Cerrado is the second largest biome in South America and accounted for the second largest contribution to carbon emissions in Brazil for the last 10 years, mainly due to land-use changes. It comprises approximately 2 million km2 and is divided into 22 ecoregions, based on environmental conditions and vegetation. The most dominant vegetation type is cerrado sensu stricto (cerrado ss), a savanna woodland. Quantifying variation of biomass density of this vegetation is crucial for climate change mitigation policies. Integrating remote sensing data with adequate allometric equations and field-based data sets can provide large-scale estimates of biomass. We developed individual-tree aboveground biomass (AGB) allometric models to compare different regression techniques and explanatory variables. We applied the model with the strongest fit to a comprehensive ground-based data set (77 sites, 893 plots, and 95,484 trees) to describe AGB density variation of cerrado ss. We also investigated the influence of physiographic and climatological variables on AGB density; this analysis was restricted to 68 sites because eight sites could not be classified into a specific ecoregion, and one site had no soil texture data. In addition, we developed two models to estimate plot AGB density based on plot basal area. Our data show that for individual-tree AGB models a) log-log linear models provided better estimates than nonlinear power models; b) including species as a random effect improved model fit; c) diameter at 30 cm above ground was a reliable predictor for individual-tree AGB, and although height significantly improved model fit, species wood density did not. Mean tree AGB density in cerrado ss was 22.9 tons ha-1 (95% confidence interval = ± 2.2) and varied widely between ecoregions (8.8 to 42.2 tons ha-1), within ecoregions (e.g. 4.8 to 39.5 tons ha-1), and even within sites (24.3 to 69.9 tons ha-1). Biomass density tended to be higher in sites close to the Amazon. Ecoregion explained 42% of biomass variation between the 68 sites (P < 0.01) and shows strong potential as a parameter for classifying regional biomass variation in the Cerrado.

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In this note we present a method for constructing constant mean curvature on surfaces in hyperbolic 3-space in terms of holomorphic data first introduced in Bianchi's Lezioni di Geometria Differenziale of 1927, therefore predating by many years the modern approaches due to Bryant, Small and others. Besides its obvious historical interest, this note aims to complement Bianchi's analysis by deriving explicit formulae for CMC-1 surfaces and comparing the various approaches encountered in the literature.

We prove that Darboux transformations commute with the Lawson correspondence and we show that the property of completeness is preserved by this commutativity. We provide examples of these results. Two applications provide families of explicitly parametrized complete surfaces of constant mean curvature 1 and − √ 5/2 in H 3 , depending on 2 parameters and 1 parameter respectively. For special choices of the parameters, we get surfaces that are periodic in one variable and in particular complete cmc surfaces or cmc1 surfaces in H 3 , with any finite or infinite number of bubbles, "segments" or embedded ends of horosphere type. Moreover, we consider Ribaucour transformations for associated linear Weingarten surfaces in space forms. We show that such a transformation is a Darboux transformation (i.e., it is conformal) if and only if the surfaces have the same constant mean curvature. We prove that Ribaucour transformations for surfaces with constant mean curvature 1 (cmc1) immersed in the hyperbolic space H 3 produce embedded ends of horosphere type.

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A geometric construction is provided that associates to a given flat front in H 3 a pair of minimal surfaces in R 3 which are related by a Ribaucour transformation. This construction is generalized associating to a given frontal in H 3 , a pair of frontals in R 3 that are envelopes of a smooth congruence of spheres. The theory of Ribaucour transformations for minimal surfaces is reformulated in terms of a complex Riccati ordinary differential equation for a holomorphic function. This enables one to simplify and extend the classical theory, that in principle only works for umbilic free and simply connected surfaces, to surfaces with umbilic points and non trivial topology. Explicit examples are included.2000 Mathematics Subject Classification: 53A35, 53C42