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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)
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 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)
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)