Abstract
It is known that minimal surfaces in Euclidean space can be represented in terms of holomorphic functions. For example, we have the well-known Weierstrass representation, where part of the holomorphic data is chosen to be the stereographic projection of the normal of the corresponding surface, and also the Björling representation, where it is prescribed a curve on the surface and the unit normal on this curve. In this work, we are interested in the holomorphic representation of minimal surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the isotropic metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only, which leads to distinct definitions of an isotropic normal. As a consequence, this may also lead to distinct forms of a Weierstrass and of a Björling representation. Here, we show how to represent simply isotropic minimal surfaces in accordance with the choice of an isotropic surface normal.
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Notes
Any conformal minimal immersion has harmonic coordinates and, consequently, each of the three coordinates can be locally seen as the real part of a holomorphic function on the plane.
By the center of the sphere \(\varSigma ^2\) we mean its focus.
The correspondence with minimal surfaces in \(\mathbb {I}^3\) is a special feature of \(\mathbb {E}_1^4\) since a zero mean curvature surface in 4d Euclidean space with zero Gaussian curvature must be a plane.
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Acknowledgements
We would like to thank useful discussions with Alev Kelleci Akbay (Firat University) and Yuichiro Sato (Tokyo Metropolitan University). This work has been financially supported by the Morá Miriam Rozen Gerber scholarship for Brazilian postdocs.
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da Silva, L.C.B. Holomorphic representation of minimal surfaces in simply isotropic space. J. Geom. 112, 35 (2021). https://doi.org/10.1007/s00022-021-00598-z
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DOI: https://doi.org/10.1007/s00022-021-00598-z