Lawson correspondence and Ribaucour transformations

Abstract:

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 $-\sqrt {5}/2$ in $\mathbb {H} ^3$, 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 $\mathbb {H}^3$, 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 $\mathbb {H}^3$ produce embedded ends of horosphere type.