Rigidity Theorems of Minimal Surfaces Foliated by Similar Planar Curves
| Autor: | Daehwan Kim, Juncheol Pyo |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Helicoid Mean curvature Minimal surface Applied Mathematics 010102 general mathematics 05 social sciences Geometry 01 natural sciences Homothetic transformation Chen–Gackstatter surface Mathematics (miscellaneous) Catenoid Scherk surface 0502 economics and business Constant-mean-curvature surface Mathematics::Differential Geometry 0101 mathematics 050203 business & management Mathematics |
| Zdroj: | Results in Mathematics. 72:1697-1716 |
| ISSN: | 1420-9012 1422-6383 |
| DOI: | 10.1007/s00025-017-0754-9 |
| Popis: | Catenoids, Riemann’s minimal surfaces, and Scherk’s surfaces (doubly periodic minimal surfaces) are classical minimal surfaces in \(\mathbb {R}^3\). The catenoid and Riemann’s minimal surface can be foliated by circles with different radii. Because the Scherk’s surface is represented by the graph of \(z(x,y)=\log \cos x-\log \cos y\), it can be foliated by curves congruent to the graph of \(z=\log {\cos x}\). In this study, we consider surfaces foliated by similar planar curves. When the surface is minimal \((H=0)\) and foliated by homothetic curves without translations, the surface is either a plane or a catenoid. In addition, a minimal surface foliated by parallel ellipses including circles is either a catenoid or a Riemann’s minimal surface. When the surface foliated by ellipses without translations has constant mean curvature, the surface is either a sphere or one of Delaunay surfaces. Finally, we prove that a nonplanar minimal surface foliated by congruent planar curves with only translations on each plane is a generalized Scherk’s surface. |
| Databáze: | OpenAIRE |
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