M. do Carmo

Rotation Hypersurfaces in Spaces of Constant Curvature

Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the helicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space H3 ; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in H3.

Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

The main result of this paper states that the traceless second fundamental tensor \({A}^{0}\)of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, \(\int{M}^{|A^0|^n{d\nu}}{M} 0,\) any such surface must be compact.

The influence of the boundary behaviour on hypersurfaces with constant mean curvature inH n+1

This paper deals with complete, properly embedded hypersurfaces M n with constant mean curvature H of the hyperbolic space H n+1, and addresses itself to the following general question. How is the behaviour of such hypersurfaces influenced by their behaviour at infinity?

Compact minimal hypersurfaces with index one in the real projective space

Abstract. Let Mn be a compact (two-sided) minimal hypersurface in a Riemannian manifold

Spherical Images of Convex Surfaces

\bar M^{n+1}

Stable hypersurfaces with constant scalar curvature.

. It is a simple fact that if

The index of constant mean curvature surfaces in hyperbolic 3-space

\bar M

A gap theorem for hypersurfaces of the sphere with constant scalar curvature one

has positive Ricci curvature then M cannot be stable (i.e. its Jacobi operator L has index at least one). If

Hypersurfaces of constant mean curvature with finite index and volume of polynomial growth

\bar M = S^{n+1}

Immersions of manifolds with non-negative sectional curvatures

is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.¶We prove that if