ABSTRACT. We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp(|x| 2 ) by using symmetrization techniques.

Dedicated to Professor Manfredo do Carmo on the occasion of his 80 th birthdayOne of the most celebrated papers by Manfredo is his joint work with Peng [19] on the classification of complete orientable stable minimal surfaces in R 3 . The authors, through a clever use of the second variation formula of the area together with some information on the conformal geometry of the surface, provided a beautiful and straightforward proof of the following

Abstract. We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point.We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.

In this paper we study sets in the n-dimensional Heisenberg group H n which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in H n . We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones.Our main result describes which are the CMC hypersurfaces of revolution in H n . The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence we classify the rotationally invariant isoperimetric sets in H n .

ABSTRACT. We consider the sub-Riemannian metric g h on S 3 provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot-Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus.We study area-stationary surfaces with or without a volume constraint in (S 3 , g h ). By following the ideas and techniques in [RR2] we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volumepreserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot-Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C 2 compact, connected, embedded surfaces in (S 3 , g h ) with empty singular set and constant mean curvature H such that H/ √ 1 + H 2 is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (S 3 , g h ).

For constant mean curvature surfaces of class C 2 immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the unique ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable.

Bovine coronavirus (BCoV) causes enteric and respiratory dis- orders in calves and dysentery in cows. In this study, 51 stool samples of calves from 10 Brazilian dairy farms were analysed by an RT-PCR that amplifies a 488-bp fragment of the hypervariable region of the spike glycoprotein gene. Maximum parsimony genealogy with a heuristic algorithm using sequences from 15 field strains studied here and 10 sequences from GenBank and bredavirus as an outgroup virus showed the existence of two major clusters (1 and 2) in this viral species, the Brazilian strains segregating in both of them. The mean nucleotide identity between the 15 Brazilian strains was 98.34%, with a mean amino acid similarity of 98%. Strains from cluster 2 showed a deletion of 6 amino acids inside domain II of the spike protein that was also found in human coronavirus strain OC43, supporting the recent proposal of a zoonotic spill- over of BCoV. These results contribute to the molecular characterization of BCoV, to the prediction of the efficiency of immunogens, and to the definition of molecular markers useful for epidemiologic surveys on coronavirus-caused diseases.

The origins of the extraordinary diversity within the Neotropics have long fascinated biologists and naturalists. Yet, the underlying factors that have given rise to this diversity remain controversial. To test the relative importance of Quaternary climatic change and Neogene tectonic and paleogeographic reorganizations in the generation of biodiversity, we examine intraspecific variation across the Heliconius cydno radiation and compare this variation to that within the closely related Heliconius melpomene and Heliconius timareta radiations. Our data, which consist of both mtDNA and genome-scan data from nearly 2250 amplified fragment length polymorphism (AFLP) loci, reveal a complex history of differentiation and admixture at different geographic scales. Both mtDNA and AFLP phylogenies suggest that H. timareta and H. cydno are probably geographic extremes of the same radiation that probably diverged from H. melpomene prior to the Pliocene-Pleistocene boundary, consistent with hypotheses of diversification that rely on geological events in the Pliocene. The mtDNA suggests that this radiation originated in Central America or the northwestern region of South America, with a subsequent colonization of the eastern and western slopes of the Andes. Our genome-scan data indicate significant admixture among sympatric H. cydno/H. timareta and H. melpomene populations across the extensive geographic ranges of the two radiations. Within H. cydno, both mtDNA and AFLP data indicate significant population structure at local scales, with strong genetic differences even among adjacent H. cydno colour pattern races. These genetic patterns highlight the importance of past geoclimatic events, intraspecific gene flow, and local population differentiation in the origin and establishment of new adaptive forms.