Möbius 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 S ^m+1 without umbilics. Four basic invariants of x under the Möbius transformation group in S ^m+1 are a Riemannian metric g called Möbius metric, a 1-form Φ called Möbius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Möbius 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 Möbius equivalent to either (i) a hypersurface with constant mean curvature and scalar curvature in S ^m+1; or (ii) the pre-image of a stereographic projection of a hypersurface with constant mean curvature and scalar curvature in the Euclidean space R ^m+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 H ^m+1. This result shows that one can use Möbius differential geometry to unify the three different classes of hypersurface with constant mean curvature and scalar curvature in S ^m+1, R ^m+1 and H ^m+1.