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Ezequiel Barbosa

Publications and source records attributed to Ezequiel Barbosa.

27 records · Page 2Linked to original sources
arXiv2017

Second-order Sobolev inequalities on a class of Riemannian manifolds with nonnegative Ricci curvature

Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying $ρΔ_g ρ\geq n- 5\geq 0$, where $Δ_g$ is the Laplace-Beltrami operator on $(M,g)$ and $ρ$ is the distance function from a given point. If $(M,g)$ supports a second-order Sobolev inequality with a constant $C>0$ close to the optimal constant $K_0$ in the second-order Sobolev inequality in $\mathbb R^n$, we show that a global volume non-collapsing property holds on $(M,g)$. The latter property together with a Perelman-type construction established by Munn (J. Geom. Anal., 2010) provide several rigidity results in terms of the higher-order homotopy groups of $(M,g)$. Furthermore, it turns out that $(M,g)$ supports the second-order Sobolev inequality with the constant $C=K_0$ if and only if $(M,g)$ is isometric to the Euclidean space $\mathbb R^n$.

math.AP↗
arXiv2016

On stable CMC hypersurfaces with free-boundary in a Euclidean Ball

In this note, we observe that if $B$ is a ball in a Euclidean space with dimension $n$, $n\geq3$, then a stable CMC hypersurface $Σ$ with free boundary in $B$ satisfies \[ nA\leq L\leq nA\left( \frac{1+\sqrt{1+4(n+1)H^2}}{2} \right)\,, \] where $L$, $A$ and $H$ denote the length of $\partial Σ$, the area of $Σ$ and the mean curvature of $Σ$, respectively. Consequently, if the boundary $\partial Σ$ is embedded then $Σ$ must be totally geodesic or starshaped with respect to the center of the ball. This result is an improvement of a theorem proved by A. Ros and E. Vergasta \cite{R-V} . In particular, if $n=3$, the only stable CMC surfaces with free boundary in $B$ are the totally geodesic disks or the spherical caps. This last result was proved very recently by I. Nunes \cite{N} using an extended stability result and a modified Hersch type balancing argument to get a better control on the genus. We don't use that modified Hersch type argument. However, we use a Nunes type Stability Lemma and a crucial result due to A. Ros and E. Vergasta.

math.DG↗
arXiv2015

Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index

Let $(M^{n+1},g,e^{-f}dμ)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in $\mathbb{R}^{n+1}$ with $2\leq n\leq 6$. We also prove some estimates on the $f$-index of $f$-minimal hypersurfaces, and give a conformal structure of $f$-minimal surface with finite $f$-index in three-dimensional smooth metric measure space.

math.DG↗
arXiv2014

Rigidity Theorems of conformal class on compact manifolds with boundary

Let M be a compact manifold with boundary. In this paper, we discuss some rigidity theorems of metrics in a same conformal class that fixes the boundary and satisfy certain integral conditions on the the scalar curvatures and the mean curvatures on the boundary. No condition on the first eigenvalues of operators is need.

math.DG↗
arXiv2014

On the limiting behavior of the Brown-York quasi-local mass in asymptotically hyperbolic manifolds

We show that the limit at infinity of the vector-valued Brown-York-type quasi-local mass along any coordinate exhaustion of an asymptotically hyperbolic $3$-manifold satisfying the relevant energy condition on the scalar curvature has the conjectured causal character. Our proof uses spinors and relies on a Witten-type formula expressing the asymptotic limit of this quasi-local mass as a bulk integral which manifestly has the right sign under the above assumptions. In the spirit of recent work by Hijazi, Montiel and Raulot, we also provide another proof of this result which uses the theory of boundary value problems for Dirac operators on compact domains to show that a certain quasi-local mass, which converges to the Brown-York mass in the asymptotic limit, has the expected causal character under suitable geometric assumptions.

math.DG↗
arXiv2014

A positive mass theorem for asymptotically flat manifolds with a non-compact boundary

We prove a positive mass theorem for $n$-dimensional asymptotically flat manifolds with a non-compact boundary if either $3\leq n\leq 7$ or if $n\geq 3$ and the manifold is spin. This settles, for this class of manifolds, a question posed in a recent paper by the first author in connection with the long-term behavior of a certain Yamabe-type flow on scalar-flat compact manifolds with boundary.

math.DG↗
arXiv2012

The k-Yamabe problem on CR manifolds

We introduce the notion of pseudohermitian k-curvature, which is a natural extension of the Webster scalar curvature, on an orientable manifold endowed with a strictly pseudoconvex pseudohermitian structure (referred here as a CR manifold) and raise the k-Yamabe problem on a compact CR manifold. When k=1, the problem was proposed and partially solved by Jerison and Lee for CR manifolds non-locally CR-equivalent to the CR sphere. For k > 1, the problem can be translated in terms of the study of a fully nonlinear equation of type complex k-Hessian. We provide some partial answers related to the CR k-Yamabe problem. We establish that its solutions with null Cotton tensor are critical points of a suitable geometric functional constrained to pseudohermitian structures of unit volume. Thanks to this variational property, we establish a Obata type result for the problem and also compute the infimum of the functional on the CR sphere. Furthermore, we show that this value is an upper bound for the corresponding one on any compact CR manifolds and, assuming the CR Yamabe invariant is positive, we prove that such an upper bound is only attained for compact CR manifolds locally CR-equivalent to the CR sphere. In the Riemannian field, recent advances have been produced in a series of outstanding works.

math.DG↗