Armando V. Corro

A CHARACTERIZATION OF THE CATENOID AND HELICOID

In this paper we show that a connected non-planar minimal surface whose asymptotic lines have the same geodesic curvature up to sign is a catenoid. As an application of this result we show that a connected non-planar minimal surface whose lines of curvature have the same geodesic curvature up to sign is a helicoid. Moreover, we show that the coordinates curves of the associate minimal surfaces to catenoid have the same geodesic curvature up to sign.

Invariantes de Laplace en hipersuperficies parametrizadas por líneas de curvatura

In this work, using the Laplace invariants theory we give other proof for the following result: A proper Dupin hypersurfaces Mn for n ≥ 4 in Rn+1 with n distinct principal curvatures and constant mobius curvature, cannot be parametrized by lines of curvature. Also, we study special classes of hypersurfaces Mn; n ≥ 3; in Rn+1, parametrized by lines of curvature with n distinct principal curvatures and we obtain a geometric relation when the Laplace invariants are vanish, we show that the foliations of Mn are umbilical hypersurfaces if and only if mijk = 0. Moreover, the foliations of Mn are Dupin hypersurfaces if and only if mij = 0.

Geodesics in minimal surfaces

Abstract—In this paper, we consider connected minimal surfaces in R3 with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper’s surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one 1-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature kg1 and kg2 of the coordinates curves satisfy αkg1 + βkg2 = 0, α, β ∈ R.

CLASSES OF HYPERSURFACES WITH VANISHING LAPLACE INVARIANTS

Consider a hypersurface M n in R n+1 with n distinct princi- pal curvatures, parametrized by lines of curvature with vanishing Laplace invariants. (1) If the lines of curvature are planar, then there are no such hyper- surfaces for n � 4, and for n = 3, they are, up to Mobius transformations, Dupin hypersurfaces with constant Mobius curvature. (2) If the principal curvatures are given by a sum of functions of sepa- rated variables, there are no such hypersurfaces for n � 4, and for n = 3, they are, up to Mobius transformations, Dupin hypersurfaces with con- stant Mobius curvature.