Minimal surfaces in euclidean 3-space and their mean curvature 1 cousins in hyperbolic 3-space

Abstract

We show that the Hopf differentials of a pair of isometric cousin surfaces, a minimal surface in euclidean 3-space and a constant mean curvature (CMC) one surface in the 3-dimensional hyperbolic space, with properly embedded annular ends, extend holomorphically to each end. Using this result, we derive conditions for when the pair must be a plane and a horosphere.

Some partial differential equations and conformal surfaces of the 4-dimensional Minkowski space

“…Then h(w) has explicitly the following components h(w) = 1 2|w| The following example is a surface which belongs to the family of the Catenoid Cousins. We reference the papers [3] and [5], where are studied the cousins surfaces and in particular the Catenoid cousins. Similarly we invite the reader to see the second example of [2] on page 341 for details.…”

Lanthanite-(Nd), Nd2(CO3)3·8H2O

“…For background to the lanthanite mineral group, see: Berzelius (1825); Blake (1853); Coutinho (1955); Shinn & Eick (1968); Ansell et al (1976);Dal Negro et al (1977); Cesbron et al (1979); Roberts et al (1980); Fujimori (1981); Svisero & Mascarenhas (1981); Nagashima et al (1986); Atencio et al (1989); Coimbra et al (1989); Graham et al (2007). For information on dawsonite, see: Corazza et al (1977).…”

“…It was not until Atencio et al (1989) analyzed a Bethlehem sample that it was found to be Nd-rich. Lanthanite-(Nd) has since been reported from other localities, including Curitiba, Brazil (Coutinho, 1955;Ansell et al, 1976;Cesbron et al, 1979;Roberts et al, 1980;Fujimori, 1981;Svisero & Mascarenhas, 1981); Saga Prefecture, Japan (Nagashima et al, 1986); Santa Isabel, São Paulo, Brazil (Coimbra et al, 1989); and Whitianga, Coromandel Peninsula, New Zealand (Graham et al, 2007). With the exception of those found in Whitianga, all lanthanite-(Nd) samples from these localities, including the one used in this study, exhibit a predominance of Nd with sub-equal La and a notable depletion of Ce.…”