Abstract. It is still an open question whether a compact embedded hypersurface in the Euclidean space R n+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 [3] the first and third authors have shown that this is true for the case of hypersurfaces in R n+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 . 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. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature (the Euclidean space R n+1 , the hyperbolic space H n+1 , and the sphere S n+1 ). In particular, we are able to extend the symmetry results given in [3] to the case of hypersurfaces with constant higher order r-mean curvature in the hyperbolic space and in the sphere.

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It is still an open question whether a compact embedded hypersurface in the Euclidean space with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of a compact constant mean curvature surface in R 3 bounded by a circle. In this paper we prove that this is true for the case of the scalar curvature. Specifically we prove that the only compact embedded hypersurfaces in the Euclidean space with constant scalar curvature and spherical boundary are the hyperplanar round balls (with zero scalar curvature) and the spherical caps (with positive constant scalar curvature). The same applies in general to the case of embedded hypersurfaces with constant r -mean curvature, with r ≥ 2.

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