The conformal Gauss map of submanifolds of the Möbius space (Q1101691)

Using E. Cartan's method of moving frames the author introduces a conformal Gauss map \(\gamma_ f\) of an immersion \(f: M^ m\to S^ n\) with values in a Grassmann manifold of the pseudo-euclidean vector space \(V^{n+2}\) underlying the Möbius space \(S^ n\); the Gauss map is associated to the second order frames. It induces a functional, the volume \(V(\gamma_ f)\) of \(\gamma_ f\) for the immersion f. Furthermore, the author introduces a generalization W(f) of the Willmore functional for surfaces and a conformal total curvature \(\tau_ c(f)\). He compares these functionals and obtains \(c_ m\tau_ c(f)/2\leq V(\gamma_ f)\leq W(f)m^{-(m-2)/2},\) \(M^ m\) compact, \(c_ m\) the volume of the euclidean unit sphere \(S^ m\). The equality of the densities of \(V(\gamma_ f)\) and \(\tau_ c(f)\) and the condition that the set of points at which \(\gamma_ f\) is not critical is connected and dense in \(M^ m\) imply that there exists a sphere \(S^{m+1}\subset S^ n\) with \(f(M^ m)\subset S^{m+1}.\) Finally the author considers the variational problem of the Willmore functional for surfaces \(f: M^ 2\to S^ n\). He obtains Euler equations generalizing that deduced by \textit{R. L. Bryant} [J. Differ. Geom. 20, 23- 53 (1984; Zbl 0555.53002)] and shows that an immersion f is a Willmore surface iff the conformal Gauss map is harmonic. For such immersions the set of umbilical points of f is a closed set with empty interior for the whole of the connected manifold \(M^ 2\).