Global classification of obstruction-flat strictly pseudoconvex boundaries in $\mathbb{C}^2$ (the strong Ramadanov conjecture in complex dimension two)

A smoothly bounded strictly pseudoconvex domain $\Omega\subset\mathbb{C}^n$ carries a unique complete Kähler–Einstein metric whose potential, the Cheng–Yau solution of Fefferman's complex Monge–Ampère equation, is generally not smooth up to the boundary: the first log-term coefficient in its Lee–Melrose asymptotic expansion, the CR obstruction density, is the exact obstruction. The source paper (Curry–Ebenfelt, 2019) proved that an embedded obstruction-flat CR 3-sphere in $\mathbb{C}^2$ is spherical provided a holomorphic vector field with approximately tangent real part exists, and left open the central question: must every smoothly bounded strictly pseudoconvex domain in $\mathbb{C}^2$ with obstruction-flat boundary be locally spherical — equivalently, does the strong form of Ramadanov's conjecture (vanishing of the boundary trace of the Bergman kernel log-term coefficient forces the boundary to be locally spherical) hold in $\mathbb{C}^2$ without any symmetry, vector-field, or closeness-to-the-ball hypothesis? The subsequent literature (rigidity of the CR 3-sphere, ellipsoid cases, counterexamples on CR hypersurfaces of real dimension $\geq 5$ inside complex manifolds) leaves the general embedded classification in $\mathbb{C}^2$ open.

Background

Let $\Omega\subset\mathbb{C}^n$, $n>1$, be a bounded domain with smooth boundary $M=\partial\Omega$. A domain is strictly pseudoconvex when its boundary is locally the level set of a strictly plurisubharmonic function; its boundary then carries a strictly pseudoconvex CR structure (a contact distribution $H\subset TM$ with a complex structure $J$ on $H$; in the embedded case $H=T M\cap J_{\mathrm{std}}T M$). Fefferman's complex Monge–Ampère (or Fefferman) equation is the Dirichlet problem $J(u):=(-1)^n\det\begin{pmatrix}u & u_{z^{\bar k}}\\ u_{z^j} & u_{z^j z^{\bar k}}\end{pmatrix}=1$ in $\Omega$, $u=0$ on $\partial\Omega$. Cheng and Yau proved it has a unique positive solution, smooth inside and $C^{n+3/2-\epsilon}$ up to the boundary, yielding the complete Kähler–Einstein metric with potential $-\log u$. Lee and Melrose showed that $u$ admits an asymptotic expansion $u\sim\rho(\eta_0+\eta_1\rho^{n+1}\log\rho+\cdots)$ in a Fefferman defining function $\rho$ (one with $J(\rho)=1+O(\rho^{n+1})$). Graham proved that the coefficients $\eta_k$ are local CR invariants of the boundary and that vanishing of the trace $b\eta_1=\eta_1|_M$ forces all $\eta_k$, $k\geq 1$, to vanish to infinite order at the boundary, so that $u$ is $C^\infty$ up to $M$. The trace $b\eta_1$ is called the obstruction function or obstruction density; a strictly pseudoconvex CR manifold with vanishing obstruction density is called obstruction flat. The unit ball has polynomial Cheng–Yau potential, so every locally spherical CR manifold (one locally CR equivalent to the unit sphere) is obstruction flat; Graham also produced many local, noncompact, non-spherical obstruction-flat hypersurfaces, so no purely local classification exists. In $\mathbb{C}^2$ (CR dimension one) obstruction flatness of $M$ is equivalent to the vanishing of the boundary trace of the log-term coefficient in the asymptotic expansion of the Bergman kernel, to being a critical point of the Burns–Epstein invariant (equivalently, of the total $Q'$-curvature, a CR analogue of the conformal $Q$-curvature) under Kuranishi wiggles (small deformations of the boundary through embedded hypersurfaces in $\mathbb{C}^2$), and to the Bach-flatness of the Fefferman space, the canonical conformal Lorentzian 4-manifold built from the CR structure. The classical Ramadanov conjecture predicts that full vanishing of the Bergman log-term coefficient characterizes the ball; it was settled affirmatively in $\mathbb{C}^2$ by Boutet de Monvel and Graham. The strong form relevant here asks whether vanishing of only the boundary trace, i.e. obstruction flatness, forces the boundary to be locally spherical. Curry and Ebenfelt (the source paper) proved: (part I) a compact strictly pseudoconvex CR 3-manifold with an infinitesimal CR symmetry that is obstruction flat is locally spherical; (part II, the source) an embedded smooth bounded strictly pseudoconvex $\Omega\subset\mathbb{C}^2$ admitting a holomorphic vector field whose real part is strictly 1-approximately tangent to $\partial\Omega$ almost everywhere (a condition automatically satisfied, e.g. by the Euler field, on a large $C^1$-open neighborhood of the unit ball) has obstruction-flat boundary only if it is locally spherical. Their proof establishes the integral identity $\int_M u^2|Q|^2\,=0$ for solutions of the CR invariant infinitesimal automorphism equation $\nabla_1\nabla_1 u+iA_{11}u=0$ (Tanaka–Webster connection $\nabla$, the canonical connection adapted to a choice of contact form; pseudohermitian torsion $A_{11}$; Cartan umbilical tensor $Q$, the CR-umbilicity invariant that vanishes exactly on locally spherical hypersurfaces), via a prolonged system solved with CR tractors (an adjoint-tractor parallelism $s_A{}^B$ with $\nabla_1 s_A{}^B=u\kappa_{10\,A}{}^B$ modulo $\boldsymbol{Z}_A\boldsymbol{Z}^B$, where $\kappa$ is the tractor curvature).

Problem Statement

Prove or disprove the following (the strong Ramadanov, or obstruction-flatness, conjecture in complex dimension two, in the embedded setting): if $\Omega\subset\mathbb{C}^2$ is a smoothly bounded strictly pseudoconvex domain whose boundary is obstruction flat — equivalently, whose Cheng–Yau solution of Fefferman's complex Monge–Ampère equation is $C^4$ (hence $C^\infty$) up to $\partial\Omega$, equivalently, for which the boundary trace of the log-term coefficient in the Bergman kernel expansion vanishes on $\partial\Omega$, equivalently, whose boundary is a critical point of the Burns–Epstein invariant (total $Q'$-curvature) under deformations in $\mathbb{C}^2$ — then $\partial\Omega$ is locally spherical, and consequently, when $\partial\Omega$ is diffeomorphic to $S^3$, $\Omega$ is biholomorphic to the unit ball $\mathbb{B}^2$. An affirmative answer must hold for all such domains without any hypothesis of CR symmetry, existence of (approximately tangent) ambient holomorphic vector fields, closeness to the ball in any topology, or restriction to ellipsoids; a negative answer must exhibit a smoothly bounded strictly pseudoconvex domain in $\mathbb{C}^2$ that is obstruction flat but not locally spherical (hence not biholomorphic to the ball).

References

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