Injectivity Radius Research Articles

Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II

Abstract We present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.

Higher-order geometric flow of hypersurfaces in a Riemannian manifold

In this paper, we consider the high-order geometric flows of a compact submanifolds [Formula: see text] in a complete Riemannian manifold [Formula: see text] with [Formula: see text], which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on [Formula: see text]. Precisely, we show that if [Formula: see text] is strictly larger than the integer part of [Formula: see text] and [Formula: see text] is an immersion for all [Formula: see text] and if [Formula: see text] is bounded by a constant which relies on the injectivity radius [Formula: see text] and sectional curvature [Formula: see text] of [Formula: see text], then [Formula: see text] must be [Formula: see text].

Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem

We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space (X,d_X). This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold Ssubset {{mathbb {R}}}^m, m>n needs to be constructed to approximate a point cloud in {{mathbb {R}}}^m. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov–Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. As an application of the main results, we give a new characterization of Alexandrov spaces with two-sided curvature bounds. Moreover, we characterize the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. We develop algorithmic procedures that solve the geometric Whitney problem for a metric space and the manifold reconstruction problem in Euclidean space, and estimate the computational complexity of these procedures. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalization of the Whitney embedding construction where approximative coordinate charts are embedded in {{mathbb {R}}}^m and interpolated to a smooth submanifold.

On the Spectra of Hyperbolic Surfaces Without Thin Handles

We obtain a sharp lower bound on the eigenvalues of the Laplace–Beltrami operator on a hyperbolic surface with injectivity radius bounded from below.

Distance difference representations of Riemannian manifolds

Let M be a complete Riemannian manifold and $$F\subset M$$ a set with a nonempty interior. For every $$x\in M$$ , let $$D_x$$ denote the function on $$F\times F$$ defined by $$D_x(y,z)=d(x,y)-d(x,z)$$ where d is the geodesic distance in M. The map $$x\mapsto D_x$$ from M to the space of continuous functions on $$F\times F$$ , denoted by $${\mathcal {D}}_F$$ , is called a distance difference representation of M. This representation, introduced recently by Lassas and Saksala, is motivated by geophysical imaging among other things. We prove that the distance difference representation $${\mathcal {D}}_F$$ is a locally bi-Lipschitz homeomorphism onto its image $${\mathcal {D}}_F(M)$$ and that for every open set $$U\subset M$$ the set $${\mathcal {D}}_F(U)$$ uniquely determines the Riemannian metric on U. Furthermore the determination of M from $${\mathcal {D}}_F(M)$$ is stable if M has a priori bounds on its diameter, curvature, and injectivity radius. This extends and strengthens earlier results by Lassas and Saksala.

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah–Singer Dirac operator / D B in L 2 depends Riesz continuously on L ∞ perturbations of local boundary conditions B . The Lipschitz bound for the map B → / D B ( 1 + / D B 2 ) − 1 2 depends on Lipschitz smoothness and ellipticity of B and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.

Shi-type estimates of the Ricci flow based on Ricci curvature

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 unknown. In particular, another Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.

Lower bound of null injectivity radius without curvature assumptions in a family of null cones

We establish a new lower bound for the null injectivity radius of a null cone. The idea is to use a function closely related to the null second fundamental form which codifies the directional expansion of the null cone along any null geodesic in it. This approach uses first derivatives of the metric instead of curvature bounds. The technique is applied to a family of null cones with the null conjugate radius less or equal to the null crossing radius, or folded due to the curvature in a precise way that we call geometric fold. This condition is necessary because there are trivial examples with bounded null injectivity radius due to global identifications. We show two examples where we compute the real null injectivity radius and the lower bound provided in this paper, in order to compare both quantities. We also give an analogous result for Riemannian manifolds.

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

AbstractWe consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

Properly immersed surfaces in hyperbolic $3$-manifolds

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N \leq -a^2 \leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by a and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface $\Sigma$ must be proper in $N$ and its total curvature must be equal to $2 \pi \chi (\Sigma)$. If $N$ is a hyperbolic $3$-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than $1$, then we prove that each end of $\Sigma$ is asymptotic (with finite positive integer multiplicity) to a totally umbilic annulus, properly embedded in $N$.

Effective distance between nested Margulis tubes

We give sharp, effective bounds on the distance between tori of fixed injectivity radius inside a Margulis tube in a hyperbolic 3 3 –manifold.

An ergodic algorithm for generating knots with a prescribed injectivity radius

The first provably ergodic algorithm for sampling the space of thick equilateral knots off-lattice, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. It is an off-lattice generalization of the pivot algorithm. This move to an off-lattice model provides a huge improvement in power and efficacy in that samples can have arbitrary values for parameters such as the thickness constraint, bending angle, and torsion, while the lattice forces these parameters into a small number of specific values. This benefit requires working in a manifold rather than a finite or countable space, which forces the use of more novel methods in Markov–Chain theory. To prove the validity of the algorithm, we describe a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This approach ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint which is used to show that the algorithm is ergodic. This ergodic sampling allows for a statistically valid method for estimating probability distributions of arbitrary functions on the space of thick knots.

Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures

In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like function with uniformly controlled gradient and Hessian on complete non-compact manifolds with bounded Ricci curvature and strictly positive injectivity radius and a general growth estimate for the norm of the soliton vector field on manifolds with bounded Ricci curvature.

Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains

AbstractThis paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

A Riemannian Geometric Approach to Output Tracking for Nonholonomic Systems

The problem of designing coordinate-invariant output tracking control laws for nonholonomic mechanical systems is addressed. The velocity constrained Euler-Lagrange equations of motion are expressed through a constrained affine connection which is compatible with the kinetic energy Riemannian metric. This formalism is used in designing an output tracking control law via backstepping, which is shown to guarantee exponential stability when the initial distance between the output and reference trajectory is within injectivity radius of the output manifold. In particular this enables almost-global tracking when the output manifold is a rank-1 symmetric space. The control law is intrinsic to the Riemannian structure, and is explicitly constructed. The control law is applied to the problem of tracking the reduced attitude of a rigid body with a nonholonomic velocity constraint. Numerical simulations illustrating the tracking performance are presented.

Embolic aspects of black hole entropy

We attempt to provide a mesoscopic treatment of the origin of black hole entropy in [Formula: see text]-dimensional spacetimes. We treat the case of horizons having space-like sections [Formula: see text] which are topological spheres, following Hawking’s and the Topological Censorship theorems. We use the injectivity radius of the induced metric on [Formula: see text] to encode the linear dimensions of the elementary cells giving rise to such entropy. We use the topological entropy of [Formula: see text] as the fundamental quantity expressing the complexity of [Formula: see text] on which its entropy depends. We point out the significance, in this context, of the Berger and Croke isoembolic inequalities.

Local estimate on convexity radius and decay of injectivity radius in a Riemannian manifold

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 [Formula: see text]: (1) the convexity radius of [Formula: see text], [Formula: see text], where [Formula: see text] is the injectivity radius of [Formula: see text] and [Formula: see text] is the focal radius of open ball centered at [Formula: see text] with radius [Formula: see text]; (2) for any two points [Formula: see text] in [Formula: see text], [Formula: see text] where [Formula: see text] is the conjugate radius of [Formula: see text]; (3) for any [Formula: see text], any (not necessarily minimizing) geodesic in [Formula: see text] has length [Formula: see text]. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.

Bounds on Harmonic Radius and Limits of Manifolds with Bounded Bakry–Émery Ricci Curvature

Under the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $$C^{\alpha } \cap W^{1, q}$$ harmonic radius for manifolds with bounded Bakry–Emery Ricci curvature when the gradient of the potential is bounded. Under these conditions, the regularity that can be imposed on the metrics under harmonic coordinates is only $$C^\alpha \cap W^{1,q}$$ , where $$q>2n$$ and n is the dimension of the manifolds. This is almost 1-order lower than that in the classical $$C^{1,\alpha } \cap W^{2, p}$$ harmonic coordinates under bounded Ricci curvature condition (Anderson in Invent Math 102:429–445, 1990). The loss of regularity induces some difference in the method of proof, which can also be used to address the detail of $$W^{2, p}$$ convergence in the classical case. Based on this lower bound and the techniques in Cheeger and Naber (Ann Math 182:1093–1165, 2015) and Wang and Zhu (Crelle’s J, http://arxiv.org/abs/1304.4490 ), we extend Cheeger–Naber’s Codimension 4 Theorem in Cheeger and Naber (2015) to the case where the manifolds have bounded Bakry–Emery Ricci curvature when the gradient of the potential is bounded. This result covers Ricci solitons when the gradient of the potential is bounded. During the proof, we will use a Green’s function argument and adopt a linear algebra argument in Bamler (J Funct Anal 272(6):2504–2627, 2017). A new ingredient is to show that the diagonal entries of the matrices in the Transformation Theorem are bounded away from 0. Together these seem to simplify the proof of the Codimension 4 Theorem, even in the case where Ricci curvature is bounded.

The local picture theorem on the scale of topology

We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian threemanifold, 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 R 3 , whose theory we also develop.

Sharp Isoperimetric Inequalities for Small Volumes in Complete Noncompact Riemannian Manifolds of Bounded Geometry Involving the Scalar Curvature

Abstract We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with $C^3$ bounded geometry in a suitable sense involving the scalar curvature function. Under $C^3$ bounds of the geometry, if the supremum of scalar curvature function $S_g<n(n-1)k_0$ for some $k_0\in \mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of the complete simply connected space form of constant sectional curvature $k_0$. This work generalizes Theorem $2$ of [12] in which the same result was proved in the case where $(M^n, g)$ is assumed to be compact. As a consequence of our result we give an asymptotic expansion in Puiseux series up to the 2nd nontrivial term of the isoperimetric profile function for small volumes, generalizing our earlier asymptotic expansion [29]. Finally, as a corollary of our isoperimetric comparison result, it is shown that for small volumes the Aubin–Cartan–Hadamard’s conjecture is true in any dimension $n$ in the special case of manifolds with $C^3$ bounded geometry, and $S_g<n(n-1)k_0$. Two different intrinsic proofs of the fact that an isoperimetric region of small volume is of small diameter. The 1st under the assumption of mild bounded geometry, that is, positive injectivity radius and Ricci curvature bounded below. The 2nd assuming the existence of an upper bound of the sectional curvature, positive injectivity radius, and a lower bound of the Ricci curvature.