Constant higher-order mean curvature hypersurfaces in Riemannian spaces

It is still an open question whether a compact embedded hypersurface in the Euclidean space Rn+1 with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in R-3. In a recent paper, Alias and Malacarne (Rev. Mat. lberoamericana 18 (2002), 431-442) have shown that this is true for the case of hypersurfaces in Rn+1 with constant scalar curvature, and more generally, hypersurfaces with constant higher-order r-mean curvature, when r >= 2. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold M, where we will consider a general geometric configuration consisting of an immersed hypersurface into M with boundary on an oriented hypersurface P of (M) over bar. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of P, as well as the geometry of P as a hypersurface of (M) over bar. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature (the Euclidean space Rn+1, the hyperbolic space Hn+1, and the sphere Sn+1). In particular, we are able to extend the symmetry results given in the recent paper mentioned above to the case of hypersurfaces with constant higher-order r-mean curvature in the hyperbolic space and in the sphere.