J. H. Lira

Federal University of Ceará | UFC ·  Departamento de Matemática

In this paper we make a detailed analysis of conservation principles in the context of a family of fourth-order gravitational theories generated via a quadratic Lagrangian. In particular, we focus on the associated notion of energy and start a program related to its study. We also exhibit examples of solutions which provide intuitions about this no...

In this paper, we analyse semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically Euclidean (AE) manifolds. In particular, electromagnetic fields give rise to this kind of system. In this context, under suitab...

In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric...

Assuming that there exists a translating soliton u∞ with speed C in a domain Ω and with prescribed contact angle on ∂Ω, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to u∞ + Ct as t →∞. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings...

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a Generalized Robertson Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the metric of such a solution is conformal to the one of the leaf of the GRW in asymptotic time. Furthermore, if t...

In the present paper, we study the coupled Einstein Constraint Equations (ECE) on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. In particular, we do not impose any specific model for infinity. First, we prove an existence criteria on compact manifolds with boundary which applies to mor...

The so called Jenkins–Serrin problem is a kind of Dirichlet problem for graphs with prescribed mean curvature that combines, at the same time, continuous boundary data with regions of the boundary where the boundary values explodes either to \(+\infty \) or to \(-\infty .\) We give a survey on the development of Jenkins–Serrin type problems over do...

In this paper we establish a natural framework for the stability of mean curvature flow solitons in warped product spaces. These solitons are regarded as stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons...

In this paper we make a detailed analysis of conservation principles in the context of a family of fourth-order gravitational theories generated via a quadratic Lagrangian. In particular, we focus on the associated notion of energy and start a program related to its study. We also exhibit examples of solutions which provide intuitions about this no...

In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric...

The reduced thin-sandwich equations (RTSE) appear within Wheeler’s thin-sandwich approach toward the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed, in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In particular, it has been shown...

We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds M × R. Moreover, we give examples of these graphs in the cases of R3 and H2 × R.

In this short note, we prove that a graphical solution to the mean curvature flow with prescribed contact angle converges to $u_\infty +Ct$, where $u_\infty$ is a translating soliton with speed $C$, as time $t$ goes to infinity. Our result holds on any smooth relatively compact subdomain $\Omega\subset N$ of a Riemannian manifold $N$ of arbitrary d...

We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough....

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on $H$ and the mean curvature of the Killing cylinder...

In this paper, we consider varifolds $\Sigma \hookrightarrow \Omega$ of arbitrary codimension and bounded mean curvature contained in an open subset of a Riemannian manifold $M$. Under mild assumptions on $\partial \Omega$ and on the curvature of $M$, we prove a barrier principle at infinity for $\Sigma$, namely we show that the distance to $\parti...

In this paper, we apply the Logistic PCA (LPCA) as a dimensionality reduction tool for visualizing patterns and characterizing the relevance of mathematics abilities from a given population measured by a large-scale assessment. We establish an equivalence of parameters between LPCA, Inner Product Representation (IPR) and the two paramenter logistic...

In this paper we analyse a type of semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically flat manifolds. In particular, electromagnetic fields give rise to this kind of systems. In this context, under suitab...

The reduced thin-sandwich equations (RTSE) appear within Wheeler's thin-sandwich approach towards the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In particular, it has been shown...

We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M\times\mathbb{R}$. Moreover, we give examples of these graphs in the cases of $\mathbb{R}^3$ and $\mathbb{H}^2\times\mathbb{R}$.

The so called Jenkins-Serrin problem is a kind of Dirichlet problem for graphs with prescribed mean curvature that combines, at the same time, continuous boundary data with regions of the boundary where the boundary values explodes either to $+\infty$ or to $-\infty.$ We give a survey on the development of Jenkins-Serrin type problems over domains...

In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R} \times P, {\rm d}t^2+g_0)$ and the Killing field is $X=\partial_t$. Similarly to what happens in the Euclidean settin...

In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R} \times P, {\rm d}t^2+g_0)$ and the Killing field is $X=\partial_t$. Similarly to what happens in the Euclidean settin...

In this paper we introduce and study a notion of mean curvature flow soliton in Riemannian ambient spaces general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces. As expected, our definition is motivated by the self-similarity of certain special solutions of...

The aim of this paper is to introduce a notion of mean curvature flow soliton general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces.

In this paper we show the validity, under certain geometric conditions, of Wheeler's thin sandwich conjecture for higher-dimensional theories of gravity. The results we present here are an extension of the results already shown by R. Bartnik and G. Fodor for the 3-dimensional. In spite of the geometric restrictions needed for the proofs, we show th...

In this paper we show the validity, under certain geometric conditions, of Wheeler's thin sandwich conjecture for higher dimensional theories of gravity. We extend the results shown by R. Bartnik and G. Fodor for the 3-dimensional case in two ways. On the one hand, we show that the results obtained by the mentioned authors are valid in arbitrary di...

The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with nonempty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According t...

We use geometric analysis of the Lorentzian distance on an n-dimensional spacetime $\bar{M}$ to derive some non-existence results and some sharp mean curvature estimates for trapped submanifolds contained in the chronological future of either a point $p\in \bar{M}$ or of a space-like achronal hypersurface ${\rm{\Sigma }}\subset \bar{M}$ . Our resul...

In this paper, we provided conditions for an entire constant mean curvature Killing graph lying inside a possible unbounded region to be necessarily a slice.

A translation operator acting in a space with a diagonal metric is introduced to describe the motion of a particle in a quantum system. We show that the momentum operator and, as a consequence, the uncertainty relation now depend on the metric. It is also shown that, for any metric expanded up to second order, this formalism naturally leads to an e...

We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. We apply these formulas to derive an anisotropic tube formula for hypersurfaces in (Formula presented.) and to give a short euclidean proof of the anisotropic Sobolev inequality. A discussion about the first variation of the anisotropic area is also incl...

We study the intrinsic and the extrinsic geometry of Clifford tori in Berger spheres, π>0, and use these results to prove several local rigidity and bifurcation results in this context. © © 2014. Published by Oxford University Press. All rights reserved. For permissions, please email: [email protected] /* */

We formulate a variational notion of anisotropic mean curvature for immersed hypersurfaces of arbitrary Riemannian manifolds. Hypersurfaces with constant anisotropic mean curvature are characterized as critical points of an elliptic parametric functional subject to a volume constraint. We provide examples of such hypersurfaces in the case of rotati...

We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds confined into either a horocylinder of N X L or a horoball of N, where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into cylinders over compact balls....

We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along its boundary. These results may be useful for modelling stationary hypersur...

We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates for stochastically complete subm...

We show that under certain curvature conditions of the ambient space an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice.

We present the area and coarea formulas for Lipschitz maps between euclidean spaces and $C^1$ maps between differentiable manifolds which are valid for general volume densities. As applications, we give a short, "euclidean" proof of the anisotropic Sobolev inequality and describe an anisotropic tube formula for hypersurfaces in $mathbb{R}^n$.

We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field. KeywordsConformal Killing graphs–Prescribed mean curvature–Quasilinear elliptic PDE

We study a Neumann problem related to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. We prove that in a particular case these graphs converge to a bounded minimal graph which contacts the cylinder over the domain orthogonally along its boundary.

We extend the interior gradient estimate due to N. Korevaar and L. Simon for solutions of the mean curvature equation from the case of Euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary data, thus extending a result...

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...

We establish a spinorial representation for surfaces immersed with prescribed mean curvature in Heisenberg space. This permits to obtain minimal immersions starting with a harmonic Gauss map whose target is either the Poincaré disc or a hemisphere of the round sphere.

In this paper we will discuss a Weierstrass type representation for minimal surfaces in Riemannian and Lorentzian 3-dimensional manifolds. Mathematics Subject Classification (2010)Primary 53C42–Secondary 53C50

A hypersurface in a Riemannian manifold is r-minimal if its (r+1)-curvature, the (r+1)th elementary symmetric function of its principal curvatures, vanishes identically. If n>2(r+1) we show that the rotationally invariant r-minimal hypersurfaces in ℝ n+1 are nondegenerate in the sense that they carry no nontrivial Jacobi fields decaying rapidly eno...

It is proved that a pair of spinors satisfying a Dirac-type equation represent surfaces immersed in Anti-de Sitter space with prescribed mean curvature. Here, we consider Anti-de Sitter space as the Lie group SU1,1 endowed with a one-parameter family of left-invariant metrics where only one of them is bi-invariant and corresponds to the isometric e...

We prove a Bonnet theorem for isometric immersions of semi-Riemannian manifolds into products of semi-Riemannian space forms. Namely, we give necessary and sufficient conditions for the existence and uniqueness (up to an isometry of the ambient space) of an isometric immersion of a semi-Riemannian manifold into a product of semi-Riemannian space fo...

We prove the existence of holomorphic quadratic differentials for surfaces with parallel mean curvature in some four-dimensional products of space forms. These differentials are then used to characterize spheres with parallel mean curvature immersed into these spaces.

It is proved that a pair of spinors satisfying a Dirac type equation represents surfaces immersed in Berger spheres with prescribed mean curvature. Using this, we prove that the Gauss map of a minimal surface immersed in a Berger sphere is harmonic. Conversely, we exhibit a representation of minimal surfaces in Berger spheres in terms of a given ha...

It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field. Résumé On démontre l'existence et unicité de graphes avec courbure moyenne prescrite dans les submersions fibrées par les solutions d'un champ de vecteurs de Killing vertical.

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.

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Rr$. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable function $\psi$ in $\bar M$, we find a closed hypersurface $\Sigma$ which is solution of an equation of the form $F...

We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into $H$-type groups.

We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.

Let $\psi$ be a given function defined on a Riemannian space. Under what conditions does there exist a compact starshaped hypersurface $M$ for which $\psi$, 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 s...

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.

The subject of this paper is properly embedded H − H- surfaces in Riemannian three manifolds of the form M 2 × R M^2\times \mathbf {R} , where M 2 M^2 is a complete Riemannian surface. When M 2 = R 2 M^2={\mathbf R}^2 , we are in the classical domain of H − H- surfaces in R 3 {\mathbf R}^3 . In general, we will make some assumptions about M 2 M^2 i...

The Christoffel problem, in its classical formulation, asks for a characterization of real functions defined on the unit sphere $S^{n-1}\subset\mathbb{R}^n$ which occur as the mean curvature radius, expressed in terms of the Gauss unit normal, of a closed convex hypersurface, i.e. the boundary of a convex body in $\mathbb{R}^n$. In this work we con...

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 o...

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 hypersu...

It is proved that an embedded hypersurface in a hemisphere of the Euclidean unit spherewith constant mean curvature and spherical boundary inherits, under certainconditions, the symmetries of its boundary. In particular, spherical caps are theonly such hypersurfaces whose boundary are geodesic spheres.

We study the existence and unicity of graphs with constant mean curvature in the Euclidean sphere \mathbbSn + 1 (a)\mathbb{S}^{n + 1} (a) of radius a. Given a compact domain \mathbbSn + 1 (a)\mathbb{S}^{n + 1} (a) and a real differentiable function \mathbbSn + 1 (a)\mathbb{S}^{n + 1} (a) considering the \mathbbSn + 1 (a)\mathbb{S}^{n + 1} (...

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space H n+1 defined over domains in geodesic spheres of H n+1 whose boundary has positive mean curvature with respect to the inward orientation.

We establish necessary and sucient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a three-step solvable Lie group. This comprises the case of immersions into complex hyperbolic spaces and Damek-Ricci spaces.