Abstract
The classical notion of the Darboux transformation of isothermic surfaces can be generalised to a transformation for conformal immersions. Since a minimal surface is Willmore, we can use the associated mathbb { C}_*-family of flat connections of the harmonic conformal Gauss map to construct such transforms, the so-called mu -Darboux transforms. We show that a mu -Darboux transform of a minimal surface is not minimal but a Willmore surface in 4-space. More precisely, we show that a mu -Darboux transform of a minimal surface f is a twistor projection of a holomorphic curve in mathbb { C}mathbb { P}^3 which is canonically associated to a minimal surface f_{p,q} in the right-associated family of f. Here we use an extension of the notion of the associated family f_{p,q} of a minimal surface to allow quaternionic parameters. We prove that the pointwise limit of Darboux transforms of f is the associated Willmore surface of f at mu =1. Moreover, the family of Willmore surfaces mu -Darboux transforms, mu in mathbb { C}_*, extends to a mathbb { C}mathbb { P}^1 family of Willmore surfaces f^mu : M rightarrow S^4 where mu in mathbb { C}mathbb { P}^1.
Highlights
A classical Darboux pair is given geometrically by a pair of conformal immersions ( f, f ) into 3-space such that there exists a sphere congruence conformally enveloping both surfaces [10]
One obtains a classical Darboux transform of an isothermic surface by a solution to a Riccati equation which is given in terms of a dual isothermic surface and a real parameter, [17]
This directly links to integrability: the parameter can be considered as the spectral parameter of an integrable system [2]
Summary
A classical Darboux pair is given geometrically by a pair of conformal immersions ( f, f ) into 3-space such that there exists a sphere congruence conformally enveloping both surfaces [10]. Parallel sections on the unit circle give again via conjugation new harmonic conformal Gauss maps and the associated surfaces for λ = eiθ ∈ S1 give the classical associated family fθ = cos θ f + sin θ f ∗ of isometric minimal surfaces where f ∗ is the conjugate minimal surface of f. This construction can be extended off the unit circle, and one obtains a generalised left-and right-associated family of minimal surfaces given by the f p,q = p f + q f ∗ and f p,q = f p + f ∗q respectively where p, q ∈ H∗ [21]. In an affine coordinate f 0,∞ are minimal surfaces in R4 with an isolated set of ends
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