J. Fabio Montenegro

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

Mean time exit and isoperimetric inequalities for minimal submanifolds of N×ℝ

\mathbb{R}^{3}

Spectrum Estimates and Applications to Geometry

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.

Complete submanifolds of

We give sharp sectional curvature estimates for complete immersed cylindrically bounded m-submanifolds φ : M → N × Rl, n+ l ≤ 2m− 1 provided that either φ is proper with the second fundamental form with certain controlled growth or M has scalar curvature with strong quadratic decay. This latter gives a non-trivial extension of the Jorge-Koutrofiotis Theorem [8]. Mathematics Subject Classification (2000): 53C42

R^n

Based on Markvorsen and Palmer’s work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of N ×R. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fundamental form.

with finite topology

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 is discrete. This gives a positive answer to a question of Yau [22].

Riemannian Submersions with Discrete Spectrum

In 1867, E. Beltrami (Ann Mat Pura Appl 1(2):329–366, 1867, [12]) introduced a second order elliptic operator on Riemannian manifolds, defined by \(\Delta ={\mathrm{{div}\,}}\circ {{\mathrm {grad}\,}}\), extending the Laplace operator on \(\mathbb {R}^{n}\), called the Laplace–Beltrami operator. The Laplace–Beltrami operator became one of the most important operators in Mathematics and Physics, playing a fundamental role in differential geometry, geometric analysis, partial differential equations, probability, potential theory, stochastic process, just to mention a few. It is in important in various differential equations that describe physical phenomena such as the diffusion equation for the heat and fluid flow, wave propagation, Laplace equation and minimal surfaces.