounded sequence of clusters. This construction is then applied to obtain manifolds with unbounded perimeter-normalized spectral gap and to show the necessity of using the injectivity radius in some known isoperimetric-type upper bounds.

Scattering Theory for the Hodge Laplacian

We prove using an integral criterion the existence and completeness of the wave operators \(W_{\pm }(\Delta _h^{(k)}, \Delta _g^{(k)}, I_{g,h}^{(k)})\) corresponding to the Hodge Laplacians \(\Delta _\nu ^{(k)}\) acting on differential \(k\)-forms, for \(\nu \in \{g,h\}\), induced by two quasi-isometric Riemannian metrics g and h on a complete open smooth manifold M. In particular, this result provides a criterion for the absolutely continuous spectra \(\sigma _{\mathrm {ac}}(\Delta _g^{(k)}) = \sigma _{\mathrm {ac}}(\Delta _h^{(k)})\) of \(\Delta _\nu ^{(k)}\) to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion requires local curvature bounds and some upper local control on the heat kernel acting on functions provided the Weitzenböck curvature endomorphism is in the Kato class, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.

Local Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry

We prove that if \(\tau \) is a large positive number, then the atomic Goldberg-type space \({\mathfrak {h}}^1(N)\) and the space \({\mathfrak {h}}_{{\mathscr {R}}_\tau }^1(N)\) of all integrable functions on N of which local Riesz transform \({\mathscr {R}}_\tau \) is integrable, are the same space on any complete noncompact Riemannian manifold N with Ricci curvature bounded from below and positive injectivity radius. We also relate \({\mathfrak {h}}^1(N)\) to a space of harmonic functions on the slice \(N\times (0,\delta )\) for \(\delta >0\) small enough.

Higher order distance-like functions and Sobolev spaces

On non-compact Riemannian manifolds, we construct distance-like functions with derivatives controlled up to some order k assuming bounds on the growth of the derivatives of the curvature up to order k−2 and on the decay of the injectivity radius. This construction extends previously known results in various directions, permitting to obtain consequences which are (in a sense) sharp. As a first main application, we give refined conditions guaranteeing the density of compactly supported smooth functions in the Sobolev space Wk,p on the manifold. Contrary to all previously known results this can be obtained also on manifolds with possibly unbounded geometry. In the particular case p=2, making use of the Weitzenböck formula for a Lichnerowicz Laplacian acting on k-covariant totally symmetric tensor fields, we can weaken the assumptions needed to obtain the density property, avoiding any condition on the highest order derivatives of the curvature. Distance-like functions are also used to obtain new disturbed Sobolev inequalities, disturbed Lp-Calderón-Zygmund inequalities and the full Omori-Yau maximum principle for the Hessian under weak assumptions.