Hypersurfaces of constant curvature in space forms

In this paper we shall discuss hypersurfaces M of space forms of constant curvature; where curvature means one of the symmetric functions of curvature associated to the second fundamental form. The values of the constant will be chosen so that the linearized equation will be an elliptic equation onM . For example, for surfaces in 3 the two possible curvatures are the mean curvature H and the Gaussian curvature K. The linearized equation for H is always elliptic and for K it is elliptic when the constant K is positive. In hyperbolic 3-space, the constant K > −1 yields an elliptic equation. Hypersurfaces of constant scalar curvature S2 will be elliptic when S2 > 0. We obtain height estimates for compact embedded hypersurfaces of m+1 whose r + 1’st symmetric function of curvature, Sr+1, is a positive constant (§6.1.): Given such a hypersurface M , with ∂M contained in a hyperplane P , then the maximum distance of a point of M to P is