Biharmonic maps and Laguerre minimal surfaces. (English) Zbl 1275.53013

Summary: A Laguerre surface is known to be minimal if and only if its corresponding isotropic map is biharmonic. The associated surface of \(\Phi\) is \(\Psi = (1 + |u|^2)\Phi\), where \(u\) lies in the unit disk. In this paper, the projection of the surface \(\Psi\) associated to a Laguerre minimal surface is shown to be biharmonic. A complete characterization of \(\Psi\) 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 the case when \(\Psi\) is a graph. Estimates of the Gaussian curvature of a Laguerre minimal surface are obtained, and several illustrative examples are given.


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A05 Surfaces in Euclidean and related spaces
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