Biharmonic Maps and Laguerre Minimal Surfaces

Abstract

A Laguerre surface is known to be minimal if and only if its corresponding isotropic map is biharmonic. For every Laguerre surface $\mathrm{\Phi }$ is its associated surface $\mathrm{\Psi }=\left(1+{\left|u\right|}^2\right)\mathrm{\Phi }$, where $\mathrm{u}$ lies in the unit disk. In this paper, the projection of the surface $\mathrm{\Psi }$ associated to a Laguerre minimal surface is shown to be biharmonic. A complete characterization of $\mathrm{\Psi }$ is obtained under the assumption that the corresponding isotropic map of the Laguerre minimal surface is harmonic. A sufficient and necessary condition is also derived for $\mathrm{\Psi }$ to be a graph. Estimates of the Gaussian curvature to the Laguerre minimal surface are obtained, and several illustrative examples are given.