The Gauss-Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

LetQ(4)( c) be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss-Kronecker curvature of a complete minimal hypersurface in Q(4)(c), c <= 0, whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M(3) of a space form Q(4)( c) with constant Gauss-Kronecker curvature K. For the case c <= 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of Q(4)( c) with K constant.

Lipid, fatty acid, protein, amino acid and ash contents in four Brazilian red algae species

Four species of marine benthic algae (Laurencia filiformis, L. intricata, Gracilaria domingensis and G. birdiae) that belong to the phylum Rhodophyta were collected in Espirito Santo State, Brazil and investigated concerning their biochemical composition (fatty acid, total lipid, soluble proteins, amino acid and ash). The total content of lipid (% dry weight) ranged from 1.1% to 6.2%: fatty acid from 0.7% to 1.0%: soluble protein from 4.6% to 18.3%, amino acid from 6.7% to 11.3% and ash from 22.5% to 38.4%. judging from their composition, the four species of algae appear to be potential sources of dietary proteins, amino acids, lipids and essential fatty acids for humans and animals. (C) 2009 Elsevier Ltd. All rights reserved.