Magnetic Topology in Coupled Binaries, Spin-orbital Resonances, and Flares

We consider topological configurations of the magnetically coupled spinning stellar binaries (e.g., merging neutron stars or interacting star–planet systems). We discuss conditions when the stellar spins and the orbital motion nearly “compensate” each other, leading to very slow overall winding of the coupled magnetic fields; slowly winding configurations allow gradual accumulation of magnetic energy, which is eventually released in a flare when the instability threshold is reached. We find that this slow winding can be global and/or local. We describe the topology of the relevant space F = T 1 S 2 as the unit tangent bundle of the two-sphere and find conditions for slowly winding configurations in terms of magnetic moments, spins, and orbital momentum. These conditions become ambiguous near the topological bifurcation points; in certain cases, they also depend on the relative phases of the spin and orbital motions. In the case of merging magnetized neutron stars, if one of the stars is a millisecond pulsar, spinning at ∼10 ms, the global resonance ω 1 + ω 2 = 2Ω (spin-plus beat is two times the orbital period) occurs approximately one second before the merger; the total energy of the flare can be as large as 10% of the total magnetic energy, producing bursts of luminosity ∼1044 erg s−1. Higher order local resonances may have similar powers, since the amount of involved magnetic flux tubes may be comparable to the total connected flux.

Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame

UDC 514.7 In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified.

Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$ . We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

Vortices over Riemann surfaces and dominated splittings

We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

Natural Paracontact Magnetic Trajectories on Unit Tangent Bundles

In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base manifolds of constant Gaussian curvature and to Kaluza–Klein type metrics on their unit tangent bundles, we give a full classification of natural paracontact slant magnetic trajectories (and geodesics).

Effective Equidistribution of the Horocycle Flow on Geometrically Finite Hyperbolic Surfaces

We prove effective equidistribution of the horocycle flow in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces.

Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.