On submanifolds with tamed second fundamental form

Abstract

We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_N\le \kappa \le 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard then $M$ has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.

10 pages

References in corpus (1)

• Eigenvalue Estimates for submanifolds of $N\times \mathbb{R}$ with locally bounded mean curvature

Cited by in corpus (4)

• On the fundamental tone of minimal submanifolds with controlled extrinsic curvature
• Density and spectrum of minimal submanifolds in space forms
• On the structure of hypersurfaces in ${\mathbb{H}}^n\times \mathbb{R}$ with finite strong total curvature
• L^2 Harmonic 1-forms on submanifolds with finite total curvature