### Artigos

Hilário Alencar, Mafredo do Carmo, Renato Tribuzy

**Resumo:** We consider surfaces M² immersed in E^{n}_{c} × R, where E^{n}_{c} is a simply connected n-dimensional complete Riemannian manifold with constant sectional curvature c = 0, and assume that the mean curvature vector of the immersion is parallel in the normal bundle. We consider further a Hopf-type complex quadratic form Q on M² , where the complex structure of M 2 is compatible with the induced metric. It is not hard to check that Q is holomorphic. We will use this fact to give a reasonable description of immersed surfaces in E^{n}_{c} × R that have parallel mean curvature vector.

H. Alencar, M. do Carmo, R. Tribuzy

**Resumo:** In 1951m H. Hopf published a theorem in a seminal paper on surfaces of constant mean curvature wich can be stated as follows. "Let a genus zero compact surface M be immersed in R³ with constant mean curvature H". Then M is isometric to the standard sphere. Hopf gave two proofs of this result. Both proof depend on the fact that any surface can given isothernal parameters(u,v)...

H. Alencar, M. do Carmo, I. Fernández R. Tribuzy

**Resumo:** This is a sequel to "A Theorem of H. Hopf and The Cauchy-Riemann inequality". Here the result if the previous paper is extended to surfaces ub three-dimensional homogeneous Riemannian manifolds whose group of isometries has dimension four and the bundle curvature is nonzero, whereas in the previous paper only the case of vanishing bundle curvature was treated

Marcus P. A. Cavalcante

**Resumo:** Given a closed subset Λ of the open unit ball B1 ⊂ R^n , n ≥ 3, we will consider a complete Riemannian metric g on B1 \ Λ of constant scalar curvature equal to n(n − 1) and conformally related to the Euclidean metric. In this paper we prove that every closed Euclidean ball B ⊂ B1 \ Λ is convex with respect to the metric g, assuming the mean curvature of the boundary ∂B1 is nonnegative with respect to the inward normal.

Krerley Oliveira

**Resumo:** We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties.

Alexander Arbieto, Carlos Matheus, Krerley Oliveira

**Resumo:** We show that, for a robust (C 2 -open) class of random non-uniformly expanding maps, there exists equilibrium states for a large class of potentials.In particular, these sytems have measures of maximal entropy. These results also give a partial answer to a question posed by Liu-Zhao. The proof of the main result uses an extension of techniques in recent works by Alves-Ara'jo, Alves-Bonatti-Viana and Oliveira.

Marcus P. A. Cavalcante, Jorge H. S. de Lira

**Resumo:** It is proved that the holomorphic quadratic differential associated to CMC surfaces in Riemannian products S2 × R and H2 × R 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 escribed. We also present some characterizations of CMC rotationally invariant discs and spheres.

Carlos Matheus, Krerley Oliveira

**Resumo:** Given a compact n-dimensional immersed Riemannian manifold M^n we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then M^n is homeomorphic to the sphere S^n. Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with small set of points of zero Gauss-Kronecker curvature are topologically the sphere minus a finite number of points. A characterization of the 2n-catenoid is obtained.

Adán J. Corcho

**Resumo:** We discuss ill-posedness issues for the initial value problem associated to the Benney system. To prove our results we use the method introduced by Kenig, Ponce and Vega to show ill-posedness for some canonical dispersive equations.

Alexander Arbieto, Carlos Matheus, Krerley Olveira

**Resumo:** O objetivo deste artigo é fazer uma rápida investigação dos trabalhos de Ennio De Giorgi acerca do problema de Plateau. Para isso, começamos com uma introdução (não tão rápida assim!) à bela história desta questão.

Hilário Alencar, Abdênago Barros, Oscar Palmas, J. Guadalupe Reyes, Walcy Santos

**Resumo:** We classify the non-extendable immersed O(m) × O(n)-invariant minimal hypersurfaces in the euclidean space R^{m+n} , m, n ≥ 3, analyzing also whether they are embedded or stable.
We show also the existence of embedded, complete, stable minimal hypersurfaces in R^{m+n}, m + n ≥ 8, m, n ≥ 3 not homeomorphic to R^{m+n−1} that are O(m) × O(n)–invariant.

José F. Alves, Krerley Oliveira, Ali Tahzibi

**Resumo:** We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of non-uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the SRB measure. In particular, we show that the SRB entropy of Viana maps varies continuously with the map.

Vinícius Mello, Luiz Velho

**Resumo:** In this paper we will develop a theory for simplicial diffeomorphims, that is, diffeomorphims that preserve the incidence relations of a simplicial complex, and analyze alternative schemes to construct them with different properties. In combining piecewise linear functions on complexes with simplicial diffeomorphisms, we propose a new representation of curves and surfaces (and hypersurfaces, in general) that is simultaneously implicit and parametric.

Hilário Alencar, Walcy Santos, Detang Zhou

**Resumo:** We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R4 with zero scalar curvature S2 , nonzero Gauss-Kronecker curvature and finite total curvature.

Marcus P. A. Cavalcante

**Resumo:** In this paper we give sufficient conditions for a bounded domain in a r-minimal hypersuface of the Euclidean space to be r-stable. The Gauss-Kronecker curvature of this hypersurfaces may be zero on a set of capacity zero.

Krerley Oliveira, Marcelo Viana

**Resumo:** We develop a Ruelle-Perron-Frobenius transfer operator approach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For H ̈lder continuous potentials not too far from constant, we prove that the transfer operator has a strictly positive Hölder continuous eigenfunction, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a non-lacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here.

A. J. Corcho, F. Linares

**Resumo:** We establish local and global results for the initial value problem associated the Schrödinger-Debye system for data in low regularity spaces. The main tool used is an optimal application of the Strichartz estimates for the linear Schrödinger operator. In the one dimensional case we also use Kato’s smoothing effect to obtain global results in fractional Sobolev spaces.