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

TL;DR

This paper proves that compact, three-dimensional, strictly pseudoconvex CR manifolds with a free circle action generally have umbilical points unless their quotient surface is a torus, where existence remains an open question.

Contribution

It establishes the existence of umbilical points on certain CR manifolds with circle symmetry, extending prior understanding and identifying cases where existence is uncertain.

Findings

Every such CR manifold with non-torus quotient has at least one orbit of umbilical points.

Compact, circular hypersurfaces in a0a0^2 have at least one circle of umbilical points.

Existence of umbilical points on tori quotients remains an open problem, but additional symmetries imply their existence.

Abstract

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.