Rigidity Theorems of Minimal Surfaces Foliated by Similar Planar Curves

Catenoids, Riemann's minimal surfaces, and Scherk's surfaces (doubly periodic minimal surfaces) are classical minimal surfaces in . 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 , it can be foliated by curves congruent to the graph of . In this study, we consider surfaces foliated by similar planar curves. When the surface is minimal 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.