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 R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document}. 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)=logcosx-logcosy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z(x,y)=\log \cos x-\log \cos y$$\end{document}, it can be foliated by curves congruent to the graph of z=logcosx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=\log {\cos x}$$\end{document}. In this study, we consider surfaces foliated by similar planar curves. When the surface is minimal (H=0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(H=0)$$\end{document} 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.