Abstract
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.
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Kim, D., Pyo, J. Rigidity Theorems of Minimal Surfaces Foliated by Similar Planar Curves. Results Math 72, 1697–1716 (2017). https://doi.org/10.1007/s00025-017-0754-9
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DOI: https://doi.org/10.1007/s00025-017-0754-9