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

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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 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 \(X:=M/U(1)\) 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.

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Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093-0112, USA

    Peter Ebenfelt

  2. Texas A&M University at Qatar, Science Program, PO Box 23874, Education City, Doha, Qatar

    Duong Ngoc Son

Authors
  1. Peter Ebenfelt
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  2. Duong Ngoc Son
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Correspondence to Peter Ebenfelt.

Additional information

Communicated by Ngaiming Mok.

Peter Ebenfelt was supported in part by the NSF Grant DMS-1301282. Duong Ngoc Son was partially supported by the Qatar National Research Fund, NPRP project 7-511-1-098. Part of the paper was done while the second author was at University of California, Irvine.

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Ebenfelt, P., Son, D.N. Umbilical points on three dimensional strictly pseudoconvex CR manifolds I: manifolds with U(1)-action. Math. Ann. 368, 537–560 (2017). https://doi.org/10.1007/s00208-016-1439-5

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  • DOI: https://doi.org/10.1007/s00208-016-1439-5

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