G. Pacelli Bessa

Eigenvalue Estimates for Submanifolds with Locally Bounded Mean Curvature

We present a method to obtain lower bounds for firstDirichlet eigenvalue in terms of vector fields with positivedivergence. Applying this to the gradient of a distance functionwe obtain estimates of eigenvalue of balls inside the cut locus and of domains Ω ⊂ M ∩ BN(p, r) in submanifolds M ⊂ϕNwith locally bounded mean curvature. Forsubmanifolds of Hadamard manifolds with bounded mean curvaturethese lower bounds depend only on the dimension of the submanifold and the bound on its mean curvature.

An Extension of Barta's Theorem and Geometric Applications

We prove an extension of a theorem of Barta and we give some geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We show that the spectrum of the Nadirashvili bounded minimal surfaces in

Stochastic completeness and volume growth

\mathbb{R}^{3}

On cylindrically bounded H-Hypersurfaces of H n × R

have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations.

An estimate for the sectional curvature of cylindricalli bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder

The Spectrum of the Martin-Morales-Nadirashvili Minimal Surfaces Is Discrete

{B(r)\times{\mathbb R}^{\ell}}