The dual superconformal surface (Q2351497)

The authors prove that a superconformal surface with arbitrary codimension in the Euclidean space has a (necessarily unique) dual superconformal surface if and only if the surface is \(S\)-Willmore. Recall that a surface \(M^2\subset\mathbb{R}^{2+n}\) is called superconformal if at any point of \(M^2\) the ellipse of curvature is a non-degenerate circle. The dual of \(M^2\) is a surface \(\tilde{M}^2\subset\mathbb{R}^{2+n}\) that induces a metric which is conformal with that of \(M^2\) and possesses a common central sphere congruence. The surface \(M^2\) is called \(S\)-Willmore when the following condition is satisfied: If \(z=x_1+ix_2\) is a local isothermal coordinate system around a point of \(M^2\), then the complex line bundle \(B(\partial_z,\partial_z)\) is parallel in the normal bundle. Here by \(B\) we denote the second fundamental form of \(M^2\). More precisely the main result of the paper under review is summarized in the following: {Theorem 1.} Let \(f:M^2\to\mathbb{R}^{n+2}\), \(n\geq 3\), be a regular locally conformally substantial superconformal surface. Then \(f\) has a dual superconformal surface if and only if it is \(S\)-Willmore. Moreover, the dual surface can be parametrized as \[ \tilde{f}=f+\frac{2}{|H|^2}(H)^{\Lambda}, \] where \(H\) denotes the mean curvature vector field of \(f\), \(\Lambda\) is the normal subbundle of rank \(n-2\) of the surface of centers perpendicular to the plane subbundle of the first normal bundle \(N^f_1\) of \(f\) orthogonal to the mean curvature vector and \((H)^{\Lambda}\) denotes taking the \(\Lambda\)-component. Furthermore, up to conformal equivalence, we have the following cases: {\parindent=6mm \begin{itemize} \item[(a)] The dual reduces to a single point if and only if \(f\) is a minimal surface. \item [(b)] The dual is obtained by composing \(f\) with an inversion and a reflection with respect to its center if and only if \(f\) is the image under stereographic projection of a minimal surface in the sphere \(\mathbb{S}^{n+2}\). \item [(c)] The dual is obtained by composing \(f\) with an inversion if and only if \(f\) is the image under stereographic projection of a minimal surface in the hyperbolic space \(\mathbb{H}^{n+2}\). \end{itemize}} It is well known that a \(S\)-Willmore surfaces is always Willmore, i.e., its mean curvature satisfies the Willmore equation \[ \Delta^{\perp}H-2|H|^2H+\sum^2_{i,j=1}\langle H, B(e_i,e_j)\rangle B(e_i,e_j)=0, \] where \(\Delta^{\perp}\) is the Laplacian acting on sections of the normal bundle of \(M^2\) and \(\{e_1,e_2\}\) is a local orthonormal frame field of \(M^2\). However, the converse is not true. In codimension \(3\) the authors prove the following result: {Theorem 2.} Any superconformal Willmore surface \(f:M^2\to\mathbb{R}^5\) is \(S\)-Willmore.