arXiv subjects

Alexandre Paiva Barreto

Publications and source records attributed to Alexandre Paiva Barreto.

7 recordsLinked to original sources

arXiv2023

Polynomial Weingarten Tubular Surfaces in Euclidean, Hyperbolic and Lorentzian 3-spaces

In this article we fully classify regular tubular surfaces in Euclidean, Lorentzian and hyperbolic 3-spaces whose Gaussian and mean curvatures $K$ and $H$ verify a polynomial relation. More precisely, we determine the set $S(Q)$ of all regular tubular surfaces whose curvatures verify a given polynomial relation $Q(K,H)=0$, and the set $Q(S)$ of all polynomial relations vanished by the curvatures of a given tubular surface $S$.

math.DG↗

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

Curvature estimates for graphs in warped products

We prove local and global upper estimates for the infimum of the mean curvature, the scalar curvature and the norm of the shape operator of graphs in a warped product space. Using these estimates, we obtain some results on pseudo-hyperbolic spaces and space forms.

math.DG↗

arXiv2014

Involutions on sapphire Sol 3-manifolds and the Borsuk-Ulam theorem for maps into $R^n$

For each sapphire Sol $3$-manifold, we classify the free involutions. For each triple $(M, τ; R^n)$ where $M$ is a sapphire Sol $3$-manifold and $τ$ is a free involution, we show if $(M, τ; R^n)$ has the Borsuk-Ulam property or not. It is known that for $n>3$ the Borsuk-Ulam property does not hold independent of the involution, so we provide a classification when $n=2$ and $3$.

math.AT↗

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↗

arXiv2012

Deformation of Hyperbolic Cone-Structures: Study of the non-Colapsing case

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence (M_{i},p_{i}) of pointed hyperbolic cone-manifolds with topological type (M,Σ), where M is a closed, orientable and irreducible 3-manifold and Σ an embedded link in M. Assuming that the lengths of the singularity remain uniformly bounded, we prove that either the sequence M_{i} collapses and M is Seifert fibered or a Sol manifold, or the sequence M_{i} does not collapse and in this case a subsequence of (M_{i},p_{i}) converges to a complete Alexandrov space of dimension 3 endowed with a hyperbolic metric of finite volume on the complement of a finite union of quasi-geodesics. We apply this result to a conjecture of Thurston and to the case where Σ is a small link in M.

math.GT↗

arXiv2012

Deformations of Hyperbolic Cone-Structures: Study of the Collapsing case

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence $(M_{i}%, p_{i}) $ of pointed hyperbolic cone-manifolds with topological type $(M,Σ) $, where $M$ is a closed, orientable and irreducible 3-manifold and $Σ$ an embedded link in $M$. If the sequence $M_{i}$ collapses and assuming that the lengths of the singularity remain uniformly bounded, we prove that $M$ is either a Seifert fibered or a $Sol$ manifold. We apply this result to a question stated by Thurston and to the study of convergent sequences of holonomies.

math.GT↗