Eigenvalue estimates for submanifolds of warped product spaces

We give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type $N^n times_f Q^q$, where $f in C^infty(N)$. We also study the essential spectrum of these minimal submanifolds.

Curvature estimates for properly immersed $phi_{h}$-bounded submanifolds

Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded submanifolds. On the other hand, Alias, Bessa and Dajczer proved sharp mean curvature estimates for properly immersed cylindrically bounded submanifolds. In this paper we prove these sectional and mean curvature estimates for a larger class of submanifolds, the properly immersed $phi$-bounded submanifolds.

Maximum principle for semi-elliptic trace operators and geometric applications

Based on ideas of L. Alias, D. Impera and M. Rigoli developed in Hypersurfaces of constant higher order mean curvature in warped products, we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature estimates and to give an extension of Alias-Impera-Rigoli Slice Theorem

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)timesR^{ell}$ in a product Riemannian manifold $N^{n-ell}timesR^{ell}$. It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature.

On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces

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 $mathbb{R}^{3}$ is discrete. This gives a positive answer to a question of Yau.

On the mean curvature of Nash isometric embeddings

J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{mathbb{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.

On submanifolds with tamed second fundamental form

We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}leq kappa leq 0$ 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.

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

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N times mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} leq kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone.

Mean time exit and isoperimetric inequalities for minimal submanifolds of $Ntimes mathbb{R}$

Based on Markvorsen and Palmers work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of $Ntimesmathbb{R}$. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fundamental form.