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.

MSC:

 32V05 CR structures, CR operators, and generalizations
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References:

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