Rigidity of lattice-symmetric proper holomorphic maps between balls at arbitrary Hölder boundary regularity

Let $f:\mathbb{B}^m\to\mathbb{B}^M$ be a proper holomorphic map between complex unit balls whose symmetry group $\mathsf{G}_f$ (the stabilizer of $f$ under the natural $\mathrm{Aut}(\mathbb{B}^m)\times\mathrm{Aut}(\mathbb{B}^M)$ action) projects to a subgroup of $\mathrm{Aut}(\mathbb{B}^m)$ containing a uniform lattice. If $f$ extends to the closure only as a Hölder continuous map, must $f$ be equivalent, under pre- and post-composition with automorphisms, to the linear embedding $z\mapsto(z,0)$? Huang, Park, Skenderi, Srimurthy, Wen, and Zimmer (arXiv:2602.05795, 2026) proved this when the projection is Zariski dense and the Hölder exponent exceeds $1/2$ (uniform lattices are Zariski dense, so this settles the large-exponent case), and Yue (1996) settled the equivalent geometric form for target dimension $M\le 2m-1$; the conjecture at arbitrary Hölder exponent — equivalently, whether every convex co-compact representation of a uniform lattice of $\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$ into $\mathrm{Isom}(\mathbb{H}^M_\mathbb{C})$ preserves a totally geodesic copy of $\mathbb{H}^m_\mathbb{C}$ — remains open. No proof or refutation is known, so the problem is recorded as open, with the sharpness of the exponent threshold $1/2$ as the crux.

Background

A proper holomorphic map is a holomorphic map for which the preimage of every compact set is compact; for maps between bounded domains this forces points to approach the boundary of the target exactly when they approach the boundary of the source. The study of such maps between complex unit balls $\mathbb{B}^m\subset\mathbb{C}^m$ and $\mathbb{B}^M\subset\mathbb{C}^M$ is a central topic of several complex variables and CR geometry (CR = Cauchy–Riemann, the geometry of real hypersurfaces like the sphere $\partial\mathbb{B}^m$). Classical rigidity takes two forms. First, boundary regularity forces algebraicity: Forstnerič (1989) proved that a proper holomorphic map $\mathbb{B}^n\to\mathbb{B}^N$ which is $C^{\,N-n+1}$-smooth up to the boundary is rational, and Cima–Suffridge (1990) showed conversely that a rational proper ball map extends $C^\infty$-smoothly to the boundary. Second, low codimension forces linearity: Faran (1986) showed that a proper holomorphic map $\mathbb{B}^n\to\mathbb{B}^N$ with $N<2n-1$ extending real-analytically to the boundary is a linear embedding up to automorphisms; Huang (1999) reached the same conclusion assuming only $C^2$ boundary smoothness; in the borderline dimension $N=2n-1$, Huang–Ji (2001) proved that for $n\ge 3$ a $C^2$-smooth proper map is either linear or equivalent to the Whitney map $z\mapsto(z',z_nz)$, so that in particular every proper holomorphic embedding is linear; and Huang–Ji–Yin (2014) established the third gap theorem, that for $n>7$ and $3n+1\le N\le 4n-7$ every proper rational map $\mathbb{B}^n\to\mathbb{B}^N$ is, up to post-composition with an automorphism, a map into a $\mathbb{B}^{3n}$ slice.

A third, more recent source of rigidity is symmetry. The automorphism group $\mathrm{Aut}(\mathbb{B}^m)$ (biholomorphic self-maps of the ball, isomorphic to the projective unitary group $\mathrm{PU}(m,1)$) acts on holomorphic maps $f:\mathbb{B}^m\to\mathbb{B}^M$ by $(\phi,\varphi)\cdot f=\varphi\circ f\circ\phi^{-1}$, and the symmetry group $\mathsf{G}_f$ is the stabilizer of $f$ under this action. D'Angelo–Xiao (2017) proved that a rational proper ball map whose symmetry group is non-compact (i.e. not contained in a compact subgroup) is the linear embedding $z\mapsto(z,0)$ up to automorphisms; Gevorgyan–Wang–Zimmer (2024) relaxed rationality to a $C^2$-smooth boundary extension. Huang–Park–Skenderi–Srimurthy–Wen–Zimmer (2026) pushes the boundary regularity down to Hölder continuity (a map is $\alpha$-Hölder if distances on the boundary are controlled by the $\alpha$-th power of distances in the source) at the cost of a stronger symmetry hypothesis: their Theorem 1.3 assumes the projection of $\mathsf{G}_f$ to $\mathrm{Aut}(\mathbb{B}^m)$ is Zariski dense — i.e. not contained in any proper real-algebraic subvariety of $\mathrm{PU}(m,1)$, which is a real algebraic group — and that $f$ extends $\alpha$-Hölder to the closures with $\alpha>1/2$. Their proof exploits a factor-of-two gap in the exponential contraction rates of a loxodromic (hyperbolic-type) automorphism along its invariant complex affine line versus off it; an $\alpha$-Hölder boundary map scales this gap by $\alpha$, so the argument collapses precisely when $\alpha\le 1/2$, which is why the threshold is exactly $1/2$ and why its sharpness is open.

The problem is equivalent to a geometric conjecture in complex hyperbolic geometry. Complex hyperbolic $m$-space $\mathbb{H}^m_\mathbb{C}$ carries a complete Kähler metric of constant holomorphic sectional curvature $-1$ and is biholomorphic to $\mathbb{B}^m$, under which $\mathrm{Aut}(\mathbb{B}^m)$ is the identity component of $\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$. A uniform lattice is a discrete cocompact subgroup of $\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$; by the Borel density theorem every such lattice is Zariski dense. A representation $\rho:\Gamma\to\mathrm{Isom}(\mathbb{H}^M_\mathbb{C})$ of a uniform lattice is convex co-compact if its orbit map is a quasi-isometry onto a convex subset. Harmonic map theory (Cao–Mok 1990, in the spirit of Siu and Sampson) associates to such a $\rho$ a $\rho$-equivariant proper holomorphic map $f:\mathbb{B}^m\to\mathbb{B}^M$ extending Hölder to the boundary, and $\rho$ preserves a totally geodesic copy of $\mathbb{H}^m_\mathbb{C}$ exactly when $f$ is the linear embedding up to automorphisms. Huang's 2001 problem list (Problem 3.2) records the resulting conjecture: a convex co-compact representation of a uniform lattice of $\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$ into $\mathrm{Isom}(\mathbb{H}^M_\mathbb{C})$ should preserve a totally geodesic $\mathbb{H}^m_\mathbb{C}$ — a complex-hyperbolic analogue of quasi-Fuchsian rigidity. Using Cao–Mok's work, Yue (1996) proved it when $M\le 2m-1$; Theorem 1.3 of the cited paper proves it whenever the equivariant boundary map has Hölder exponent exceeding $1/2$. The remaining open territory is arbitrary Hölder exponent (in particular $\alpha\le 1/2$) and, in the geometric formulation, target dimensions $M\ge 2m$ in general.

Problem Statement

Determine the truth of Conjecture 1.5 of Huang–Park–Skenderi–Srimurthy–Wen–Zimmer (2026), stated here in the non-trivial range $2\le m<M$ (the equivalent geometric Conjecture 1.4 is stated for $2\le m\le M$, the case $m=M$ being degenerate): Suppose $2\le m<M$ and $f:\mathbb{B}^m\to\mathbb{B}^M$ is a proper holomorphic map between complex unit balls which extends to a Hölder continuous map $\overline{\mathbb{B}^m}\to\overline{\mathbb{B}^M}$ (any Hölder exponent, with no lower bound). If the image of the natural projection of the symmetry group $\mathsf{G}_f$ (the stabilizer of $f$ under the action $(\phi,\varphi)\cdot f=\varphi\circ f\circ\phi^{-1}$ with $\phi\in\mathrm{Aut}(\mathbb{B}^m)$, $\varphi\in\mathrm{Aut}(\mathbb{B}^M)$) to $\mathrm{Aut}(\mathbb{B}^m)$ contains a uniform lattice (a discrete cocompact subgroup), must there exist automorphisms $\phi_1\in\mathrm{Aut}(\mathbb{B}^m)$ and $\phi_2\in\mathrm{Aut}(\mathbb{B}^M)$ such that $\phi_2\circ f\circ\phi_1(z)=(z,0)$ for all $z\in\mathbb{B}^m$? Equivalently (Conjecture 1.4, allowing $2\le m\le M$): must every convex co-compact representation $\rho:\Gamma\to\mathrm{Isom}(\mathbb{H}^M_\mathbb{C})$ of a uniform lattice $\Gamma<\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$ preserve a totally geodesic copy of $\mathbb{H}^m_\mathbb{C}$ in $\mathbb{H}^M_\mathbb{C}$? The case of Hölder exponent $\alpha>1/2$ is already settled affirmatively by Theorem 1.3 of the cited paper (together with the Borel density theorem), and the case $M\le 2m-1$ by Yue (1996); the open core is arbitrary Hölder exponent — in particular whether the threshold $\alpha>1/2$ is sharp, i.e. whether a counterexample (a lattice-symmetric proper ball map not equivalent to $z\mapsto(z,0)$) exists at exponent $\alpha\le 1/2$.

References

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