We give a sectional curvature estimate for cylindrically bounded, properly immersed submanifolds of product manifolds N × R ℓ , provided the norm of the second fundamental form grows at most linearly.
We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As… (More)
We show that the spectrum of complete minimal submanifolds properly immersed in a ball of R n are discrete. In particular, the Martin-Morales complete minimal surfaces properly immersed in a ball of R 3 have discrete spectrum. This gives a partial answer to a question of Yau .
We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of R 3 is discrete. This gives a positive answer to a question of Yau .
J. Nash proved in  that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B R N (1) of the Euclidean space R N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding.
We extend the Chern-Heinz inequalities about mean curvature and scalar curvature of graphs of C 2-functions to leaves of transversally oriented codimension one C 2-foliations of Riemannian manifolds. That extends partially Salavessa's work on mean curvature of graphs and generalize results of Barbosa-Kenmotsu-Oshikiri  and Barbosa-Gomes-Silveira … (More)