Thomas Ivey

Ricci solitons on compact three-manifolds

Abstract In this short article we show that there are no compact three-dimensional Ricci solitons other than spaces of constant curvature. This generalizes a result obtained for surfaces by Hamilton [4]. The proof involves a careful analysis of the ODE for the curvature which is associated to the Ricci flow.

KNOT TYPES, HOMOTOPIES AND STABILITY OF CLOSED ELASTIC RODS

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the m-fold covered circle to the n-fold covered circle for any m,n relatively prime. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion. Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centerlines.

BÄCKLUND TRANSFORMATIONS AND KNOTS OF CONSTANT TORSION

The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vortex filament flow (also known as the Localized Induction Equation). In doing so, we obtain analytic constant-torsion representatives for a large number of knot types.

Remarks on KdV-type Flows on Star-Shaped Curves

Abstract We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into R P 1 . We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwarzian KdV equation as its projectivization. (For both flows, the curvature evolves by the KdV equation.) Using algebro-geometric methods and the relation of group-based moving frames to AKNS-type representations, we construct examples of closed solutions of Pinkall’s flow associated with periodic finite-gap KdV potentials.

Finite-Gap Solutions of the Vortex Filament Equation: Genus One Solutions and Symmetric Solutions

AbstractFor the class of quasiperiodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction, and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such as Euler elastica, constant torsion curves, and self-intersecting filaments. We also prove generalizations of these connections to higher genus.

Local existence of Ricci Solitons.

The Ricci flow ∂g/∂t = −2Ric(g) is an evolution equation for Riemannian metrics. It was introduced by Richard Hamilton, who has shown in several cases ([7], [8], [9]) that the flow converges, up to re-scaling, to a metric of constant curvature. However, “soliton” solutions to the flow give examples where the Ricci flow does not uniformize the metric, but only changes it by diffeomorphisms. Soliton solutions are generated by initial data satisfying

Stability of small-amplitude torus knot solutions of the localized induction approximation

We study the linear stability of small-amplitude torus knot solutions of the localized induction approximation equation for the motion of a thin vortex filament in an ideal fluid. Such solutions can be constructed analytically through the connection with the focusing nonlinear Schrodinger equation using the method of isoperiodic deformations. We show that these (p, q) torus knots are generically linearly unstable for p q, in contrast with an earlier linear stability study by Ricca (1993 Chaos 3 83–95; 1995 Chaos 5 346; 1995 Small-scale Structures in Three-dimensional Hydro and Magneto-dynamics Turbulence (Lecture Notes in Physics vol 462) (Berlin: Springer)). We also provide an interpretation of the original perturbative calculation in Ricca (1995), and an explanation of the numerical experiments performed by Ricca et al (1999 J. Fluid Mech. 391 29–44), in light of our results.

SURFACES WITH ORTHOGONAL FAMILIES OF CIRCLES

The lines of curvature on a cyclide of Dupin are circular arcs. A surface which carries two orthogonal families of circular arcs must arise as an integral surface of an overdetermined exterior differential system. We show that the only solutions of this system are the cyclides of Dupin.

Connecting geometry, topology and spectra for finite-gap NLS potentials

Using the connection between closed solution curves of the vortex filament flow and quasiperiodic solutions of the nonlinear Schrodinger equation (NLS), we relate the knot types of finite-gap solutions to the Floquet spectra of the corresponding NLS potentials, in the special case of small amplitude curves close to multiply covered circles.

Knot types, Floquet spectra, and finite-gap solutions of the vortex filament equation

Using the connection between closed solution curves of the vortex filament equation and the periodic problem for the nonlinear Schrodinger equation (NLS), we investigate the possibility of relating the knot types of finite-gap solution to the Floquet spectra of the corresponding NLS potentials, in the special case of small amplitude curves close to multiply-covered circles.