Bulletin of the Brazilian Mathematical Society, New Series

Gavrilov, Lubomir

doi: 10.1007/s00574-011-0001-zpmid: N/A

We find an upper bound to the maximal number of limit cycles, which bifurcate from a hamiltonian two-saddle loop of an analytic vector field, under an analytic deformation.

Mora, Leonardo

doi: 10.1007/s00574-011-0002-ypmid: N/A

We present a real multidimensional version of the Schwarz Lemma on a bounded convex domain D of ℝ n endowed with the Hilbert metric. We provide as an application an extension of a Birkhoff’s Theorem on mappings contracting the Hilbert metric.

Marmi, Stefano; Sauzin, David

doi: 10.1007/s00574-011-0003-xpmid: N/A

We discuss the quasianalytic properties of various spaces of functions suit-able for one-dimensional small divisor problems. These spaces are formed of functions [InlineMediaObject not available: see fulltext.]1-holomorphic on certain compact sets K j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K j ’s.

Fernández-Pérez, Arturo

doi: 10.1007/s00574-011-0004-9pmid: N/A

Let F(z) = Re(P(z)) + h.o.t be such that M = (F = 0) defines a germ of real analytic Levi-flat at 0 ∈ ℂ n , n ≥ 2, where P (z) is a homogeneous polynomial of degree k with an isolated singularity at 0 ∈ ℂ n and Milnor number µ. We prove that there exists a holomorphic change of coordinate ϕ such that ϕ(M) = (Re(h) = 0), where h(z) is a polynomial of degree µ + 1 and j 0 k (h) = P.

Mendes de Jesus, C.; Moraes, S.; Romero-Fuster, M.

doi: 10.1007/s00574-011-0005-8pmid: N/A

We associate weighted graphs to stable Gauss maps on orientable closed surfaces immersed in 3-space and prove that any bipartite weighted graph can be associated to some stable Gauss map.

Gervasio Colares, A.; Palmas, Oscar

doi: 10.1007/s00574-011-0006-7pmid: N/A

In this paper we define closed partially conformal vector fields and use them to give a characterization of Riemannian manifolds which admit this kind of fields as some special warped products foliated by (n − 1)-umbilical hypersurfaces. Examples are described in space forms. In particular, closed partially conformal vector fields in Euclidean spaces are associated to the most simple foliations given by hyperspheres, hyperplanes or coaxial cylinders. Finally, for manifolds admitting such vector fields, we impose conditions for a hypersurface to be (n − 1)-umbilical, or, in particular, a leaf of the corresponding foliation.

Liu, Huili; Umehara, Masaaki; Yamada, Kotaro

doi: 10.1007/s00574-011-0007-6pmid: N/A

In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem for wave fronts in space forms. As an application, we show that the following four objects are essentially the same: conformally flat n-manifolds (n ≥ 3) with admissible singular points (i.e. admissible GCF-manifolds) frontals as hypersurfaces in the lightcone Q + n+1 frontals as hypersurfaces in the hyperbolic space H n+1 spacelike frontals as hypersurfaces in the de Sitter space S 1/n+1.

Henk, Martin; Hernández Cifre, María

doi: 10.1007/s00574-011-0008-5pmid: N/A

We investigate the roots of relative Steiner polynomials of convex bodies. In dimension 3 we give a precise description of their location in the complex plane and we study the analogous problem in higher dimensions. In particular, we show that the roots (in the upper half plane) form a convex cone; for dimensions ≤ 9 this cone is completely contained in the (open) left half plane, which is not true in dimensions ≥ 12. Moreover, we characterize certain special families of convex bodies by means of properties of their roots.