Existence Of Geodesics Research Articles

Hofer's metric in compact Lie groups

In this article, we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M . Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite-dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.

Geodesics near a curvature singularity of stationary and axially symmetric space–times

In this work we study the local behavior of geodesics in the neighborhood of a curvature singularity contained in stationary and axially symmetric space–times. Apart from these properties, the metrics we shall focus on will also be required to admit a quadratic first integral for their geodesics. In particular, we search for the conditions on the geometry of the space–time for which null and time-like geodesics can reach the singularity. These conditions are determined by the equations of motion of a freely-falling particle. We also analyze the possible existence of geodesics that do not become incomplete when encountering the singularity in their path. The results are stated as criteria that depend on the inverse metric tensor along with conserved quantities such as energy and angular momentum. As an example, the derived criteria are applied to the Plebański–Demiański class of space–times. Lastly, we propose a line element that describes a wormhole whose curvature singularities are, according to our results, inaccessible to causal geodesics.

On geodesic flows with symmetries and closed magnetic geodesics on orbifolds

Let $Q$ be a closed manifold admitting a locally free action of a compact Lie group $G$. In this paper, we study the properties of geodesic flows on $Q$ given by suitable G-invariant Riemannian metrics. In particular, we will be interested in the existence of geodesics that are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.

Baire categorical aspects of first passage percolation

In the previous decades, the theory of first passage percolation became a highly important area of probability theory. In this work, we will observe what can be said about the corresponding structure if we forget about the probability measure defined on the product space of edges and simply consider topology in the terms of residuality. We focus on interesting questions arising in the probabilistic setup that make sense in this setting, too. We will see that certain classical almost sure events, as the existence of geodesics have residual counterparts, while the notion of the limit shape or time constants gets as chaotic as possible.

The degenerate special Lagrangian equation

This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in Cn. Existence of geodesics in the space of positive Lagrangians is an important step in a program for proving existence and uniqueness of special Lagrangians. Moreover, it would imply certain cases of the strong Arnold conjecture from Hamiltonian dynamics.We show the DSL is degenerate elliptic. We introduce a space–time Lagrangian angle for one-parameter families of graph Lagrangians, and construct its regularized lift. The superlevel sets of the regularized lift define subequations for the DSL in the sense of Harvey–Lawson. We extend the existence theory of Harvey–Lawson for subequations to the setting of domains with corners, and thus obtain solutions to the Dirichlet problem for the DSL in all branches. Moreover, we introduce the calibration measure, which plays a rôle similar to that of the Monge–Ampère measure in convex and complex geometry. The existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the DSL have a well-defined length in the space of positive Lagrangians.

Geodesics in Asymmetric Metric Spaces

Abstract In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We define midpoints, convexity, and quasi-midpoints, but without assuming the space be intrinsic.We distinguish all along those results that need a stronger separation hypothesis. Eventually we discuss how the newly developed theory impacts the most important results, such as the existence of geodesics, and the renowned Hopf-Rinow (or Cohn-Vossen) theorem.

Existence of geodesics in Gödel type space-times

Existence of geodesics in Gödel type space-times

Existence of permanent and breaking waves for a shallow water equation: a geometric approach

The existence of global solutions and the phenomenon of blow-up of a solution in finite time for a recently derived shallow water equation are studied. We prove that the only way a classical solution could blow-up is as a breaking wave for which we determine the exact blow-up rate and, in some cases, the blow-up set. Using the correspondence between the shallow water equation and the geodesic flow on the manifold of diffeomorphisms of the line endowed with a weak Riemannian structure, we give sufficient conditions for the initial profile to develop into a global classical solution or into a breaking wave. With respect to the geometric properties of the diffeomorphism group, we prove that the metric spray is smooth infering that, locally, two points can be joined by a unique geodesic. The qualitative analysis of the shallow water equation is used to to exhibit breakdown of the geodesic flow despite the existence of geodesics that can be continued indefinitely in time.

On the existence of geodesics on stationary Lorentz manifolds with convex boundary

In this paper we consider the problem of the existence and multiplicity for geodesics not touching the boundary of a stationary Lorentz manifold having convex boundary. A physical example of a stationary (and nonstatic) Lorentz manifold having convex boundary is the stationary, axisymmetric, asymptotically flat, gravitational field outside a rotating massive object, whenever its angular speed is small and its mean radius is close to the Schwarzschild radius.

Category of loop spaces of open subsets in euclidean space

has been studied in several recent papers (e.g. [l, 2, 5, 6, 91) in situations where the potential function V is singular, in the sense that it is defined and C’ only on an open subset U of Euclidean k-space Rk and not on the whole of Rk. If E = WGT2(R, Rk) is the Sobolev space of T-periodic functions from R to Rk and E(U) is the subset of E of functions which map into U, then the topology of E(U) plays a crucial role in these studies. In particular, the knowledge that the Lusternik-Schnirelmann category of E(U) is infinite, when used in conjunction with min-max methods, implies the existence of infinitely many critical points. Another subset E,,(U), which consists of those functions in E(U) which take the origin to a base point in U, plays a dual role. First of all, in problems concerning the existence of geodesics in U from one subset A of U to another B of U (see e.g. [3]), the category of E,(U) plays a role. Secondly, the general inequality cat E(U) 2 cat E,,(U) (see [6]), when U is connected and simply connected, can be applied to show that cat E(U) = cat E,,(U), thus providing a tool for computing cat E(U). The topological analogue of E(U) is the space of free loops AU = C’[S’, U] of continuous maps from the circle S’ to U and the analogue of E,(U) is the space of based loops RU which consists of maps in AU which take a base point in S’ to a base point in U. AU has the same homotopy type as E(U) and S2U has the same homotopy type as E,,(U). Thus, homotopy type invariants (e.g. category) of E(U) and E,(U) may be studied through AU and SZU, respectively. The following classical theorem of Serre [l l] is useful in this regard and is stated in terms of a general space X.

The existence of geodesics joining two given points

The existence of geodesics joining two given points