Surfaces in $$\mathbb {R}^7$$ R 7 obtained from harmonic maps in $$S^6$$ S 6

Isothermic surfaces obtained from harmonic maps in S6

“…Proposition 3.1. [3] Let I F and II F be the first and the second fundamental forms of F : Σ → R 7 , respectively. Let N be a vector field of the normal bundle T F ⊥ .…”

“…Let e 2α be the common conformal factor of ϕ and F . One can check [3] that the mean curvature vector of F is given by…”

“…It is a well-known fact that any non-conformal harmonic map ϕ from a simplyconnected Riemann surface Σ into the round 2-sphere S 2 is the Gauss map of a constant Gauss curvature surface, F : Σ → R 3 , and of two parallel constant mean curvature surfaces, F ± = F ± ϕ : Σ → R 3 ; the surface F integrates the closed 1-form ω = ϕ × * dϕ, where × denotes the standard cross product of R 3 .…”

“…This means that we can integrate on simply-connected domains in order to obtain a map F : Σ → R 7 . If ϕ is a conformal harmonic immersion, then F is a conformal immersion; and, in contrast with the 3-dimensional case, where F is necessarily a totally umbilical surface, F can exhibit a wide variety of geometrical behaviors in the 7-dimensional case [3].…”

“…Recall [4] that a surface in R n is isothermic if, away from umbilic points, it admits conformal curvature line (CCL) coordinates, that is, conformal coordinates with respect to which each second fundamental form is diagonal. In this short note, we prove that the immersions F in R 7 obtained from superconformal [1,3] harmonic maps in S 3 = S 6 ∩ W by integration of the corresponding 1-forms ω, with W a 4-dimensional subspace of R 7 , are isothermic. We will also see that the isothermic surfaces so obtained include a certain class of constant mean curvature surfaces in R 3 that can be deformed isometrically through isothermic surfaces into non-spherical pseudo-umbilical surfaces in R 7 .…”