Extension of Krust theorem and deformations of minimal surfaces
| Autor: | Akamine, Shintaro, Fujino, Hiroki |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space $\mathbb{E}^3$ is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space $\mathbb{L}^3$. In this article, we introduce a new deformation family that continuously connects minimal surfaces in $\mathbb{E}^3$ and maximal surfaces in $\mathbb{L}^3$, and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformation families containing the associated family and the L\'opez-Ros deformation. Furthermore, minimal surfaces in the isotropic 3-space $\mathbb{I}^3$ appear in the middle of the above deformation family. We also prove another type of Krust's theorem for this family, which implies that the graphness of such minimal surfaces in $\mathbb{I}^3$ strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory. Comment: 20 pages, 10 figures. Subsection 5.4 has been added and some typographical errors have been fixed |
| Databáze: | arXiv |
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