The Invariant Szegő metric on strongly pseudoconvex domains

Abstract: The Fefferman--Szegő metric (g_{\operatorname{FS}}^Ω) on a (C^\infty)-smooth bounded strongly pseudoconvex domain (Ω\subset\mathbb C^n) is an invariant metric defined via the Fefferman surface measure. For this metric, we first establish the vanishing of its (L^2)-Dolbeault cohomology outside the middle degree: (\dim H^{p,q}_2(Ω)=0) if (p+q\ne n), while (\dim H^{p,q}_2(Ω)=\infty) if (p+q=n). We also prove that the metric has (C^\infty)-bounded geometry. Using this analytic property, we obtain several rigidity results. In particular, if the Fefferman--Szegő metric is a gradient Kahler--Ricci soliton, then (Ω) is biholomorphic to the unit ball (\mathbb B^n). Moreover, if the metric has constant scalar curvature, then it is Einstein, and again (Ω) is biholomorphic to (\mathbb B^n). We also give a Ramadanov-type criterion in terms of the Fefferman--Szegő invariant function. Finally, in dimension (n=2), assuming the existence of a Kahler immersion into a finite-dimensional ball that maps boundary to boundary transversally, we show that the logarithmic term of the Fefferman--Szegő kernel vanishes to infinite order. Consequently, the boundary is locally spherical; if, in addition, (Ω) is simply connected, then (Ω) is biholomorphic to (\mathbb B^2).

• The paper establishes that the Fefferman–Szegő metric is complete and has smooth bounded geometry, while its L²-Dolbeault cohomology vanishes outside the middle degree and is infinite-dimensional when p + q = n.
• The paper proves rigidity results showing that gradient Kähler–Ricci soliton or constant-scalar-curvature Fefferman–Szegő metrics force the domain to be biholomorphic to the unit ball.
• For n ≥ 4, the paper derives a quantitative Ramadanov-type criterion in which the order-ρ² behavior of the Fefferman–Szegő invariant detects local sphericity, and shows that certain finite-dimensional Kähler immersions force infinite-order vanishing of the kernel’s logarithmic term.

Overview

This paper studies the Fefferman–Szegő metric $g_{\mathrm{FS}}$ on a $C^\infty$-smooth bounded strongly pseudoconvex domain $\Omega \subset \mathbb{C}^n$, a Kähler metric obtained from the diagonal of the Fefferman–Szegő kernel, i.e., the reproducing kernel of the Hardy space $\mathrm{H}^2(\partial\Omega)$ defined with respect to the Fefferman surface measure $\sigma_{\mathrm{F}}$. The Fefferman surface measure was introduced precisely to remedy the failure of Euclidean surface measure to transform covariantly under biholomorphisms; under an (T)-biholomorphism (a biholomorphism extending as a $C^2$-diffeomorphism whose complex Jacobian determinant admits a global holomorphic branch of $(\det J_\mathbb{C}F)^{n/(n+1)}$), the kernel transforms with the standard Jacobian factor and the induced metric is invariant. The authors establish four main results: a complete computation of the $L^2$-Dolbeault cohomology of $(\Omega, g_{\mathrm{FS}})$, $C^\infty$-bounded geometry of the metric together with rigidity theorems (gradient Kähler–Ricci soliton and constant scalar curvature characterizations of the ball), a Ramadanov-type criterion for local sphericity in terms of the Fefferman–Szegő invariant function for $C^\infty$0, and a Kähler-immersion criterion forcing infinite-order vanishing of the logarithmic term of the Szegő kernel.

$C^\infty$1-Dolbeault cohomology

The first theorem computes the $C^\infty$2-cohomology of the $C^\infty$3-complex on square-integrable forms with respect to $C^\infty$4: for $C^\infty$5,

$C^\infty$6

The vanishing half follows from Donnelly's criterion: since $C^\infty$7 is complete (by Barrett–Lee) and its Kähler form satisfies $C^\infty$8 with $C^\infty$9, it suffices to show that $\Omega \subset \mathbb{C}^n$0 is bounded in supremum norm. The proof of this bound is the analytic core of the section. Writing $\Omega \subset \mathbb{C}^n$1 with $\Omega \subset \mathbb{C}^n$2 and using the asymptotics $\Omega \subset \mathbb{C}^n$3 and $\Omega \subset \mathbb{C}^n$4 for $\Omega \subset \mathbb{C}^n$5 ($\Omega \subset \mathbb{C}^n$6 for $\Omega \subset \mathbb{C}^n$7), the authors split tangent directions at each boundary point into two regimes according to whether the normal singularity $\Omega \subset \mathbb{C}^n$8 or the tangential singularity $\Omega \subset \mathbb{C}^n$9 dominates the metric. In both regimes they obtain uniform upper bounds on $\mathrm{H}^2(\partial\Omega)$0, with the ratio converging to $\mathrm{H}^2(\partial\Omega)$1 along non-tangential directions. The same dichotomy yields the comparability

$\mathrm{H}^2(\partial\Omega)$2

which verifies Ohsawa's hypothesis with $\mathrm{H}^2(\partial\Omega)$3, $\mathrm{H}^2(\partial\Omega)$4 and gives infinite dimensionality of $\mathrm{H}^2(\partial\Omega)$5 in the middle degree, since strict pseudoconvexity makes every boundary point regular and non-degenerate. This extends the classical Donnelly–Fefferman picture for the Bergman metric to the Fefferman–Szegő setting in all dimensions $\mathrm{H}^2(\partial\Omega)$6.

Bounded geometry and rigidity

The second group of results establishes that $\mathrm{H}^2(\partial\Omega)$7 has $\mathrm{H}^2(\partial\Omega)$8-smooth bounded geometry: all covariant derivatives of the Riemann curvature tensor are uniformly bounded, and the injectivity radius is positive. The argument adapts Zimmer's framework of bounded intrinsic geometry. Since $\mathrm{H}^2(\partial\Omega)$9 has bounded intrinsic geometry, there exist holomorphic embeddings $\sigma_{\mathrm{F}}$0 with $\sigma_{\mathrm{F}}$1, extending smoothly to the boundary, such that $\sigma_{\mathrm{F}}$2. Pulling back the Szegő kernel defines "local kernels" $\sigma_{\mathrm{F}}$3 on $\sigma_{\mathrm{F}}$4, and the key estimate—proved via the invariance of the $\sigma_{\mathrm{F}}$5-function $\sigma_{\mathrm{F}}$6 and Zimmer's comparison of Bergman kernels on $\sigma_{\mathrm{F}}$7 and $\sigma_{\mathrm{F}}$8—is that $\sigma_{\mathrm{F}}$9 and all its derivatives are uniformly bounded independently of $C^2$0. Since curvature components of a Kähler metric are universal polynomial expressions in derivatives of $C^2$1 and of the pulled-back metric, this yields $C^2$2 for every $C^2$3; positive injectivity radius then follows from Liu–Sun–Yau via $C^2$4.

Two rigidity consequences follow. First, combining bounded geometry with the negative pinching of $C^2$5 near the boundary (from the boundary asymptotics $C^2$6 and $C^2$7), Sha's soliton theorem implies that any gradient Kähler–Ricci soliton structure on $C^2$8 forces the metric to be Kähler–Einstein, hence by Yuan's theorem $C^2$9. Since the ball trivially carries the soliton structure with $(\det J_\mathbb{C}F)^{n/(n+1)}$0, the characterization is an if-and-only-if statement: there are no non-trivial gradient Kähler–Ricci solitons among Fefferman–Szegő metrics on smooth strongly pseudoconvex domains. Second, if $(\det J_\mathbb{C}F)^{n/(n+1)}$1 has constant scalar curvature, the identity $(\det J_\mathbb{C}F)^{n/(n+1)}$2 shows that $(\det J_\mathbb{C}F)^{n/(n+1)}$3 is harmonic with respect to $(\det J_\mathbb{C}F)^{n/(n+1)}$4; the maximum principle with the known boundary value forces $(\det J_\mathbb{C}F)^{n/(n+1)}$5, which by Yuan's criterion gives Kähler–Einstein and hence $(\det J_\mathbb{C}F)^{n/(n+1)}$6. These extend Yuan's Kähler–Einstein rigidity to the soliton and cscK settings, paralleling results of Sha and others for the Bergman metric.

A Ramadanov-type criterion

The third result addresses the Engliš–Zhang/Lu–Tian conjecture for the Fefferman–Szegő kernel: does infinite-order vanishing of the logarithmic term $(\det J_\mathbb{C}F)^{n/(n+1)}$7 in the Boutet de Monvel–Sjöstrand expansion $(\det J_\mathbb{C}F)^{n/(n+1)}$8 imply local sphericity of $(\det J_\mathbb{C}F)^{n/(n+1)}$9? For $L^2$0, the paper proves a sharp quantitative version. Using the Hirachi–Komatsu–Nakazawa asymptotics $L^2$1, where $L^2$2 is the Chern–Moser curvature invariant, the authors construct the auxiliary defining function $L^2$3 and compute its Monge–Ampère operator through a sequence of row and column operations on the determinant defining $L^2$4. The outcome is

$L^2$5

where $L^2$6 is a nonzero multiple of $L^2$7. Consequently, $L^2$8 is locally spherical if and only if $L^2$9 at every boundary point. This gives a biholomorphic-invariant characterization of local sphericity purely in terms of the Fefferman–Szegő invariant function, proved without recourse to Fu–Wong's result used in earlier approaches. The restriction to $(\Omega, g_{\mathrm{FS}})$0 is essential and explicit: the derivation requires $(\Omega, g_{\mathrm{FS}})$1, which fails in dimensions 2 and 3 where the expansion of $(\Omega, g_{\mathrm{FS}})$2 itself contains lower-order terms; whether the invariant characterizes sphericity for $(\Omega, g_{\mathrm{FS}})$3 remains open.

Kähler immersions into finite-dimensional balls

The final theorem, inspired by Palmieri's work on Bergman metrics, treats Kähler immersions $(\Omega, g_{\mathrm{FS}})$4 with $(\Omega, g_{\mathrm{FS}})$5, assuming $(\Omega, g_{\mathrm{FS}})$6 extends smoothly past the boundary, maps $(\Omega, g_{\mathrm{FS}})$7 into $(\Omega, g_{\mathrm{FS}})$8 transversally, and maps the exterior into the exterior. Via Calabi's diastasis criterion—isometry is equivalent to preservation of diastasis potentials—the identity $(\Omega, g_{\mathrm{FS}})$9 yields

$C^\infty$0

Substituting the Fefferman expansion of $C^\infty$1 and using Fu–Wong's lemma (applicable because $C^\infty$2 is a positive integer and $C^\infty$3 is smooth up to the boundary) forces $C^\infty$4 to vanish to infinite order on $C^\infty$5; transversality ensures $C^\infty$6 is a genuine defining function, so $C^\infty$7 vanishes to infinite order. For $C^\infty$8, the Boutet de Monvel/Ebenfelt resolution of the conjecture then gives local sphericity, and simple connectivity plus Chern–Ji yields $C^\infty$9. The hypothesis is natural in the sense that holomorphic immersions into complex hyperbolic space $C^\infty$00 always exist for $C^\infty$01, $C^\infty$02; the constraint is immersion into a finite-dimensional ball with the specific scaling $C^\infty$03. Notably, under these hypotheses Conjecture 1.1's assumption is satisfied, so a general proof of that conjecture would upgrade the conclusion to all dimensions.

Limitations and open questions

Several restrictions are stated plainly in the paper. The rigidity theorems require $C^\infty$04-smooth boundaries, and the authors explicitly ask whether this can be relaxed to lower regularity. The Ramadanov-type criterion is confined to $C^\infty$05 by the asymptotic expansion technique, leaving dimensions 2 and 3 open for the invariant-function characterization. The Kähler immersion result delivers the full biholomorphic-to-ball conclusion only for $C^\infty$06 with simple connectivity; in higher dimensions it establishes only infinite-order vanishing of the log term, conditional on the still-open Engliš–Zhang conjecture for $C^\infty$07. Finally, the soliton rigidity relies on Sha's theorem, which presupposes completeness, $C^\infty$08-bounded geometry, and boundary Ricci pinching—conditions verified here but not removable within the current argument.

Conclusion

The paper consolidates the Fefferman–Szegő metric as a canonical object on strongly pseudoconvex domains whose analytic behavior parallels, and in some respects refines, that of the Bergman metric: its $C^\infty$09-cohomology exhibits the same middle-dimensional pattern, its bounded geometry supports soliton and cscK rigidity with no non-trivial gradient solitons, and its invariant function detects Chern–Moser curvature quantitatively at order $C^\infty$10 for $C^\infty$11. The remaining gaps—low regularity boundaries, dimensions 2 and 3 of the Ramadanov problem, and the higher-dimensional immersion-to-ball question—are clearly delineated and reduce, in part, to the unresolved log-term conjecture for the Fefferman–Szegő kernel.