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 S.-S. Chern and J. K. Moser in 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, {\it provided} that the Riemann surface $X:=M/U(1)$ is not a torus. In particular, every compact, circular and strictly pseudoconvex hypersurface in $\mathbb C^2$ 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.