Umbilical points on three dimensional strictly pseudoconvex CR manifolds. I: Manifolds with \(\mathrm{U}(1)\)-action. (English) Zbl 1375.32056

The paper is concerned about the existence of umbilical points on compact, \(3\)-dimensional, strictly pseudoconvex CR manifolds.
S. S. Chern and J. K. Moser [Acta Math. 133, 219–271 (1975; Zbl 0302.32015)] proved that any germ \((M,p)\) of a Levi-nondegenerate hypersurface can be brought into a convergent normal form, called the Chern-Moser normal form, by applying biholomorphisms. This normal form solves the biholomorphic equivalence problem for this class of hypersurfaces up to the action of the automorphisms of the hyperquadric model of \((M,p)\). More precisely, there are holomorphic coordinates \((z,w)\) in \(\mathbb C^2\) with \(w = u + i v\) such that \((M,p)\) can be expressed as a graph as follows: \[ v = z \bar z + c_{4,2}(u) z^4 \bar z^2 + c_{2,4}(u) z^2 \bar z^4 + \sum_{k +\ell \geq 7} c_{k,\ell}(u)z^k \bar z^\ell, \] where \(\min(k,\ell) \geq 2\) and \(c_{k,\ell} = \bar c_{\ell,k}\).
The coefficient \(c_{4,2}(0)\) has the property that \(c_{4,2}(0) \neq 0\) is a biholomorphic invariant and is referred to as Cartan’s \(6\)-th order tensor or umbilical tensor of \(M\). The point \(p \in M\) is called umbilical if \(c_{2,4}(0) = 0\).
The works of S. M. Webster [Duke Math. J. 104, No. 3, 463–475 (2000; Zbl 0971.32019)] and X. Huang and S. Ji [Trans. Am. Math. Soc. 359, No. 3, 1191–1204 (2007; Zbl 1122.32027)] provide a study of umbilical points of real ellipsoids. See also the survey by P. Ebenfelt [Int. J. Math. 28, No. 9, Article ID 1740001 (2017; Zbl 1378.32021)] about umbilical points of \(3\)-dimensional CR manifolds. In the paper of Chern-Moser [loc. cit.] it was asked whether there exist compact manifolds without umbilical points which are diffeomorphic to the sphere. The paper under review provides a negative answer to this question assuming that \(M\) admits additional symmetries:
Theorem 1. Let \(M\) be a smooth, compact hypersurface in \(\mathbb C^2\) that bounds a complete circular domain. Then the set of umbilical points on \(M\) contains at least one circle.
A domain \(D \subset \mathbb C^n\) is complete circular if \(z \in D\) implies that \(\{u z : u \in \overline{\mathbb D}\}\) is contained in \(D\), where \(\overline{\mathbb D} \subset \mathbb C\) is the closed unit disk. The theorem follows from a more general result:
Theorem 2. For \(M\) a compact, strictly pseudoconvex, three-dimensional CR manifold, assume that there is a free action of \(\mathrm{U}(1)\) on \(M\) by CR automorphisms such that the action is everywhere transverse to the CR tangent spaces of \(M\). If the compact surface \(X := M/\mathrm{U}(1)\) is not a torus, then the set of umbilical points contains at least one \(\mathrm{U} (1)\)-orbit.
In this setting one can identify \(M\) as the unit circle bundle in a holomorphic line bundle \(L\) over a Riemann surface \(X = M/\mathrm{U}(1)\). If one denotes \(\pi: M \rightarrow X\), then for an umbilical point \(p \in M\) the circle given by the \(\mathrm{U}(1)\)-orbit \(\pi^{-1}(z_0)\) is umbilical for \(z_0 = \pi(p) \in X\). The theorem is proved by showing an index formula which relates the index of an isolated umbilical circle \(\pi^{-1}(z_0)\) for \(z_0 \in X\) (defined via Cartan’s \(6\)-th order tensor) with the Euler characteristic of \(X\), provided the set of umbilical points of \(M\) consists of isolated circles. Another sufficient condition for the existence of umbilical points is given in terms of \(\mathrm{Aut}(M)\), the group of CR automorphisms of \(M\).
Theorem 3. Under the assumptions of Theorem 2, if \(\dim_{\mathbb R} \mathrm{Aut}(M) \geq 2\), then the set of umbilical points contains at least one \(\mathrm{U} (1)\)-orbit.
To prove this theorem it is required to study the case when \(X\) is a torus. The additional vector field allows to show the existence of umbilical points, using the description of Cartan’s \(6\)-th order tensor in terms of a second covariant derivative of the Gauss curvature on \(X\). Furthermore in this paper the authors study umbilical points of in some sense curved Hessians of functions on Riemann surfaces and provide a collection of open problems in these directions.


32V05 CR structures, CR operators, and generalizations
Full Text: DOI arXiv


[1] Bland, J; Duchamp, T, Moduli for pointed convex domains, Invent. Math., 104, 61-112, (1991) · Zbl 0731.32010
[2] Calabi, E.: Extremal Kähler metrics. In: Seminar on differential geometry, vol. 102 of Ann. of Math. Stud., pp. 259-290. Princeton Univ. Press, Princeton (1982) · Zbl 0379.53016
[3] Cartan, É, Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes, Ann. Mat. Pura Appl., 11, 17-90, (1933) · Zbl 0005.37304
[4] Chern, SS; Moser, JK, Real hypersurfaces in complex manifolds, Acta Math., 133, 219-271, (1974) · Zbl 0302.32015
[5] Epstein, CL, CR-structures on three-dimensional circle bundles, Invent. Math., 109, 351-403, (1992) · Zbl 0786.32013
[6] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994) (reprint of the 1978 original) · Zbl 0836.14001
[7] Hamburger, H, Beweis einer carathéodoryschen vermutung, Teil I. Ann. Math., 2, 63-86, (1940) · Zbl 0023.06902
[8] Helgason, S.: Differential Geometry and Symmetric Spaces. Pure and Applied Mathematics, vol. XII. Academic Press, New York-London (1962) · Zbl 0111.18101
[9] Huang, X; Ji, S, Every real ellipsoid in \({\mathbb{C}}^2\) admits CR umbilical points, Trans. Am. Math. Soc., 359, 1191-1204, (2007) · Zbl 1122.32027
[10] Isaev, AV, On a family of real hypersurfaces in a complex quadric, Differ. Geom. Appl., 33, 259-266, (2014) · Zbl 1286.32004
[11] Isaev, A, Analogues of rossi’s map and E. cartan’s classification of homogeneous strongly pseudoconvex 3-dimensional hypersurfaces, J. Lie Theory, 16, 407-426, (2006) · Zbl 1127.53043
[12] Ivanov, VV, An analytic conjecture of Carathéodory, Sibirsk. Mat. Zh., 43, 314-405, (2002) · Zbl 1056.53003
[13] Jacobowitz, H.: An Introduction to CR Structures. Mathematical Surveys and Monographs, vol. 32. American Mathematical Society, Providence (1990) · Zbl 0712.32001
[14] Jacobowitz, H.: Private Communication (2015) · Zbl 0781.32014
[15] Lamel, B; Mir, N; Zaitsev, D, Lie group structures on automorphism groups of real-analytic CR manifolds, Am. J. Math., 130, 1709-1726, (2008) · Zbl 1165.32017
[16] Lee, JM, CR manifolds with noncompact connected automorphism groups, J. Geom. Anal., 6, 79-90, (1996) · Zbl 0859.32003
[17] Lempert, L, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France, 109, 427-474, (1981) · Zbl 0492.32025
[18] Lempert, L, On three-dimensional Cauchy-Riemann manifolds, J. Am. Math. Soc., 5, 923-969, (1992) · Zbl 0781.32014
[19] Loboda, AV, On the sphericity of rigid hypersurfaces in \({ C}^2\), Mat. Zametki, 62, 391-403, (1997)
[20] Rossi, H.: Attaching analytic spaces to an analytic space along a pseudoconcave boundary. In: Proceedings of the Conference on Complex Analysis in Minneapolis, pp. 242-256. Springer, Berlin (1965) · Zbl 0143.30301
[21] Smyth, B; Xavier, F, Real solvability of the equation \(∂ ^2_{\overline{z}}ω =ρ g\) and the topology of isolated umbilics, J. Geom. Anal., 8, 655-671, (1998) · Zbl 0983.53004
[22] Webster, SM, Pseudo-Hermitian structures on a real hypersurface, J. Differ. Geom., 13, 25-41, (1978) · Zbl 0379.53016
[23] Webster, SM, Holomorphic differential invariants for an ellipsoidal real hypersurface, Duke Math. J., 104, 463-475, (2000) · Zbl 0971.32019
[24] Webster, SM, A remark on the Chern-Moser tensor, Houston J. Math., 28, 433-435, (2002) · Zbl 1024.32017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.