Minimal Surfaces Obtained by Ribaucour Transformations

Abstract

We consider Ribaucour transformations between minimal surfaces and we relate such transformations to generating planar embedded ends. Applying Ribaucour transformations to Enneper's surface and to the catenoid, we obtain new families of complete, minimal surfaces, of genus zero, immersed in R3, with infinitely many embedded planar ends or with any finite number of such ends. Moreover, each surface has one or two nonplanar ends. A particular family is obtained from the catenoid, for each pair (n,m), n≠m, such that nm0 is an irreducible rational number. For any such pair, we get a 1-parameter family of finite total curvature, complete minimal surfaces with n+2 ends, n embedded planar ends and two nonplanar ends of geometric index m, whose total curvature is −4π(n+m). The analytic interpretation of a Ribaucour transformation as a Backlund type transformation and a superposition formula for the nonlinear differential equation ΔΦ= e-2Φ is included.