arXiv subjects

Francisco Fontenele

Publications and source records attributed to Francisco Fontenele.

9 recordsLinked to original sources

arXiv2021

On regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb R^{n+1}$, $n\geq 2$, defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb R^{n+1}$, $n\geq 2$, defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

math.DG↗

arXiv2020

A conformal characterization of manifolds of constant sectional curvature

A special case of the main result states that a complete $1$-connected Riemannian manifold $(M^n,g)$ is isometric to one of the models $\mathbb R^n$, $S^n(c)$, $\mathbb H^n(-c)$ of constant curvature if and only if every $p\in M^n$ is a non-degenerate maximum of a germ of smooth functions whose Riemannian gradient is a conformal vector field.

math.DG↗

arXiv2018

Rotational Surfaces with second fundamental form of constant length

We obtain an infinite family of complete non embedded rotational surfaces in $\mathbb R^3$ whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homothety and a rigid motion, a member of that family. In particular, the round sphere and the circular cylinder are the only complete embedded rotational surfaces in $\mathbb R^3$ with second fundamental form of constant length.

math.DG↗

arXiv2017

A characterization of round spheres in space forms

Let $\mathbb Q^{n+1}_c$ be the complete simply-connected $(n+1)$-dimensional space form of curvature $c$. In this paper we obtain a new characterization of geodesic spheres in $\mathbb Q^{n+1}_c$ in terms of the higher order mean curvatures. In particular, we prove that the geodesic sphere is the only complete bounded immersed hypersurface in $\mathbb Q^{n+1}_c,\;c\leq 0,$ with constant mean curvature and constant scalar curvature. The proof relies on the well known Omori-Yau maximum principle, a formula of Walter for the Laplacian of the $r$-th mean curvature of a hypersurface in a space form, and a classical inequality of Gårding for hyperbolic polynomials.

math.DG↗

arXiv2014

On the complexity of isometric immersions of hyperbolic spaces in any codimension

Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a Lipschitz map $F:M^m\to\mathbb R^n$, where $M^m$ is a Hadamard manifold whose curvature lies between negative constants. The main result of this paper is that $F$ must perform a substantial compression: For every $r>0$ and integer $k\geq 2$ there exist $k$ geodesic balls of radius $r$ in $M^m$ that are arbitrarily far from each other, but whose images under $F$ are bunched together arbitrarily close in the Hausdorff sense of $\mathbb R^n$. In particular, every isometric embedding $\mathbb H^m\to\mathbb R^n$ of hyperbolic space must have a complex asymptotic behavior, regardless of how high the codimension is. Hence, there is no truly simple way to realize $\mathbb H^m$ isometrically inside any Euclidean space.

math.DG↗

arXiv2013

Some remarks on the Pigola-Rigoli-Setti version of the Omori-Yau maximum principle

We prove that the hypotheses in the version of the Omori-Yau maximum principle that was given by Pigola-Rigoli-Setti are logically equivalent to the assumption that the manifold carries a $C^2$ proper function whose gradient and Hessian (Laplacian) are bounded. In particular, this result extends the scope of the original Omori-Yau principle, formulated in terms of lower bounds for curvature.

math.DG↗

arXiv2009

Heinz type estimates for graphs in Euclidean space

Let $M^n\subset\mathbb R^{n+1}$ be the graph of a $C^2$-real valued function defined in a closed ball of $\mathbb R^n$. In this work, we obtain upper bounds for $\inf_M|H|$ and $\inf_M|R|$, where $H$ and $R$ are, respectively, the mean curvature and the scalar curvature of $M^n$, generalizing estimates given by Heinz in the case $n=2$ [Math. Annalen 129, 451-454, 1955]. Under the assumption that $M^n$ has negative Ricci curvature, we also obtain an upper bound for $\inf_M|A|$, where $|A|$ is the length of the second fundamental form. As a consequence of this latter estimate one obtains that $\inf |A|=0$ for all entire graphs with negative Ricci curvature in Euclidean space. This gives a partial answer to a question raised by Smith-Xavier [Invent. Math. 90, 443-450, 1987].

math.DG↗

arXiv2009

Good shadows, dynamics, and convex hulls

The Ekeland variational principle implies what can be regarded as a strong version, in the $C^1$ category, of the Yau minimum principle: under the appropriate hypotheses {\it every} minimizing sequence admits a {\it good shadow}, a second minimizing sequence that has good properties and is asymptotic to the original one. Using arguments from dynamical systems, we give another proof of this result and also establish, with the aid of Gromov's theorem on monotonicity of volume ratios, a special case of a conjecture claiming the existence of good shadows in the original $C^2$ setting of the Yau minimum principle. The interest in having an abundance of good shadows stems from the fact that this is a desirable property if one wants to refine the applications of the asymptotic minimum principle, as it allows for information to be localized at infinity. These ideas are applied in this paper to the study of the convex hulls of complete submanifolds of Euclidean $n$-space that have controlled Grassmanian-valued Gauss maps.

math.DG↗

arXiv2009

A Riemannian Bieberbach estimate

The Bieberbach estimate, a pivotal result in the classical theory of univalent functions, states that any injective holomorphic function $f$ on the open unit disc $D$ satisfies $|f"(0)|\leq 4 |f'(0)|$. We generalize the Bieberbach estimate by proving a version of the inequality that applies to all injective smooth conformal immersions $f : D\to \Bbb R^n, n\geq 2$. The new estimate involves two correction terms. The first one is geometric, coming from the second fundamental form of the image surface $f(D)$. The second term is of a dynamical nature, and involves certain Riemannian quantities associated to conformal attractors. Our results are partly motivated by a conjecture in the theory of embedded minimal surfaces.

math.DG↗