CR umbilical points on unit circle bundles over Riemann surfaces: the torus case and Loewner-type index bounds

On a compact, strictly pseudoconvex, three dimensional CR manifold $M$ with a free CR circle action transverse to the CR tangent spaces, $M$ is the unit circle bundle of a positive holomorphic line bundle over the Riemann surface $X=M/U(1)$, and Ebenfelt and Son proved a Poincaré–Hopf type index formula forcing at least one umbilical circle whenever $X$ is not a torus. Three interlocked questions from that work remain open: (a) the degenerate case where $X$ is a torus — does there exist a positively curved line-bundle metric on a torus whose unit circle bundle has no CR-umbilical points, equivalently a doubly periodic function $u$ with $Pu\neq 0$ everywhere for an explicit fourth order operator $P$; (b) whether the CR umbilic quadratic differential, built from the second covariant $z$-derivative of the base Gauss curvature, is locally a flat Hessian near its isolated zeros; and (c) whether the index of an isolated CR-umbilical orbit is always at most $1$, the CR analogue of Loewner's conjecture. Later work of Ebenfelt, Son and Zaitsev constructed compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^2$ without umbilics, settling the ambient Chern–Moser existence question, but those examples are Grauert tubes with a non-transverse circle action and leave the line-bundle questions (a)–(c) untouched.

Background

A CR (Cauchy–Riemann) manifold of hypersurface type is a real manifold $M^{2n+1}$ with a distinguished complex subbundle $T^{1,0}M$ of the complexified tangent bundle; for $n=1$ these are the boundaries of domains in $\mathbb{C}^2$ and their abstract analogues. $M$ is strictly pseudoconvex when its Levi form (the intrinsic Hermitian form on $T^{1,0}M$ coming from $d$ of the contact form) is positive definite. In their 1974 memoir Chern and Moser asked whether every compact, strictly pseudoconvex, three dimensional CR manifold must carry umbilical points — points where the manifold can be osculated by a sphere to higher than expected order, detected in dimension three by the vanishing of a sixth-order curvature tensor discovered by É. Cartan. Huang and Ji proved that every real ellipsoid in $\mathbb{C}^2$ has such points, and for more than eight decades Cartan's homogeneous examples (and their covers) were the only known compact non-umbilical ones. Ebenfelt and Son (2017) studied $M$ admitting a free $U(1)$-action by CR automorphisms, transverse to the CR tangent spaces. By a theorem of Epstein, such an $M$ is the unit circle bundle of a positive holomorphic line bundle $\pi:L\to X$ over the Riemann surface $X=M/U(1)$ with a positively curved Hermitian metric $h$, and the umbilical locus is the union of circles $\pi^{-1}(z_0)$ over the zeros of $r=Pu$, where the function $u$ is determined by $h$ through $e^{u}=-D\bar D\log h$ (with $D=\partial/\partial z$, $\bar D=\partial/\partial\bar z$) and $P$ is the fourth order operator

$$Pu=D^3\bar D u-3(Du)D^2\bar D u+2(Du)^2D\bar D u-(D^2u)D\bar D u=e^{2u}D\bigl(e^{-u}D\bigl(e^{-u}D\bar D u\bigr)\bigr)=-\tfrac{e^{2u}}{2}K_{;zz},$$

$K_{;zz}$ denoting the second covariant $z$-derivative of the Gauss curvature $K$ of the metric $ds^2=e^{u}|dz|^2$ on $X$. They proved a Poincaré–Hopf type index formula $\sum_k \iota(\mathcal O_k)=\chi(X)$ for the indices of isolated umbilical orbits — the index $\iota(\mathcal O)$ of an isolated umbilical orbit being $-\tfrac12$ times the winding number (degree) of $Pu/|Pu|$ along a small positively oriented loop around the corresponding zero of $Pu$ — whence umbilical points exist whenever $X$ is not a torus. The construction parallels classical surface theory, where Carathéodory's conjecture (every sphere embedded in $\mathbb{R}^3$ has at least two umbilics) reduces to Loewner's still-open conjecture that the index of an isolated umbilical point of the flat Hessian of a smooth function is at most $1$; a quadratic differential is called flat Hessian if, in suitable local coordinates, it is $D^2f$ up to a scalar factor. Ebenfelt and Son showed (their Proposition 4.4) that general curved Hessians have complete flexibility — their indices can be any half-integer — so no Loewner-type bound can hold without extra structure. Subsequently Ebenfelt, Son and Zaitsev (2018) settled the ambient Chern–Moser question by exhibiting the compact strictly pseudoconvex boundaries $M_\varepsilon=\partial\{ (\log|z|)^2+(\log|w|)^2<\varepsilon^2\}\subset\mathbb{C}^2$, diffeomorphic to $T^3$, with no umbilical points at all; but these are Grauert tubes — neighborhoods of the flat torus in its complexified tangent bundle, realized as domains in $\mathbb{C}^2$ — whose natural circle action is free yet not transverse to the CR tangent spaces, so they do not arise as unit circle bundles in positively curved line bundles and the circle-bundle questions below survive. Ebenfelt's survey of the area and the later circle-bundle literature (obstruction flatness, ellipsoid umbilical loci, Grauert-tube examples) likewise do not touch them.

Problem Statement

Decide the three questions left open by Ebenfelt–Son (2017) for the umbilical theory of unit circle bundles in positively curved holomorphic line bundles over compact Riemann surfaces, which jointly complete the existence theory behind their index formula $\sum_k\iota(\mathcal O_k)=\chi(X)$: (a) the torus case (their Questions 4.9 and 4.10) — do there exist a holomorphic line bundle $\pi:L\to X$ over a torus $X=\mathbb{C}/\Lambda$ and a positively curved Hermitian metric $h$ on $L$ such that the unit circle bundle $M\subset L$ has no CR-umbilical points; equivalently, do there exist a complex number $\omega$ with $\mathrm{Im}\,\omega\neq 0$ and a smooth function $u(z,\bar z)$ on $\mathbb{C}$, periodic with periods $1$ and $\omega$, such that $Pu\neq 0$ everywhere, where

$$Pu=D^3\bar D u-3(Du)D^2\bar D u+2(Du)^2D\bar D u-(D^2u)D\bar D u=e^{2u}D(e^{-u}D(e^{-u}D\bar D u))$$

with $D=\partial/\partial z$, $\bar D=\partial/\partial\bar z$; (b) the local flat-Hessian question (their Question 4.3) — near an isolated zero $z_0$ of $K_{;zz}$, where $K$ is the Gauss curvature of $ds^2=e^u|dz|^2$, is the CR umbilic quadratic differential $\Xi_K=e^{2\varphi}K_{;zz}\,dz\otimes dz$ (where $e^{2\varphi}|dz|^2$ is the base metric, i.e. $2\varphi=u$) flat Hessian, i.e. do there exist a local coordinate system and smooth real-valued functions $f$, $\rho$ with $D^2f=\rho\,K_{;zz}$ near $z_0$; and (c) the CR-Loewner index bound — is the index $\iota(\mathcal O)$ of every isolated CR-umbilical orbit, defined as $-\tfrac12$ times the degree of $Pu/|Pu|$ along a small positively oriented loop around the corresponding zero of $Pu$, at most $1$? Part (a) is precisely the degenerate $\chi(X)=0$ case of the index formula in which it permits a non-umbilical bundle; a positive answer to (b) combined with Loewner's conjecture would imply (c), which combined with the index formula would sharpen the existence theorem to a minimum of two umbilical orbits whenever $X\cong\mathbb{CP}^1$ (e.g. boundaries of complete circular domains in $\mathbb{C}^2$, i.e. domains containing $tz$ whenever they contain $z$, for every $|t|\le 1$); note that by Ebenfelt–Son's Proposition 4.4 no such bound can hold for general curved Hessians.

References

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