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We consider the variational problem defined by the functional H 2 −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. In this paper we study the variational problem for the functional W(S, f) = H 2 − K K dA on… (More)

Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are presented: the Umehara-Yamada perturbation, the classical and Bryant's Weierstrass type representations , and the duality for cmc-1 surfaces are interpreted in… (More)

- Emilio Musso, Lorenzo Nicolodi
- Journal of Mathematical Imaging and Vision
- 2009

We prove that any subset of ℝ2 parametrized by a C 1 periodic function and its derivative is the Euclidean invariant signature of a closed planar curve. This solves a problem posed by Calabi et al. (Int. J. Comput. Vis. 26:107–135, 1998). Based on the proof of this result, we then develop some cautionary examples concerning the application of signature… (More)

In this article we study constrained variational problems in one independent variable defined on the space of integral curves of a Frenet system in a homogeneous space G/H. We prove that if the Lagrangian is G-invariant and coisotropic then the extremal curves can be found by quadratures. Our proof is constructive and relies on the reduction theory for… (More)

The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the deformation theory of submanifolds in… (More)

- Emilio MUSSO
- 2012

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… (More)

We study an analogue of the classical Bäcklund transformation for L-iso-thermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. We show how to construct the Bianchi–Darboux transforms of an L-isothermic surface by solving an integrable linear differential system. We then establish a permutability theorem for iterated Bianchi–Darboux… (More)

- Emilio Musso, Lorenzo Nicolodi
- SIAM J. Control and Optimization
- 2008

We establish a correspondence between Darboux's special isother-mic surfaces of type (A, 0, C, D) and the solutions of the second order p.d.e. Φ∆Φ − |∇Φ| 2 + Φ 4 = s, s ∈ R. We then use the classical Darboux transformation for isothermic surfaces to construct a Bäcklund transformation for this equation and prove a superposition formula for its solutions. As… (More)

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… (More)