Searcharxiv⌕ Search

arXiv subjects

Jorge H. S. de Lira

Publications and source records attributed to Jorge H. S. de Lira.

7 recordsLinked to original sources
arXiv2021

A barrier principle at infinity for varifolds with bounded mean curvature

Our work investigates varifolds $Σ\subset M$ in a Riemannian manifold, with arbitrary codimension and bounded mean curvature, contained in an open domain $Ω$. Under mild assumptions on the curvatures of $M$ and on $\partial Ω$, also allowing for certain singularities of $\partial Ω$, we prove a barrier principle at infinity, namely we show that the distance of $Σ$ to $\partial Ω$ is attained on $\partial Σ$. Our theorem is a consequence of sharp maximum principles at infinity on varifolds, of independent interest.

math.DG↗
arXiv2013

The Dirichlet Problem for Curvature Equations in Riemannian Manifolds

We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to extend some of the existence theorems of Caffarelli, Nirenberg and Spruck [4] and Ivochkina, Trundinger and Lin [19] to more general curvature functions and less convex domains.

math.DG↗
arXiv2008

Examples of scalar-flat hypersurfaces in $\mathbb{R}^{n+1}$

Given a hypersurface $M$ of null scalar curvature in the unit sphere $\mathbb{S}^n$, $n\ge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $\Rr^{n+1}$ as a normal graph over a truncated cone generated by $M$. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.

math.DG↗
arXiv2007

Uniqueness of starshaped compact hypersurfaces with prescribed $m$-th mean curvature in hyperbolic space

Let $ψ$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $ψ$, when evaluated on $M$, coincides with the $m-$th elementary symmetric function of principal curvatures of $M$ for a given $m$? The corresponding existence and uniqueness problems in Euclidean space have been investigated by several authors in the mid 1980's. Recently, conditions for existence were established in elliptic space and, most recently, for hyperbolic space. However, the uniqueness problem has remained open. In this paper we investigate the problem of uniqueness in hyperbolic space and show that uniqueness (up to a geometrically trivial transformation) holds under the same conditions under which existence was established.

math.DG↗
arXiv2005

Examples and structure of CMC surfaces in some Riemannian and Lorentzian homogeneous spaces

It is proved that the holomorphic quadratic differential associated to CMC surfaces in Riemannian products $\mathbb{S}^2\times\Rr$ and $\mathbb{H}^2\times \Rr$ discovered by U. Abresch and H. Rosenberg could be obtained as a linear combination of usual Hopf differentials. Using this fact, we are able to extend it for Lorentzian products. Families of examples of helicoidal CMC surfaces on these spaces are explicitly described. We also present some characterizations of CMC rotationally invariant discs and spheres. Finally, after establish some height and area estimates, we prove the existence of constant mean curvature Killing graphs.

math.DG↗
arXiv2003

Constant higher order mean curvature hypersurfaces in Riemannian spaces

It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in R^3. In a recent paper the first and third authors have shown that this is true for the case of hypersurfaces in R^{n+1} with constant scalar curvature, and more generally, hypersurfaces with constant higher order r-mean curvature, when r>1. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold, where we will consider a general geometric configuration consisting of an immersed hypersurface with boundary on an oriented hypersurface P. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of P, as well as the geometry of P. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature. In particular, we are able to extend the previous symmetry results to the case of hypersurfaces with constant higher order r-mean curvature in the hyperbolic space and in the sphere.

math.DG↗