Using the notion of the ellipse of curvature we study compact surfaces in high dimensional space forms. We obtain some inequalities relating the area of the surface and the integral of the square of the norm of the mean curvature vector with topological invariants. In certain cases, the ellipse is a circle; when this happens, restrictions on the Gaussian and normal curvatures give us some rigidity results.
Abstract. We give conditions which oblige properly embedded constant mean curvature one surfaces in hyperbolic 3-space to intersect. Our results are inspired by the theorem that two disjoint properly immersed minimal surfaces in R 3 must be planes.The half-space theorem says that a properly immersed minimal surface in R 3 that is disjoint from a plane (thus in a half-space) is a plane. The strong halfspace theorem says that two disjoint properly immersed minimal surfaces in R 3 are planes. The latter is deduced from the former by finding a plane between the two surfaces. These theorems are due to D. Hoffman and W. Meeks [H-M].In this paper we establish results of this nature in hyperbolic 3-space for mean curvature one surfaces and horospheres.Let N denote a horosphere of H 3 , and B the horoball of H 3 bounded by N ; the mean curvature of N is one and the mean curvature vector of N points into B. Let C be the other connected component of H 3 \N . In the same spirit we shall establish:
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