A characterization of quadric constant mean curvature hypersurfaces of spheres

Let $\phi:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let us denote by $\ell_v:M\to\mathbb{R}$ the function given by $\ell_v(x)=\phi(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by $f_v(x)=\nu(x),v$, where $\nu:M\to\mathbb{S}^{n}$ is a Gauss map. We will prove that if $M$ has constant mean curvature, and, for some $v\ne{\bf 0}$ and some real number $\lambda$, we have that $\ell_v=\lambda f_v$, then, $\phi(M)$ is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface $M^n$ in $\mathbb{S}^{n+1}$ which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to $2n+4$.Comment: Final version (February 2008). To appear in the Journal of Geometric Analysi

Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson-Walker spacetimes

In this paper we analyze the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete noncompact case. In that case, our approach is based on the use of a generalized version of the Omori-Yau maximum principle for trace type differential operators, recently given in [3].Comment: To appear in Mathematical Proceedings of the Cambridge Philosophical Societ

Comparison theory of Lorentzian distance with applications to spacelike hypersurfaces

In this paper we summarize some comparison results for the Lorentzian distance function in spacetimes, with applications to the study of the geometric analysis of the Lorentzian distance on spacelike hypersurfaces. In particular, we will consider spacelike hypersufaces whose image under the immersion is bounded in the ambient spacetime and derive sharp estimates for the mean curvature of such hypersurfaces under appropriate hypotheses on the curvature of the ambient spacetime. The results in this paper are part of our recent work [1], where complete details and further related results may be found

Comparison theory of Lorentzian distance with applications to spacelike hypersurfaces

Abstract. In this note we summarize some comparison results for the Lorentzian distance function in spacetimes, with applications to the study of the geometric analysis of the Lorentzian distance function on spacelike hypersurfaces. In particular, we will consider spacelike hypersufaces whose image under the immersion is bounded in the ambient spacetime and derive sharp estimates for the mean curvature of such hypersurfaces under appropriate hypotheses on the curvature of the ambient spacetime. The results in this note are part of our recent paper Keywords: Lorenzian distance function, Hessian and Laplacian comparison results, spacelike hypersurface, mean curvature, Omori-Yau maximum principle. PACS: 04.20.Cv,02.40.Vh THE LORENTZIAN DISTANCE FUNCTION Consider M n+1 an (n + 1)-dimensional spacetime, and let p, q be points in M. Using the standard terminology and notation from Lorentzian geometry, one says that q is in the chronological future of p, written p q, if there exists a future-directed timelike curve from p to q. Similarly, q is in the causal future of p, written p < q, if there exists a future-directed causal (i.e., nonspacelike) curve from p to q. Obviously, p q implies p < q. As usual, p ≤ q means that either p < q or p = q. For a subset S ⊂ M, one defines the chronological future of S as I + (S) = {q ∈ M : p q for some p ∈ S}, and the causal future of S as J + (S) = {q ∈ M : p ≤ q for some p ∈ S}. Thus S ∪ I + (S) ⊂ J + (S). In particular, the chronological future I + (p) and the causal future J + (p) of a point p ∈ M are Given a point p ∈ M, one can define the Lorentzian distance function from p by d p (q) = d (p, q). In order to guarantee the smoothness of d p as a function on M, one needs to restrict this function on certain special subsets of M. Consider v is a future-directed timelike unit vector} the fiber of the unit future observer bundle of M at p, and set s p : COMPARISON RESULTS FOR THE LORENTZIAN DISTANCE FROM A POINT where ∇ 2 stands for the Hessian operator on M. Observe where ∆ stands for the (Lorentzian) Laplacian operator on M. On the other hand, under the assumption that the sectional curvatures of the timelike planes of M are bounded from below by a constant c, we get the following result. where ∇ 2 stands for the Hessian operator on M. The proofs of Lemma 2, Lemma 3 and Lemma 4 follow from the fact that where γ is the radial future directed unit timelike geodesic from p to q and J is the Jacobi field along γ with J(0) = 0 and J(s) = x, and it is strongly based on the maximality of the index of Jacobi fields. For the details, see [1, Section 3]. SPACELIKE HYPERSURFACES CONTAINED IN I + (p) Consider ψ : Σ n → M n+1 a spacelike hypersurface immersed into a spacetime M. Since M is time-oriented, there exists a unique future-directed timelike unit normal field N globally defined on Σ. Let A stand for the shape operator of Σ with respect to N. We will assume that there exists a point p ∈ M such that I + (p) = / 0 and that for every tangent vector field X ∈ T Σ, where ∇ 2 r and ∇ 2 u stand for the Hessian of r and u in M and Σ, respectively. Assume now that K M (Π) ≤ c (resp. K M (Π) ≥ c) for all timelike planes in M, and that u < π/ √ −c on Σ when c < 0. Then by the Hessian comparison results for r given in Lemma 2 (resp. Lemma 4), one gets tha

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature.Comment: First version (December 2008). Final version, including new title (February 2009). To appear in Mathematische Annale

The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian 3-dimensional Heisenberg spaces ${\cal H}={\cal H}(\tau)$. Each such ${\cal H}$ is the total space of a Riemannian submersion onto the Euclidean plane $\mathbb{R}^2$ with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in ${\cal H}$ with respect to the Riemannian submersion over certain domains $\Omega\subset\mathbb{R}^2$ taking on prescribed boundary values

Anales de Edafología y Agrobiología Tomo 36 Número 1-2

Duración efectiva del índice de sequedad, por M. P. Garmendía y J. Garmendía.-- Adsorción y evolución de manganeso en arcillas, por O. Carpena, I. Tovar, A. Lax y F. Costa.-- Ecología de leguminosas en relación con algunos factores ambientales en Guadalajara. I. Aspectos florísticos y relación con la clase ele suelo, por M. Morey.-- Variaciones del contenido de nitrógeno en una plantación de Lolium perenne,por Esther Simón Martínez.-- Predicción de temperaturas máximas diarias, por E. Hernández, J. A. Hernández, J. F. Sánchez y J. Garmendía.-- Efectos del almacenaje sobre las propiedades físicas y biológicas de muestras tamizadas de suelos orgánicos, por F. Díaz-Fierros Viqueiro.-- Morfometría del cuarzo y circón aplicada al estudio genético de un suelo policíclico, por M. C. Villar Celorio.-- Contribución al estudio de la terra rossa española. II. Mineralogía de la fracción arcilla, por L. J. Alías, M. Nieto y J. Albaladejo.-- Entisoles del Campo de Cartagena (Murcia). Características generales y mineralógicas, por L. J. Alías y R. Ortiz Silla.-- Estudio sobre la composición química de variedades de almendra del sureste español, por F. Romojaro, J. F. García y F. J. López Andreu.-- Componentes del plátano canario y sus variaciones durante la maduración, por A. Carlos Blesa, M. A. Rodríguez Raymond y A. Maestre.-- Contribución al estudio de la platanera canaria. Relación entre la actividad respiratoria y la maduración de los plátanos, por A. Carlos Blesa, M. A Rodríguez Raymoud, C. D. Lorenzo e Isabel López.-- Notas.-- Reestructuración del C. S. I. C.-- Nombramiento del Prof. Casas Peláez como Presidente del C. S. I. C. 1.-- Carta del Presidente del C. S. I. C. al personal del mismo.-- Nombramiento del Prof. Snárez y Suárez como Director general de Educación Bastea.-- 6.° Curso Internacional de Fertilidad de Suelos y Nutrición Vegetal.—8ª Reunión Internacional de Micromorfología de Suelos.—19ª Conferencia General de la UNESCO.-- Sociedad Española de Ciencia del Suelo.-- Nombramiento de Secretario del Centro de Edafología y Biología Aplicada de Salamanca.-- Dimisión del Director del Centre de Edafología y Biología Aplicada del Cuarto (Sevilla).-- Propuesta de Director del Centro de Edafología y Biología Aplicada del Cuarto.-- Viaje del Prof. Troncoso.-- Grupo Español de Trabajo del Cuaternario.-- Autorizaciones para realizar función docente.-- Invitaciones a Profesores extranjeros.-- Programa de cooperación internacional con lberoamérica: bolsas de estudio y viaje .-- Conferencia del Prof. Salerno.-- Viaje realizado a Hispanoamérica por el Dr. D. Francisco Girela Vilchez.-- Creación del Centro de Formaci.ón y Promoción de Personal del C. S. I. C. (C. F. P. P.)Peer reviewed2019-08.- CopyBook.- Libnova.- Biblioteca ICA