Deformation Minimal Bending of Compact Manifolds: Case of Simple Closed Curves

The problem of minimal distortion bending of smooth compact embedded connected Riemannian $n$-manifolds $M$ and $N$ without boundary is made precise by defining a deformation energy functional $\Phi$ on the set of diffeomorphisms \diff(M,N). We derive the Euler-Lagrange equation for $\Phi$ and determine smooth minimizers of $\Phi$ in case $M$ and $N$ are simple closed curves.Comment: Typos corrected to match the final version of the paper, which has appeared in Opuscula Mathematica in January, 200

Distortion Minimal Morphing I: The Theory For Stretching

We consider the problem of distortion minimal morphing of $n$-dimensional compact connected oriented smooth manifolds without boundary embedded in $\R^{n+1}$. Distortion involves bending and stretching. In this paper, minimal distortion (with respect to stretching) is defined as the infinitesimal relative change in volume. The existence of minimal distortion diffeomorphisms between diffeomorphic manifolds is proved. A definition of minimal distortion morphing between two isotopic manifolds is given, and the existence of minimal distortion morphs between every pair of isotopic embedded manifolds is proved

On the continuation of an invariant torus in a family with rapid oscillations

This is the published version, also available here: http://dx.doi.org/10.1137/S0036141098338740.A persistence theorem for attracting invariant tori for systems subjected to rapidly oscillating perturbations is proved. The singular nature of these perturbations prevents the direct application of the standard persistence results for normally hyperbolic invariant manifolds. However, as is illustrated in this paper, the theory of normally hyperbolic invariant manifolds, when combined with an appropriate continuation method, does apply

Tidal Dynamics of Relativistic Flows Near Black Holes

DOI: 10.1002/andp.200410126 http://arxiv.org/PS_cache/astro-ph/pdf/0404/0404170v3.pdfWe point out novel consequences of general relativity involving tidal dynamics of ultrarelativistic relative motion. Specifically, we use the generalized Jacobi equation and its extension to study the force-free dynamics of relativistic flows near a massive rotating source. We show that along the rotation axis of the gravitational source, relativistic tidal effects strongly decelerate an initially ultrarelativistic flow with respect to the ambient medium, contrary to Newtonian expectations. Moreover, an initially ultrarelativistic flow perpendicular to the axis of rotation is strongly accelerated by the relativistic tidal forces. The astrophysical implications of these results for jets and ultrahigh energy cosmic rays are briefly mentioned