M�bius geometry of hypersurfaces with constant mean curvature and scalar curvature

Abstract

Let \(x:{\bf M}^m\rightarrow {\bf S}^{m+1}\) be a hypersurface in the (m+1)-dimensional unit sphere Sm+1 without umbilics. Four basic invariants of x under the Mobius transformation group in Sm+1 are a Riemannian metric g called Mobius metric, a 1-form Φ called Mobius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Mobius second fundamental form. In this paper, we prove the following classification theorem: let \(x:{\bf M}^m\rightarrow {\bf S}^{m+1}\) be a hypersurface, which satisfies (i) Φ≡0, (ii) A+λg+μB≡0 for some functions λ and μ, then λ and μ must be constant, and x is Mobius equivalent to either (i) a hypersurface with constant mean curvature and scalar curvature in Sm+1; or (ii) the pre-image of a stereographic projection of a hypersurface with constant mean curvature and scalar curvature in the Euclidean space Rm+1; or (iii) the image of the standard conformal map of a hypersurface with constant mean curvature and scalar curvature in the (m+1)-dimensional hyperbolic space Hm+1. This result shows that one can use Mobius differential geometry to unify the three different classes of hypersurface with constant mean curvature and scalar curvature in Sm+1, Rm+1 and Hm+1.