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Browsing by Author "Rosenberg, H."

Now showing 1 - 3 of 3
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    Some Remarks on Deformations of Minimal Surfaces
    (1983) Rosenberg, H.; Toubiana, E.
    We consider complete minimal surfaces (c.m.s.'s) in R3 and their deformations. M1 is an deformation of M0 if M1 is a graph over M0 in an tubular neighborhood of M1 and M1 is - C1 close to M0. A c.m.s. M0 is isolated if all minimal surfaces M1, which are sufficiently small deformations of M0, are congruent to M0. Many of the classical minimal surfaces in R3 are known to be isolated [2]; however, no example was known of a nonisolated minimal surface.
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    Symmetry of Constant Mean Curvature Hypersurfaces in Hyperbolic Space
    (1984) Levitt, G.; Rosenberg, H.
    In a recent paper, M. Do Carmo and B. Lawson studied hypersurfaces M of constant mean curvature in hyperbolic space [2]. They use the Alexandrov reflection technique to study M given the asymptotic boundary ðM. For example, one of their theorems says M is a horoshpere when ðM reduces to a point. They also prove a Bernstein type theorem for minimal graphs. In this paper we shall extend their results to other boundary conditions. We prove an embedded M, of constant mean curvature, with ðM a subset of a codimension one sphere S, either is invariant by reflection in the hyperbolic hyperplane containg S or is a hypersphere. In the former case M is a "bigraph" over H: it meets any geodesic orthogonal to H either not at all or transversaly in two points (one on each side of H) or tangentially on H.
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    Topology and Differentiability of Labyrinths in the Disc and Annulus
    (1984) Levitt, G.; Rosenberg, H.
    The study of differential equations in the plane which are locally of the form ðy / ðx =F(x,y), gives rise to labyrinths. They are limit sets of bounded solutions to this equation. This is made precise in [Ro], where the singularities considered are thorns and tripods. In part I of this paper, we shall extend the results of [Ro] to differential equations with n-prong singularities, in the disc and annulus. For the disc, the story is not essentially different from the previous case. However, for the annulus, the study is quite different and more complicated. In both cases, we obtain a topological structure theorem for solutions of the equation.