Lorenzo Nicolodi

A variational problem for surfaces in Laguerre geometry

We consider the variational problem defined by the functional ∫ H−K K dA on immersed surfaces in Euclidean space. Using the invariance of the functional under the group of Laguerre transformations, we study the extremal surfaces by the method of moving frames.

Laguerre geometry of surfaces with plane lines of curvature

We study surfaces with plane lines of curvature in the framework of Laguerre geometry and provide explicit representation formulae for these surfaces in terms of a potential function. As an application, we explicitly integrate allL- minimal surfaces with plane curvature lines.

Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space

An overview of the various transformations of isothermic surfaces and their interrelations is given using aquaternionic formalism. Applications to the theory of cmc-1 surfaces inhyperbolic space are given and relations between the two theories are discussed. Within this context, we give Möbius geometric characterizations for cmc-1 surfaces in hyperbolic space and theirminimal cousins.

THE BIANCHI–DARBOUX TRANSFORM OF L-ISOTHERMIC SURFACES

We study an analogue of the classical Backlund transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. First we show how to construct the Bianchi–Darboux transforms of an L-isothermic surface by solving an integrable linear differential system, then we establish a permutability theorem for iterated Bianchi–Darboux transforms.

Hamiltonian flows on null curves

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the Korteweg–de Vries (KdV) hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the trajectories of a certain particle model on null curves described by a Lagrangian linear in the curvature. In addition, we show that the curvature of a null curve which evolves by similarities can be computed in terms of the solutions of the second Painleve equation.

On the Cauchy problem for the integrable system of Lie minimal surfaces

In this paper we apply the Cartan-Kahler theory of exterior differential systems to solve the Cauchy problem for the integrable system of Lie minimal surfaces and discuss the underlying geometry. One purpose for this work is to show how methods and language from the theory of exterior differential systems may prove to be useful in the study of real analytic initial value problems, especially for gaining insight into the geometric aspects of the initial conditions and the solutions.

Closed trajectories of a particle model on null curves in anti-de Sitter 3-space

We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.

Reduction for Constrained Variational Problems on 3-Dimensional Null Curves

We consider the optimal control problem for null curves in de Sitter 3-space defined by a functional which is linear in the curvature of the trajectory. We show how techniques based on the method of moving frames and exterior differential systems, coupled with the reduction procedure for systems with a Lie group of symmetries, lead to the integration by quadratures of the extremals. Explicit solutions are found in terms of elliptic functions and integrals.

The geometric Cauchy problem for the membrane shape equation

We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surface area. Our approach uses the method of moving frames and techniques from the theory of exterior differential systems.

Reduction for the projective arclength functional

Abstract We consider the variational problem for curves in real projective plane defined by the projective arclength functional and discuss the integrability of its stationary curves in a geometric setting. We show how methods from the subject of exterior differential systems and the reduction procedure for Hamiltonian systems with symmetries lead to the integration by quadratures of the extrema. A scheme of integration is illustrated.