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Antonio Caminha

Publications and source records attributed to Antonio Caminha.

6 recordsLinked to original sources
arXiv2020

A maximum principle related to volume growth and applications

In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold $M$ for which there exists a bounded vector field $X$ such that $\langle\nabla f,X\rangle\geq 0$ on $M$ and $\mathrm{div} X\geq af$ outside a suitable compact subset} of $M$, for some constant $a>0$, under the assumption that $M$ has either polynomial or exponential volume growth. We then use it to obtain some straightforward applications to smooth functions and, more interestingly, to Bernstein-type results for hypersurfaces immersed into a Riemannian manifold endowed with a Killing vector field, as well as to some results on the existence and size of minimal submanifolds immersed into a Riemannian manifold endowed with a conformal vector field.

math.DG↗
arXiv2017

On the structure of complete kählerian manifolds furnished with closed conformal vector fields

We show that if a connected compact kählerian surface $M$ with nonpositive gaussian curvature is furnished with a closed conformal vector field $ξ$ whose singular points are isolated, then $M$ is isometric to a flat torus and $ξ$ is parallel. We also consider the case of a connected complete kählerian manifod $M$ of complex dimension $n>1$ and furnished with a nontrivial closed conformal vector field $ξ$. In this case, it is well known that the singularities of $ξ$ are automatically isolated and the nontrivial leaves of the distribution generated by $ξ$ and $Jξ$ are totally geodesic in $M$. Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of $M$ along it is nonpositive, we show that $ξ$ is parallel and $M$ is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete kählerian manifolds of complex dimension $n-1$. In particular, the the universal covering of $M$ is isometric to a riemannian product having $\mathbb R^2$ as a factor. We also present a generic example showing that one cannot get rid of the hypothesis on the nonpositivity of the holomorphic sectional curvature along at least one such leaf.

math.DG↗
arXiv2014

CMC hypersurfaces of semi-Riemannian groups

In this paper, we study the geometry of a connected oriented cmc Riemannian hypersurface $M$ of a semi-Riemannian group $G$ of Lie algebra $\mathfrak g$ and index 0 or 1. If $G$ is Riemannian and $M$ is compact and transversal to an element of $\mathfrak g$, we show that it is a lateral class of a closed embedded Lie subgroup of $G$; we also do this if $G$ is Lorentzian, provided $M$ has sufficiently large mean curvature. If $G$ is Riemannian semisimple and $M$ is compact, we prove that $M$ has degenerate Gauss map and minimal relative nullity at least 1. We also extend the above results to the case where $M$ is complete and noncompact. For a Riemannian $G$, we show that a minimal $M$ is either transversal to an element of $\mathfrak g$, hence stable, or has degenerate Gauss map and minimal relative nullity at least 1; for $M$ cmc and transversal to an element of $\mathfrak g$, if we ask the immersion to be proper and have bounded second fundamental form, then $M$ is also a lateral class of a closed embedded Lie subgroup of $G$, provided a certain growing condition on the size of the corresponding Gauss map is satisfied. Finally, for a Lorentzian group $G$, with sectional curvatures bounded from above on Lorentzian planes, we extend a result of Y. Xin, proving that a complete $M$ is totally umbilical, provided it is transversal to a timelike element of $\mathfrak g$, has large enough mean curvature and bounded hyperbolic Gauss map.

math.DG↗