On the analytic and geometric aspects of obstruction flatness

Let $\Omega\subset\mathbb{C}^{n}$ be a smoothly bounded, strongly pseudoconvex domain and consider the Dirichlet problem

If $u$ is a solution of (1.1), then $-\log u$ is the Kähler potential of a complete Kähler–Einstein metric in $\Omega$ with negative Ricci constant. The existence and uniqueness of a solution $u\in C^{\infty}(\Omega)$ to (1.1) was established by S.-Y. Cheng and S.-T. Yau [CY80]. Prior to that, C. Fefferman [Fef76] had constructed approximate solutions $\rho\in C^{\infty}(\overline{\Omega})$ to (1.1) that satisfy $J(\rho)=1+O(\rho^{n+1})$, and shown that such $\rho$ are unique modulo $O(\rho^{n+2})$. Fefferman’s approximate solutions $\rho$ are called Fefferman defining functions and the exact solution $u$ is referred to as the Cheng–Yau solution. Subsequently, J. Lee and R. Melrose [LM82] showed that the Cheng–Yau solution $u$ admits an asymptotic expansion of the form

where each $\eta_{k}$ is in $C^{\infty}(\overline{\Omega})$ and $\rho$ is a Fefferman defining function. It follows from (1.2) that, in general, the Cheng–Yau solution $u$ can only possess a finite degree of boundary smoothness; namely, $u$ is in $C^{n+2-\varepsilon}(\overline{\Omega})$ for any $\varepsilon>0$. C. R. Graham discovered that in the expansion (1.2), the restriction of $\eta_{1}$ to the boundary, $\eta_{1}|_{\partial\Omega}$, turns out to be precisely the obstruction to $C^{\infty}$ boundary regularity of the Cheng–Yau solution. To be more precise, in [Gra87a] Graham proved that if $\eta_{1}|_{\partial\Omega}$ vanishes identically on an open subset $U\subset\partial\Omega$, then $\eta_{k}$ vanishes to infinite order on $U$ for every $k\geq 1$. For this reason, $\eta_{1}|_{\partial\Omega}$ is called the obstruction function.

Graham also showed (op. cit.) that, for any $k\geq 1$, the coefficients $\eta_{k}$ mod $O(\rho^{n+1})$ are independent of the choice of Fefferman defining function $\rho$ and are determined near a point $p\in\partial\Omega$ by the local CR geometry of $\partial\Omega$ near $p$. As a consequence, the $\eta_{k}$ mod $O(\rho^{n+1})$, for $k\geq 1$, are local CR invariants that can be defined on any strongly pseudoconvex CR hypersurface in a complex manifold. In particular, the obstruction function $\mathcal{O}:=\eta_{1}|_{\partial\Omega}$ is a local CR invariant that can be defined on any such strongly pseudoconvex CR hypersurface $M$ (see Graham [Gra87a]). If the obstruction function $\mathcal{O}$ vanishes at $p\in M$, then $p$ is called an obstruction flat point. If $\mathcal{O}$ vanishes identically on $M$, then $M$ is said to be obstruction flat. We note that the definition of obstruction flat point only requires the obstruction function $\mathcal{O}$ to vanish rather than being flat, but the terminology is motivated by Theorem 3.2 below.

The most basic examples of obstruction flat hypersurfaces are the unit sphere in $\mathbb{C}^{n}$ and, more generally, any spherical CR hypersurface; recall that a CR hypersurface $M$ is called spherical if, at every $p\in\Sigma$, there is a neighborhood of $p$ that is CR diffeomorphic to an open piece of the sphere. Graham [Gra87a, Gra87b] showed that locally there are many non-spherical but obstruction flat hypersurfaces. There are, however, no known examples of smoothly bounded, strongly pseudoconvex domains $\Omega\subset\mathbb{C}^{n}$ whose boundaries are non-spherical but obstruction flat. In the more general context of compact strongly pseudoconvex CR manifolds (always of hypersurface type in this paper) $M=M^{2n-1}$ of dimension $2n-1$, there are non-spherical but obstruction flat examples for $n\geq 3$ (i.e., $\dim M\geq 5$), as was recently demonstrated by the authors [EXX22], but there are no known examples of such for $n=2$ ($\dim M=3$). Indeed, the examples constructed in [EXX22] for $n\geq 3$ are unit spheres in negative line bundles over compact Kähler manifolds of dimension $n-1$ and, for $n=2$, non-spherical but obstruction flat examples cannot exist in this context by a result of the first-named author [Ebe18].

If a compact, strongly pseudoconvex, 3-dimensional CR manifold $M$ bounds a domain in a complex manifold, the obstruction function $\mathcal{O}$ coincides (modulo a universal non-zero constant) with the boundary trace of the log term $\psi_{B}$ in Fefferman’s asymptotic expansion of the Bergman kernel. It has been conjectured that every such compact, strongly pseudoconvex, 3-dimensional CR manifold $M$ which is obstruction flat is also spherical. This is often referred to as the Strong Ramadanov Conjecture, so named in accordance with the classical Ramadanov Conjecture, which asserts that $M$ is spherical if $\psi_{B}$ vanishes to infinite order along $M$. The Ramadanov Conjecture for boundaries $M$ of domains in 2-dimensional complex manifolds has been established in [Gra87b, BdM93], where it suffices to assume that $\psi_{B}$ vanishes to second order along $M$. Progress and further discussion of the Strong Ramadanov Conjecture can be found in [Ebe18, CE19, CE21a, CE21b].

In this paper, we investigate analytic and geometric properties of obstruction flatness of strongly pseudoconvex CR hypersurfaces of dimension $2n-1$. Our first two results concern local aspects. Theorem 3.2 asserts that any strongly pseudoconvex CR hypersurface $M\subset\mathbb{C}^{n}$ can be osculated at a given point $p\in M$ by an obstruction flat one up to order $2n+4$ generally and $2n+5$ if and only if $p$ is an obstruction flat point. We recall that osculation by spherical CR hypersurfaces is possible up to order $4$ ($6$ for $n=2$) generally and $5$ ($7$ for $n=2$) if and only if $p$ is CR umbilical. In Theorem 4.1, we show that locally there are non-spherical but obstruction flat CR hypersurfaces with transverse symmetry for $n=2$. For $n\geq 3$, there are even compact ones, as demonstrated by the examples constructed in [EXX22], but for $n=2$, any compact obstruction flat $M$ with a transverse symmetry is necessarily spherical [Ebe18]. Graham’s local construction of non-spherical but obstruction flat CR hypersurfaces $M$ does not easily lend itself to ensuring that $M$ has a transverse symmetry. Thus, Theorem 4.1 clarifies the necessity of compactness for the result in [Ebe18]. The final main result in this paper concerns the existence of obstruction flat points on compact, strongly pseudoconvex, 3-dimensional CR hypersurfaces. The analogous question of existence of umbilical points was raised by J. K. Moser and S.-S. Chern [CM74]. Theorem 5.1 asserts that the unit sphere in a negative line bundle over a Riemann surface $X$ always has at least one circle of obstruction flat points. The corresponding result for umbilical points in this context was established in [ES17], provided that $X$ is not a torus.

The paper is organized as follows. In section 2, we review the Chern–Moser normal forms for strongly pseudoconvex CR hypersurfaces. In Section 3, we discuss the osculation result in Theorem 3.2. Section 4 is dedicated to Theorem 4.1 and Section 5 to Theorem 5.1.

We conclude this introduction by giving two alternative characterizations of obstruction flat points. The first is given by the following proposition, which is a simple consequence of the Lee-Melrose expansion (1.2) above.

In this section, we recall the Chern–Moser normal form for strongly pseudoconvex hypersurfaces. Let $M\subset\mathbb{C}^{n}$ be a strongly pseudoconvex hypersurface and assume for a moment that $M$ is real analytic. In [CM74], it was proved that given a point $p\in M$, there exists a coordinate chart $(z,w)=(z_{1},\cdots,z_{n-1},w)\in\mathbb{C}^{n}$, vanishing at $p$, in which $M$ is defined by a convergent power series of the form

where $w=u+iv$ and $\alpha,\beta$ are multi-indices in $\mathbb{Z}_{\geq 0}^{n-1}$. The coefficients $A_{\alpha\bar{\beta}}^{l}\in\mathbb{C}$ satisfy $\overline{A_{\alpha\bar{\beta}}^{l}}=A_{\beta\bar{\alpha}}^{l}$ and certain trace conditions. If we introduce

then the trace conditions can be formulated as follows: For each $l$,

where $\Delta$ is the standard Laplace operator in $z$. Equation (2.1) is called a Chern-Moser normal form of $M$ at $p$. Chern-Moser normal forms are unique modulo an action of the finite dimensional Lie group of automorphisms of the (spherical) hyperquadric $2u=|z|^{2}$ that fix the origin. In a Chern–Moser coordinate chart, one assigns weights to the coordinates as follows: $\operatorname{wt}(z_{j})=1$, for $j=1,\ldots,n-1$, and $\operatorname{wt}(w)=2$. For a product, the weight is the sum of weights of all factors. When $M$ is merely smooth, one can find, for any $m\geq 2$, holomorphic charts such that $M$ is given in Chern–Moser normal form modulo terms of weight $m+1$. Alternatively, one can find formal charts such that $M$ is given by a formal power series of the form (2.1).

When $n=2$, the above conditions yield $A_{2\bar{2}}^{l}=A_{2\bar{3}}^{l}=A_{3\bar{2}}^{l}=A_{3\bar{3}}^{l}=0$ for any $l\geq 0$, and (the symmetric tensor associated to) the first nontrivial term $A_{2\bar{4}}^{0}$ is called E. Cartan’s 6th order tensor. In the case $n\geq 3$, (the symmetric tensor associated to) $A_{2\bar{2}}^{0}$ is called the Chern-Moser curvature tensor. In each case, if the aforementioned tensor vanishes, then the reference point $p\in M$ is called a CR umbilical point. The CR hypersurface is spherical near $p$ if and only if the tensor vanishes in a neighborhood of $p$. Furthermore, Graham [Gra87a] showed that, in the case $n=2$, the obstruction tensor $\mathcal{O}$ equals $4A^{0}_{44}$.

We first recall the definition of osculating a hypersurface in $\mathbb{C}^{n}$. Let $M$ be a strongly pseudoconvex hypersurface in $\mathbb{C}^{n}$ with $p\in\mathbb{C}^{n}$. Let $(z,w)=(z_{1},\cdots,z_{n-1},w)\in\mathbb{C}^{n}$ be a coordinate chart, vanishing at $p$, in which $M$ is given as a graph of the form

where $w=u+iv$ and $\phi$ is $O_{wt}(3)$, i.e., vanishes to weighted order at least 3 at $(z,v)=(0,0)$; here, the weights of $z$ and $w$ are as in Section 2. Now, let $M^{\prime}$ be another hypersurface and assume that, in the coordinate system $(z,w)$, it is given as a graph of the form (3.1) with graphing function $|z|^{2}+\phi^{\prime}(z,\bar{z},v)$.

It is well-known that when $n=2$, $M$ can be osculated by a spherical hypersurface to weighted order 6 at $p$, and to $7$th order if and only if $M$ is CR umbilical at $p$. When $n\geq 3$, $M$ can be osculated by a spherical hypersurface to weighted order 4 at $p$, and to $5$th order if and only if $M$ is CR umbilical at $p$. We shall show that a similar phenomenon holds with osculation by a spherical hypersurface replaced by osculation by an obstruction flat one. Of particular note is that osculation by an obstruction flat hypersurface can be achieved to a significantly higher weighted order, increasing with the dimension $n$, than osculation by a spherical one. This, of course, reflects the fact that, locally, being obstruction flat is weaker than being spherical.

Recall that, by Prop 4.14 in [Gra87a], the collection of (locally) obstruction flat, strongly pseudoconvex CR hypersurfaces properly contains that of spherical ones. We will show that the proper containment still holds even if we restrict to CR hypersurfaces with transverse symmetry. By [EXX22] (see Corollary 1.3 there and the discussion right after it), for any odd integer $m\geq 5$, there are even compact, obstruction flat and non-spherical $m$-dimensional CR hypersurfaces with transverse symmetry. By [Ebe18], however, in the $3$-dimensional case such compact CR manifolds do not exist. Our next result shows that, locally, there are (many) obstruction flat and non-spherical $3$-dimensional CR hypersurfaces with transverse symmetry.

Here $K_{;\bar{z}\bar{z}}$ and $K_{;\bar{z}\bar{z}zz}$ stand for the repeated covariant derivatives with respect to $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial\bar{z}}$: Let $\nabla$ be the Levi-Civita connection for $(X,g)$, by denoting $\nabla_{z}:=\nabla_{\frac{\partial}{\partial z}}$ and $\nabla_{\bar{z}}:=\nabla_{\frac{\partial}{\partial\bar{z}}}$, we then have $K_{;\bar{z}\bar{z}}=\nabla_{\bar{z}}\nabla_{\bar{z}}K$ and $K_{;\bar{z}\bar{z}zz}=\nabla_{z}\nabla_{z}\nabla_{\bar{z}}\nabla_{\bar{z}}K$.

The following lemma on the regularity of $h$ and $\phi$ for obstruction flat hypersurfaces in $\mathbb{C}^{2}$ will not be directly used in the proof of Proposition 4.1. We record it here for completeness.

Let $\mathcal{C}_{0}^{\omega}(\mathbb{R})$ be the set of germs of real valued $C^{\omega}$ (real analytic) functions at the origin in $\mathbb{R}$ and $\mathcal{C}_{0}^{\omega}(\mathbb{C})$ be the set of germs of real valued $C^{\omega}$ functions at the origin in $\mathbb{C}$. Given any $\phi=\phi(z,\bar{z})\in\mathcal{C}_{0}^{\omega}(\mathbb{C})$, it is $C^{\omega}$ on some open neighborhood $V\subset\mathbb{C}$ containing the origin, and we define a Kähler metric $\omega$ on $V$ by (4.1). Then we solve $\partial_{z}\partial_{\bar{z}}\log h(z,\bar{z})=e^{\phi(z,\bar{z})}$ to find a positive function $h$ on $V$. We then define a germ of strongly pseudoconvex hypersurface $M_{\phi,h}$ as in the equation of (4.3) for $(z,\xi)\in V\times\mathbb{C}$. Note that while the choice of $h$ is not unique, any other choice $\widetilde{h}$ must be such that $\log h$ and $\log\widetilde{h}$ differ by a harmonic function on $V$. Consequently, $h=\widetilde{h}|e^{f}|^{2}$ for some holomorphic function $f$ near the origin in $\mathbb{C}$. This implies that, by shrinking $V$ if needed, the hypersurfaces $M_{\phi,h}$ and $M_{\phi,\widetilde{h}}$ are CR diffeomorphic via the map $(z,\xi)\rightarrow(z,e^{f}\xi)$. For this reason, we will simply denote $M_{\phi,h}$ by $M_{\phi}$. Set

The following questions originate in the work of S.-S. Chern and J. K. Moser [CM74]. Are there compact strongly pseudoconvex 3-dimensional CR hypersurfaces without CR umbilical points? Can such a CR hypersurface exist in $\mathbb{C}^{2}$? The answers to both questions turn out to be affirmative: An example in $\mathbb{CP}^{2}$ was actually found earlier by E. Cartan in [Car33] but it is not embeddable into $\mathbb{C}^{2}$. More recently, D. Son, D. Zaitsev and the first-named author [ESZ18] constructed an example in $\mathbb{C}^{2}$. On the other hand, when $M$ is the circle bundle of a negative line bundle over a compact Riemann surface $X$, D. Son and the first-named author [ES17] proved that $M$ has at least a circle of CR umbilical points provided that $X$ is not a torus. Here, we prove an analogous result for obstruction flat points.