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J. Fabio Montenegro

Publications and source records attributed to J. Fabio Montenegro.

14 recordsLinked to original sources
arXiv2021

Maximizing the first eigenvalue of the Jacobi operator

We consider the Jacobi operator, defined on a closed oriented hypersurfaces immersed in the Euclidean space with the same volume of the unit sphere. We show a local generalization for the classical result of the Willmore functional for the Euclidean sphere. As a consequence, we prove that the first eigenvalue of the Jacobi operator in the Euclidean sphere is a local maximum and this result is a global one in the closed oriented surfaces space of $\mathbb{R}^3$ and genus zero.

math.DG↗
arXiv2019

Foliation by free boundary constant mean curvature leaves

Let $M$ be a Riemannian manifold of dimension $n+1$ with smooth boundary and $p\in \partial M$. We prove that there exists a smooth foliation around $p$ whose leaves are submanifolds of dimension $n$, constant mean curvature and its arrive perpendicular to the boundary of M, provided that $p$ is a nondegenerate critical point of the mean curvature function of $\partial M$.

math.DG↗
arXiv2012

Essential spectrum of a class of Riemannian manifolds

In this paper we consider a family of Riemannian manifolds, not necessarily complete, with curvature conditions in a neighborhood of a ray. Under these conditions we obtain that the essential spectrum of the Laplacian contains an interval. The results presented in this paper allow to determine the spectrum of the Laplace operator on unlimited regions of space forms, such as horoball in hyperbolic space and cones in Euclidean space.

math.DG↗
arXiv2011

An estimate for the sectional curvature of cylindrically bounded submanifolds

We give sharp sectional curvature estimates for complete immersed cylindrically bounded $m$-submanifolds $ϕ:M\to N\times\mathbb{R}^{\ell}$, $n+\ell\leq 2m-1$ 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 [7]

math.DG↗
arXiv2010

Riemannian submersions with discrete spectrum

We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the total space is discrete if and only if the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.

math.DG↗
arXiv2009

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.

math.DG↗
arXiv2008

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.

math.DG↗
arXiv2008

Eigenvalues Estimates for the p-Laplace Operator on Manifolds

We obtain geometric estimates for the first eigenvalue and the fundamental tone of the p-laplacian on manifolds in terms of admissible vector fields. Also, we defined a new spectral invariant and we show its relation with the geometry of the manifold.

math.DG↗
arXiv2006

On compact CMC-Hypersurfaces of $N\times \mathbb{R}$

Let ${\mathscr F}(N\times \mathbb{R})$ be the set of all closed $H$-hypersurfaces $M\subset N\times \mathbb{R}$, where $N$ is a simply connected complete Riemannian $n$-manifold with sectional curvature $K_{N}\leq -κ^{2}<0$. We show that ${\Hm}(N\times \mathbb{R})=\inf_{M\in {\mathscr F}(N\times \mathbb{R})}\{| H_{M}| \}\geq (n-1)κ/n $.

math.DG↗
arXiv2006

Complete subamanifolds of $\mathbb{R}^{n}$ with finite topology

We show that a complete $m$-dimensional immersed submanifold $M$ of $\mathbb{R}^{n}$ with $a(M)<1$ is properly immersed and have finite topology, where $a(M)\in [0,\infty]$ is an scaling invariant number that gives the rate that the norm of the second fundamental form decays to zero at infinity. The class of submanifolds $M$ with $a(M)<1$ contains all complete minimal surfaces in $\mathbb{R}^{n}$ with finite total curvature, all $m$-dimensional minimal submanifolds $M $ of $ \mathbb{R}^{n}$ with finite total scalar curvature $\smallint_{M}| α|^{m} dV<\infty $ and all complete 2-dimensional complete surfaces with $\smallint_{M}| α|^{2} dV<\infty $ and nonpositive curvature with respect to every normal direction, since $a(M)=0$ for them.

math.DG↗
arXiv2005

On Cheng's Eigenvalue Comparison Theorems

We prove Cheng's eigenvalue comparison theorems for geodesic balls within the cut locus under weaker geometric hypothesis, and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity becomes isometric rigidity under upper sectional curvature bounds or lower Ricci curvature bounds. We construct examples of smooth metrics showing that our results are true extensions of Cheng's theorem. We also construct a family of complete smooth metrics on the Euclidean space non-isometric to the constant sectional curvature k metrics of the simply connected space forms of constant sectional curvature k such that the geodesic balls of radius r have the same first eigenvalue and the geodesic spheres have the same mean curvatures. In the end we construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone positive that generalize Veeravalli's examples.

math.DG↗
arXiv2005

An Extension of Barta's Theorem and Geometric Applications

We prove an extension of a theorem of Barta then we make few 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 prove an stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove a generalization of a result of Kazdan-Kramer about existence of solutions of certain quasi-linear elliptic equations.

math.DG↗
arXiv2002

Eigenvalue estimates for submanifolds with locally bounded mean curvature

We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean curvature of the submanifold. For submanifolds fo Hadamard manifolds these lower bounds depend only on the dimension and mean curvature of the submanifold.

math.DG↗