Four-manifolds with positive curvature

In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval $[\frac{3\sqrt{3}-5}{4},\,1].$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the $4$-sphere.Comment: Revised versio

Estimates for Minimal Volume and Minimal Curvature on 4-dimensional compact manifolds

In a remarkable article published in 1982, M. Gromov introduced the concept of minimal volume, namely, the minimal volume of a manifold $M^n$ is defined to be the greatest lower bound of the total volumes of $M^n$ with respect to complete Riemannian metrics whose sectional curvature is bounded above in absolute value by 1. While the minimal curvature, introduced by G. Yun in 1996, is the smallest pinching of the sectional curvature among metrics of volume 1. The goal of this article is to provide estimates to minimal volume and minimal curvature on 4-dimensional compact manifolds involving some differential and topological invariants. Among these ones, we get some sharp estimates for minimal curvature.Comment: revised versio

Volume functional of compact 44-manifolds with a prescribed boundary metric

We prove that a critical metric of the volume functional on a $4$-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.Comment: To appear in The Journal of Geometric Analysi

Geometric Inequalities for Critical Metrics of the Volume Functional

The goal of this article is to investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifold. In addition, localized version estimates to the mean curvature and area of the boundary of critical metrics are also obtained.Comment: Fixed typo

An Analogy of the Quantum Hall Condutivity in a Lorentz-symmetry Violation Setup

We investigated some influences of unconventional physics, such Lorentz-symmetry violation, for quantum mechanical systems. In this context, we calculated a important contribution for Standard Model Extension. In the non-relativistic limit, we obtained a analogy of the Landau levels and the quantum Hall conductivity related to this contribution for low energy systems.Comment: 12 pages, no figure

On Various Types of Shadowing for Geometric Lorenz Flows

We show that Lorenz flows have neither limit shadowing property nor average shadowing property nor the asymptotic average shadowing property where the reparametrizations related to these concepts relies on the set of increasing homeomorphisms with bounded variation.Comment: 18 pages, 7 figure

Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M.$ Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form $\Bbb{R}^{4},$ $\Bbb{H}^{4}$ or $\Bbb{S}^{4}$. Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that $M$ has divergence-free Bach tensor.Comment: To appear in The Journal of Geometric Analysi

Critical metrics of the volume functional on compact three-manifolds with smooth boundary

We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manifolds. In addition, we obtain a B\"ochner type formula which enables us to show that a Miao-Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in $\Bbb{S}^3.$Comment: to appear in The Journal of Geometric Analysi

Bounds on volume growth of geodesic balls for Einstein warped products

The purpose of this note is to provide some volume estimates for Einstein warped products similar to a classical result due to Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. To do so, we make use of the approach of quasi-Einstein manifolds which is directly related to Einstein warped product. In particular, we present an obstruction for the existence of such a class of manifolds.Comment: To appear in Proc. of the American Math. So

Lorentz symmetry violation and an Analog of Landau levels

Within the context of Lorentz violating extended electrodynamics, we study an analog of Landau quantization for a system where a neutral particle moves in the presence of an electromagnetic field and a constant four-vector that breaks Lorentz symmetry. The nonrelativistic Hamiltonian associated to this system is obtained using the Foldy-Wouthuysen transformation for a Dirac spinor. The degenerated energy spectrum is obtained for a time-like and a space-like parameter Lorentz-breaking vector. The energy dependence of the cyclotron rotation direction in terms of supersymmetric quantum mechanics is analyzed.Comment: 11 pages, no figures, v2: new title,new versio