Umbilical points on three dimensional strictly pseudoconvex CR manifolds I: manifolds with U(1)-action

The question of existence of umbilical points, in the CR sense, on compact, three dimensional, strictly pseudoconvex CR manifolds was raised in the seminal paper by Chern and Moser (Acta Math 133:219–271, 1974). In the present paper, we consider compact, three dimensional, strictly pseudoconvex CR manifolds that possess a free, transverse action by the circle group U(1). We show that every such CR manifold M has at least one orbit of umbilical points, provided that the Riemann surface X:=M/U(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X:=M/U(1)$$\end{document} is not a torus. In particular, every compact, circular and strictly pseudoconvex hypersurface in C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb C^2$$\end{document} has at least one circle of umbilical points. The existence of umbilical points in the case where X is a torus is left open in general, but it is shown that if such an M has additional symmetries, in a certain sense, then it must possess umbilical points as well.