Family Of Geodesics Research Articles

Holomorphic maps acting as Kobayashi isometries on a family of geodesics

Holomorphic maps acting as Kobayashi isometries on a family of geodesics

Estimates for low Steklov eigenvalues of surfaces with several boundary components

R\'esum\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $$\sigma _1$$ σ 1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

The Covariance Metric in the Blaschke Locus

We prove that the Blaschke locus has the structure of a finite dimensional smooth manifold away from the Teichmüller space and study its Riemannian manifold structure with respect to the covariance metric introduced by Guillarmou, Knieper and Lefeuvre in Guillarmou et al. in (Ergod Theory Dyn Syst 43:974–1022, 2021). We also identify some families of geodesics in the Blaschke locus arising from Hitchin representations for orbifolds and show that they have infinite length with respect to the covariance metric.

Geodesics on adjoint orbits of SL(n, ℝ)

In this paper, we examine the geodesics on adjoint orbits of [Formula: see text] that are equipped with [Formula: see text]-invariant metrics, where [Formula: see text] is the maximal compact subgroup. Our primary technique involves translating this problem into a geometric problem in the tangent bundle of certain [Formula: see text]-flag manifolds and describing the geodesic equations with respect to the Sasaki metric on the tangent bundle. Additionally, we utilize tools from Lie Theory to obtain explicit descriptions of families of geodesics. We provide a detailed analysis of the case for [Formula: see text].

Thick families of geodesics and differentiation

The differentiation theory of Lipschitz functions on metric spaces taking values in a Banach space with the Radon-Nikodým property (RNP), originally developed by Cheeger—Kleiner, has proven to be a powerful tool to prove non-bi-Lipschitz embeddability of metric spaces into these Banach spaces. Important examples of metric spaces to which this theory applies include nonabelian Carnot groups and Laakso spaces. In search of a metric characterization of the RNP, Ostrovskii found another class of spaces that do not bi-Lipschitz embed into RNP spaces, namely spaces containing thick families of geodesics. Our first result is that any metric space containing a thick family of geodesics also contains a subset and a probability measure on that subset which satisfies a weakened form of RNP Lipschitz differentiability. A corollary is a new nonembeddability result. Our second main result is that, if the metric space is a non-RNP Banach space, a subset consisting of a thick family of geodesics can be constructed to satisfy true RNP differentiability. An intriguing question is whether this differentiation criterion, or some weakened form of it such as the one we prove in the first result, actually characterizes general metric spaces non-bi-Lipschitz embeddable into RNP Banach spaces.

Conjugate and cut points in ideal fluid motion

Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume preserving diffeomorphisms having sufficiently strong positive curvatures which ‘pull’ nearby flows together. Physically, they indicate a form of (transient) stability in the configuration space of particle positions: a family of flows starting with the same configuration deviate initially and subsequently re-converge (resonate) with each other at some later moment in time. Here, we first establish existence of conjugate points in an infinite family of Kolmogorov flows—a class of stationary solutions of the Euler equations—on the rectangular flat torus of any aspect ratio. The analysis is facilitated by a general criterion for identifying conjugate points in the group of volume preserving diffeomorphisms. Next, we show non-existence of conjugate points along Arnold stable steady states on the annulus, disk and channel. Finally, we discuss cut points, their relation to non-injectivity of the exponential map (impossibility of determining a flow from a particle configuration at a given instant) and show that the closest cut point to the identity is either a conjugate point or the midpoint of a time periodic Lagrangian fluid flow.

Generalizing Stretch Lines for Surfaces with Boundary

Abstract In 1986, William Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmüller space of closed or punctured surfaces. We extend his theory to the Teichmüller space of surfaces with boundary endowed with the arc distance. We construct a large family of geodesics for the Teichmüller space of a surface with boundary, generalizing Thurston’s stretch lines. We prove that the Teichmüller space of a surface with boundary is a geodesic and Finsler metric space with respect to the arc distance. As a corollary, we find a new class of geodesics in the Teichmüller space of a closed surface that are not stretch lines in the sense of Thurston.

Uniqueness results for bodies of constant width in the hyperbolic plane

Abstract Following Santaló’s approach, we prove several characterizations of a disc among bodies of constant width, constant projection lengths, or constant section lengths on given families of geodesics.

On Discrete Conjugate Semi-Geodesic Nets

Abstract We introduce two canonical discretizations of nets on generic surfaces, which consist of an one-parameter family of geodesics and its associated family of conjugate lines. The two types of “discrete conjugate semi-geodesic nets” constitute discrete focal nets of circular nets, which mimics the classical connection between surfaces parametrized in terms of curvature coordinates and their focal surfaces on which one corresponding family of coordinate lines are geodesics. The discrete surfaces constructed in this manner are termed circular-geodesic and conical-geodesic nets, respectively, and may be characterized by compact angle conditions. Geometrically, circular-geodesic nets are constructed via normal lines of circular nets, while conical-geodesic nets inherit their name from their intimately related conical nets, which also discretize curvature line parametrized surfaces. We establish a variety of properties of these discrete nets, including their algebraic and geometric 3D consistency, with the latter playing an important role in (integrable) discrete differential geometry.

Traversable wormhole magnetic monopoles from Dymnikova metric

We study a traversable wormhole originated by a transformation over the 4D Dymnikova metric, which describes analytic black holes (BH). By using a transformation of coordinates which is adapted from the one used in the Einstein-Rosen bridge, we study a specific family of geodesics in which a test particle with non-zero electric charge induces an effective magnetic monopole, that is perceived by observers outside the wormhole. Because the Riemannian geometry cannot explain the presence of magnetic monopoles, we propose a torsional geometry in order to explore the possibility that magnetic monopoles can be geometrically induced. We obtain an expression that relates torsion and magnetic fields jointly with a Dirac-like expression for magnetic and electric charges, such that torsion makes it possible to define a fundamental length that provides a magnetic field and a spacetime discretization.

Rotating black holes in higher order gravity

We develop a new technique for finding black hole solutions in modified gravity that have "stealth" hair, i.e., hair whose only gravitational effect is to tune the cosmological constant. We consider scalar-tensor theories in which gravitational waves propagate at the speed of light, and show that Einstein metrics can be painted with stealth hair provided there exists a family of geodesics always normal to spacelike surfaces. We also present a novel scalar-dressed rotating black hole that has finite scalar field at both the black hole and cosmological event horizons.

The exact dynamical Chern–Simons metric for a spinning black hole possesses a fourth constant of motion: a dynamical-systems-based conjecture

The recent gravitational wave observations by the LIGO/Virgo collaboration have allowed the first tests of general relativity in the extreme gravity regime, when comparable-mass black holes and neutron stars collide. Future space-based detectors, such as the Laser Interferometer Space Antenna, will allow tests of Einstein’s theory with gravitational waves emitted when a small black hole falls into a supermassive one in an extreme mass-ratio inspiral. One particular test that is tailor-made for such inspirals is the search for chaos in extreme gravity. We here study whether chaos is present in the motion of test particles around spinning black holes of parity-violating modified gravity, focusing in particular on dynamical Chern–Simons gravity. We develop a resummation strategy that restores all spin terms in the general relativity limit, while retaining up to fifth-order-in-spin terms in the dynamical Chern–Simons corrections to the Kerr metric. We then calculate Poincaré surfaces of section and rotation numbers of a wide family of geodesics of this resummed metric. We find evidence for geodesic chaos, portrayed by thin chaotic layers surrounded by deformed invariant tori. This chaotic layers shrink in size as terms of higher-order in spin are included in the dynamical Chern–Simons corrections to the Kerr metric. Our numerical findings suggest that the geodesics of the as-of-yet unknown exact solution for spinning black holes in this theory may be integrable, and that there may thus exist a fourth integral of motion associated with this exact solution. The studies presented here begin to lay the foundations for chaotic tests of general relativity with the observation of extreme mass ratio inspirals with the Laser Interferometer Space Antenna.

Homogeneous variational problems and Lagrangian sections

Abstract We define a canonical line bundle over the slit tangent bundle of a manifold, and define a Lagrangian section to be a homogeneous section of this line bundle. When a regularity condition is satisfied the Lagrangian section gives rise to local Finsler functions. For each such section we demonstrate how to construct a canonically parametrized family of geodesics, such that the geodesics of the local Finsler functions are reparametrizations.

Overspinning a Kerr black hole: The effect of the self-force

We study the scenario in which a massive particle is thrown into a rapidly rotating Kerr black hole in an attempt to spin it up beyond its extremal limit, challenging weak cosmic censorship. We work in black-hole perturbation theory, and focus on nonspinning, uncharged particles sent in on equatorial orbits. We first identify the complete parameter-space region in which overspinning occurs when backreaction effects from the particle's self-gravity are ignored. We find, in particular, that overspinning can be achieved only with particles sent in from infinity. Gravitational self-force effects may prevent overspinning by radiating away a sufficient amount of the particle's angular momentum (``dissipative effect''), and/or by increasing the effective centrifugal repulsion, so that particles with suitable parameters never get captured (``conservative effect''). We analyze the full effect of the self-force, thereby completing previous studies by Jacobson and Sotiriou (who neglected the self-force) and by Barausse, Cardoso and Khanna (who considered the dissipative effect on a subset of orbits). Our main result is an inequality, involving certain self-force quantities, which describes a necessary and sufficient condition for the overspinning scenario to be overruled. This ``censorship'' condition is formulated on a certain one-parameter family of geodesics in the limit of an extremal Kerr geometry. We find that the censorship condition is insensitive to the dissipative effect (within the first-order self-force approximation used here), except for a subset of perfectly fine-tuned orbits, for which a separate censorship condition is derived. We do not obtain here the self-force input needed to evaluate either of our two conditions, but discuss the prospects for producing the necessary data using state-of-the-art numerical codes.

Monge problem in metric measure spaces with Riemannian curvature-dimension condition

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X,d,m) enjoying the Riemannian curvature-dimension condition RCD∗(K,N), with N<∞. For the first marginal measure, we assume that μ0≪m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge–Kantorovich problem, attain the same minimal value.Moreover we prove a structure theorem for d-cyclically monotone sets: neglecting a set of zero m-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.

Metric spaces nonembeddable into Banach spaces with the Radon–Nikodým property and thick families of geodesics

We show that a geodesic metric space which does not admit bilipschitz embeddings into Banach spaces with the Radon-Nikod\'ym property does not necessarily contain a bilipschitz image of a thick family of geodesics. This is done by showing that any thick family of geodesics is not Markov convex, and comparing this result with results of Cheeger-Kleiner, Lee-Naor, and Li. The result contrasts with the earlier result of the author that any Banach space without the Radon-Nikod\'ym property contains a bilipschitz image of a thick family of geodesics.

Radon–Nikodým property and thick families of geodesics

Banach spaces without the Radon–Nikodým property are characterized as spaces containing bilipschitz images of thick families of geodesics defined as follows. A family T of geodesics joining points u and v in a metric space is called thick if there is α>0 such that for every g∈T and for any finite collection of points r1,…,rn in the image of g, there is another uv-geodesic g˜∈T satisfying the conditions: g˜ also passes through r1,…,rn, and, possibly, has some more common points with g. On the other hand, there is a finite collection of common points of g and g˜ which contains r1,…,rn and is such that the sum of maximal deviations of the geodesics between these common points is at least α.

Semi-local inversion of the geodesic ray transform in the hyperbolic plane

The inversion of the ray transform on the hyperbolic plane has applications in geophysical exploration and in medical imaging techniques (such as electrical impedance tomography). The geodesic ray transform has been studied in more general geometries and including attenuation, but all of the available inversion formulas require knowledge of the ray transform for all the geodesics. In this paper we present a different inversion formula for the ray transform on the hyperbolic plane, which has the advantage of only requiring knowledge of the ray transform in a reduced family of geodesics. The required family of geodesics is directly related to the set where the original function is to be recovered.

De Sitter geodesics: reappraising the notion of motion

The de Sitter spacetime is transitive under a combination of translations and proper conformal transformations. Its usual family of geodesics, however, does not take into account this property. As a consequence, there are points in de Sitter spacetime which cannot be joined to each other by any one of these geodesics. By taking into account the appropriate transitivity properties in the variational principle, a new family of maximizing trajectories is obtained, whose members are able to connect any two points of the de Sitter spacetime. These geodesics introduce a new notion of motion, given by a combination of translations and proper conformal transformations, which may possibly become important at very-high energies, where conformal symmetry plays a significant role.

Optimal Transport with Branching Distance Costs and the Obstacle Problem

We address the Monge problem in metric spaces with a geodesic distance: $(X,d)$ is a Polish space and $d_N$ is a geodesic Borel distance which makes $(X,d_N)$ a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along a family of geodesics. We introduce three assumptions on a given $d_{N}$-monotone transference plan $\pi$ which imply, respectively, strong consistency of disintegration, continuity of the conditional probabilities of the first marginal, and a regularity property for the geometry of chain of transport rays. We show that this regularity is sufficient for the construction of a transport map with the same transport cost of $\pi$. We apply these results to the Monge problem in $\mathbb{R}^{d}$ with smooth, convex, and compact obstacle obtaining the existence of an optimal map, provided the first marginal is absolutely continuous with respect to the d-dimensional Lebesgue measure.