Finite-volume quotient rigidity for bounded pseudoconvex domains with $C^{1,1}$ boundary (Zimmer's Conjecture 1.6)

Zimmer's Conjecture 1.6 (Indiana Univ. Math. J. 70 (2021), 2653-2676; arXiv:1910.05288) asks whether two endpoint theorems on ball rigidity for bounded domains admit a common extension: the co-compact action theorem at $C^{1,1}$ boundary regularity, and the finite-volume quotient theorem at $C^2$ regularity. Precisely: if a bounded pseudoconvex domain $\Omega\subset\mathbb{C}^d$ with $C^{1,1}$ boundary carries a discrete group $\Gamma\leq\operatorname{Aut}(\Omega)$ acting freely such that $\Gamma\backslash\Omega$ has finite volume with respect to the Bergman, Kahler-Einstein, or Kobayashi-Eisenman volume, must $\Omega$ be biholomorphic to the unit ball? No proof, no counterexample, and no partial result covering non-convex pseudoconvex domains below $C^2$ regularity is known: the conjecture remains open. The known neighboring cases - $C^2$ pseudoconvex and convex $C^{1,\epsilon}$ (Zimmer), convex with irreducibility hypotheses (Liu-Wu), co-compact $C^{1,1}$ (Zimmer), and two-dimensional generic convex piecewise $C^2$ (Pu-Wang) - leave exactly this borderline-regularity finite-volume core unresolved.

Background

Several complex variables has no analogue of the uniformization theorem for Riemann surfaces: simply connected domains in $\mathbb{C}^d$ with $d\geq 2$ are not classified up to biholomorphism (a biholomorphism is a bijective holomorphic map with holomorphic inverse), so a long-running program instead characterizes the unit ball $\mathbb{B}^d$ by the dynamics of its biholomorphism group. For a domain $\Omega$, write $\operatorname{Aut}(\Omega)$ for its automorphism group; when $\Omega$ is bounded, this group is a Lie group acting properly on $\Omega$. Say that $\Omega$ covers a compact manifold if some discrete subgroup $\Gamma\leq\operatorname{Aut}(\Omega)$ acts freely, properly discontinuously, and co-compactly, i.e. the quotient $\Gamma\backslash\Omega$ is compact.

A classical theorem of Wong and Rosay (1977/1979) asserts that a bounded domain in $\mathbb{C}^d$ with $C^2$ boundary that covers a compact manifold is biholomorphic to the ball. Its proof localizes at a strongly pseudoconvex boundary point - a point where the Levi form (the complex Hessian of a defining function, restricted to the complex tangent space of the boundary) is positive definite; every bounded $C^2$ domain has at least one such point, and co-compactness transports every point of $\Omega$ into its neighborhood. The conclusion genuinely depends on regularity: by a theorem of Borel, every bounded symmetric domain - a bounded domain whose automorphism group is a semisimple Lie group acting transitively, such as the ball or the polydisk - admits a torsion-free lattice (a discrete subgroup of finite quotient volume, here with compact quotient) acting freely and co-compactly, yet among bounded symmetric domains only the ball has $C^{1,1}$ boundary.

Two extensions of the Wong-Rosay theorem were completed by Zimmer. First, lowering boundary regularity in the co-compact case: any bounded domain with $C^{1,1}$ boundary (the boundary is locally the graph of a differentiable function whose gradient is Lipschitz) that covers a compact manifold is biholomorphic to the ball (Zimmer, Indiana Univ. Math. J. 2021). Below $C^2$ the Levi form is not classically defined and the domain need not possess any strongly pseudoconvex boundary point, so localization fails; the proof instead rescales the domain, in the manner introduced by Frankel for convex domains covering varieties, to obtain a limiting domain containing a one-parameter subgroup of automorphisms, then applies the Frankel-Nadel solution of a conjecture of Kazhdan (a bounded domain admitting a compact quotient and a non-compact one-parameter group of automorphisms is a bounded symmetric domain) to conclude that $\Omega$ is symmetric, and finally rules out symmetric domains of real rank at least two by a volume-contraction analysis of the rescaled geometry. Second, replacing co-compactness by finite volume in the $C^2$ category, answering a question of Yau recorded in Wong's paper: if a bounded pseudoconvex domain with $C^2$ boundary admits a discrete $\Gamma\leq\operatorname{Aut}(\Omega)$ acting freely such that $\Gamma\backslash\Omega$ has finite volume, then $\Omega$ is the ball (Zimmer, J. Differential Geom. 2021). Pseudoconvexity is the standard regularity class of the field - for bounded domains with $C^2$ boundary it is equivalent to the Levi form being positive semidefinite, and in general it means the domain admits a continuous plurisubharmonic defining function. The three invariant volumes appearing there are: the Bergman volume, the volume form of the Bergman metric (the invariant Kahler metric induced by the reproducing kernel of the square-integrable holomorphic functions on $\Omega$); the Kahler-Einstein volume, the volume form of the complete Kahler metric with Ricci tensor $-(d+1)g$ that exists on every bounded pseudoconvex domain; and the Kobayashi-Eisenman volume, the invariant pseudo-volume form generated by the Kobayashi metric, the canonical pseudodistance that every holomorphic map contracts.

Finite-volume quotients are the natural generality for this program: the Teichmueller space of a compact surface of genus at least two is biholomorphic to a bounded domain and has a finite-volume quotient by a finite-index torsion-free subgroup of its mapping class group, which acts freely; and a result of Griffiths implies that when the quotient of a bounded domain by such a free discrete automorphism group is a smooth quasi-projective variety (a Zariski-open subset of a projective variety), that quotient has finite Kobayashi-Eisenman volume. The rigidity statement therefore constrains which bounded domains can model Teichmueller spaces and universal covers of quasi-projective varieties. In the convex case, Zimmer's finite-volume theorem also covers $C^{1,\epsilon}$ boundaries for any $\epsilon>0$, and Liu and Wu proved finite-volume quotient rigidity for convex domains under irreducibility or complex-dimension-two hypotheses. What no known result covers is the crossing of the two extensions: a finite-volume quotient hypothesis at only $C^{1,1}$ boundary regularity for general, non-convex pseudoconvex domains. Zimmer's Conjecture 1.6 posits that the conclusion is nevertheless still the ball.

Problem Statement

Let $\Omega\subset\mathbb{C}^d$ be a bounded pseudoconvex domain whose boundary is of class $C^{1,1}$, and let $\Gamma\leq\operatorname{Aut}(\Omega)$ be a discrete group of biholomorphic automorphisms acting freely on $\Omega$ (no non-identity element of $\Gamma$ has a fixed point in $\Omega$). Suppose the quotient $\Gamma\backslash\Omega$ has finite volume with respect to at least one of the three natural $\operatorname{Aut}(\Omega)$-invariant volume forms: the Bergman volume, the Kahler-Einstein volume, or the Kobayashi-Eisenman volume. Must $\Omega$ be biholomorphic to the unit ball $\mathbb{B}^d$?

This is Conjecture 1.6 of Zimmer (Indiana Univ. Math. J. 70 (2021), 2653-2676; arXiv:1910.05288). Each of the three volume hypotheses is separately meant to suffice, and the question is exactly whether the co-compactness hypothesis of Zimmer's Theorem 1.1 can be relaxed to finite quotient volume at the same $C^{1,1}$ boundary regularity. The conjecture is posed for all dimensions $d$ (the substantive difficulty is $d\geq 2$), with no convexity, higher boundary regularity, or $\Gamma$-restriction hypotheses beyond those stated.

References

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