Weierstrass' theorem in weighted Sobolev spaces with derivatives: announcement of results.
Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials: a survey.
Isoperimetric inequalities and Dirichlet functions of Riemann surfaces.
We prove that if a Riemann surface has a linear isoperimetric inequality and verifies an extra condition of regularity, then there exists a non-constant harmonic function with finite Dirichlet integral in the surface. We prove too, by an example, that the implication is not true without the condition of regularity.
Two remarks on Riemann surfaces.
We study the relationship between linear isoperimetric inequalities and the existence of non-constant positive harmonic functions on Riemann surfaces. We also study the relationship between growth conditions of length of spheres and the existence and the existence of Green's function on Riemann surfaces.
Quasi-equivalence of compacta and spaces of components.
Let X, Y be two compacta with Sh(X) = Sh (Y). Then, the spaces of components of X, Y are homeomorphic. This does not happen, in general, when X, Y are quasi-equivalent. In this paper we give a sufficient condition for the existence of a homeomorphism between the spaces of components of two quasi-equivalent compacta X, Y which maps each component in a quasi-equivalent component.
A new characterization of Gromov hyperbolicity for negatively curved surfaces.
In this paper we show that to check Gromov hyperbolicity of any surface of constant negative curvature, or Riemann surface, we only need to verify the Rips condition on a very small class of triangles, namely, those obtained by marking three points in a simple closed geodesic. This result is, in fact, a new characterization of Gromov hyperbolicity for Riemann surfaces.
The hyperbolicity constant of infinite circulant graphs
If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant...
Isoperimetric inequalities in Riemann surfaces of infinite type.
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