Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy

The equation of a motion of curves in the projective plane is deduced. Local flows are defined in terms of polynomial differential functions. A family of local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The integration of the congruence curves is discussed. Local motions defined by the traveling wave cnoidal solutions of the fifth-order Kaup-Kupershmidt equation are described

Tableaux over Lie algebras, integrable systems, and classical surface theory

Starting from suitable tableaux over finite dimensional Lie algebras, we provide a scheme for producing involutive linear Pfaffian systems related to various classes of submanifolds in homogeneous spaces which constitute integrable systems. These include isothermic surfaces, Willmore surfaces, and other classical soliton surfaces. Completely integrable equations such as the G/G_0-system of Terng and the curved flat system of Ferus-Pedit may be obtained as special cases of this construction. Some classes of surfaces in projective differential geometry whose Gauss-Codazzi equations are associated with tableaux over sl(4,R) are discussed.Comment: 16 pages, v3: final version; changes in the expositio

The spinor representation of constant mean curvature one surfaces in the Hyperbolic space

We review and comment on some aspects of the spinor representation for constant mean curvature one surfaces in hyperbolic space developed by Bobenko-Pavlyukevich-Springborn in [1]. The relations with the Bryant representation are addressed and some examples are discusse

Critical Robertson-Walker universes

The integral of the energy density function $\mathfrak m$ of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant $\Lambda$ gives rise to an action functional on the space of scale functions of RW spacetime metrics. This paper studies closed RW spacetimes which are critical for this functional, subject to volume-preserving variations (critical RW spacetimes). A complete classification of critical RW spacetimes is given and explicit solutions in terms of Weierstrass elliptic functions and their degenerate forms are computed. The standard energy conditions (weak, dominant, and strong) as well as the cyclic property of critical RW spacetimes are discussed.Comment: 18 pages, LaTe

Closed Trajectories of the conformal arclength functional

The purpose of this report is to give a brief overview of some results about the geometry of closed critical curves of a conformally invariant functional for space curve

Symplectic Applicability of Lagrangian Surfaces

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered