Willmore hypersurfaces with constant Möbius curvature in R n+1

Abstract

For an immersed hypersurface \({f : M^n \rightarrow R^{n+1}}\) without umbilical points, one can define the Mobius metric g on f which is invariant under the Mobius transformation group. The volume functional of g is a generalization of the well-known Willmore functional, whose critical points are called Willmore hypersurfaces. In this paper, we prove that if a n-dimensional Willmore hypersurfaces \({(n \geq 3)}\) has constant sectional curvature c with respect to g, then c = 0, n = 3, and this Willmore hypersurface is Mobius equivalent to the cone over the Clifford torus in \({S^{3} \subset R^{4}}\) . Moreover, we extend our previous classification of hypersurfaces with constant Mobius curvature of dimension \({n \ge 4}\) to n = 3, showing that they are cones over the homogeneous torus \({S^1(r) \times S^1(\sqrt{1 - r^2}) \subset S^3}\) , or cylinders, cones, rotational hypersurfaces over certain spirals in the space form R 2, S 2, H 2, respectively.