Rigidity of proper holomorphic maps between unit balls with non-compact symmetry at low Hölder boundary regularity

A 2026 preprint of Huang–Park–Skenderi–Srimurthy–Wen–Zimmer proves that a proper holomorphic map $f:\mathbb{B}^m\to\mathbb{B}^M$ ($2\le m<M$) whose symmetry group $G_f$ projects to a Zariski-dense subgroup of $\mathrm{Aut}(\mathbb{B}^m)$ and which extends to the closed ball as an $\alpha$-Hölder map with $\alpha>1/2$ is, up to automorphisms, the linear totally geodesic embedding $z\mapsto(z,0)$. The paper leaves open the conjectural core: whether the same rigidity holds at arbitrary Hölder exponent when the symmetry projection contains a uniform lattice (Conjecture 1.5), equivalently whether every convex co-compact representation of a uniform lattice in $\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$ into $\mathrm{Isom}(\mathbb{H}^M_\mathbb{C})$ preserves a totally geodesic copy of $\mathbb{H}^m_\mathbb{C}$ (Conjecture 1.4, a conjecture recorded in Huang's 2001 problem list with Yue's 1996 theorem covering $M\le 2m-1$), together with the companion questions whether mere non-compactness of the symmetry projection suffices at Hölder regularity and whether the exponent threshold $1/2$ is sharp. The problem is to prove or refute, in the concrete form given in the Problem Statement below, the rigidity conjecture for symmetry-rich proper ball maps of low boundary regularity. The lattice-equivariant case (Conjectures 1.4/1.5 of the source) in the region reached by neither Yue's theorem nor the new Hölder theorem — dimensions $M\ge 2m$ combined with Hölder exponents $0<\alpha\le 1/2$ — is the geometric core of this conjecture, and the sharpness of the exponent $1/2$ is its companion optimality question; the task requires resolving the conjecture itself, not settling these surrounding questions separately.

Background

Let $\mathbb{B}^m\subset\mathbb{C}^m$ denote the complex Euclidean unit ball. A holomorphic map $f:\mathbb{B}^m\to\mathbb{B}^M$ is proper if preimages of compact sets are compact; classifying proper holomorphic maps between balls, and in particular deciding when such a map is forced to be the trivial embedding $z\mapsto(z,0)$, is a long-standing program in several complex variables. Two classical rigidity phenomena are known: sufficient smoothness of the boundary extension forces a proper ball map to be rational (a quotient of tuples of polynomials, which then extends smoothly to the closed ball), and in low codimension sufficient smoothness forces linearity. A third, more recent source of rigidity is symmetry. The automorphism group $\mathrm{Aut}(\mathbb{B}^m)\cong PU(m,1)$ acts on holomorphic maps by $(\varphi,\phi)\cdot f=\phi\circ f\circ\varphi^{-1}$ for $\varphi\in\mathrm{Aut}(\mathbb{B}^m)$, $\phi\in\mathrm{Aut}(\mathbb{B}^M)$, and the symmetry group $G_f$ of $f$ is the stabilizer of $f$ under this action; the trivial embedding has maximal symmetry. D'Angelo–Xiao (2017, Corollary 3.2) proved that if the symmetry group $G_f$ is non-compact and $f$ is rational, then $f$ is the trivial embedding up to automorphisms; Gevorgyan–Wang–Zimmer (2024) lowered the regularity requirement to a $C^2$-smooth boundary extension. (Non-compactness of the projection of $G_f$ to $\mathrm{Aut}(\mathbb{B}^m)$ implies non-compactness of $G_f$ itself, since the continuous image of a compact group is compact, so both results apply a fortiori whenever the projection is non-compact.) The 2026 preprint of Huang–Park–Skenderi–Srimurthy–Wen–Zimmer lowers the regularity further: an $\alpha$-Hölder boundary extension (i.e. $|f(x)-f(y)|\le C|x-y|^\alpha$ on the closed ball) with $\alpha>1/2$, at the cost of strengthening non-compactness to Zariski density of the projection image in the real-algebraic group $\mathrm{Aut}(\mathbb{B}^m)$ — meaning that the smallest real-algebraic subset of $\mathrm{Aut}(\mathbb{B}^m)$ containing the projection image is the whole group (Theorem 1.3). Its proof shows that $f$ maps complex affine lines into complex affine lines, applies the Fundamental Theorem of Affine Geometry — which characterizes fractional linear transformations as exactly the maps with this line-preserving property — to deduce that $f$ is rational, and then invokes the D'Angelo–Xiao theorem; the line-to-line propagation uses the dynamics of loxodromic automorphisms (those with exactly two boundary fixed points), whose contraction rates differ by a factor of two between the complex line through the fixed points and its complement — this factor-two gap is exactly why the argument requires $2\alpha>1$. The question has an equivalent geometric form. Complex hyperbolic $m$-space $\mathbb{H}^m_\mathbb{C}$, the simply connected Kähler manifold of constant holomorphic sectional curvature $-1$, is biholomorphic to $\mathbb{B}^m$ with its Bergman metric, and $\mathrm{Aut}(\mathbb{B}^m)$ is the identity component of $\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$. A uniform lattice is a discrete subgroup $\Gamma$ of a Lie group whose quotient space is compact (hence of finite volume); by the Borel density theorem, a lattice in $\mathrm{Aut}(\mathbb{B}^m)$ is Zariski dense. A representation $\rho:\Gamma\to\mathrm{Isom}(\mathbb{H}^M_\mathbb{C})$ is convex co-compact when it is discrete and the quotient of the convex hull of its limit set (the accumulation set of an orbit on the boundary at infinity) is compact. Harmonic-map theory (Cao–Mok) produces a $\rho$-equivariant proper holomorphic map $f:\mathbb{B}^m\to\mathbb{B}^M$ that extends Hölder-continuously to the closed ball, and $\rho$ preserves a totally geodesic copy of $\mathbb{H}^m_\mathbb{C}$ (a copy whose geodesics are geodesics of the ambient space) exactly when this $f$ is the trivial embedding up to automorphisms. The rigidity conjecture recorded in Huang's 2001 problem list — stated as Conjecture 1.4 in the preprint — asserts that every convex co-compact representation of a uniform lattice $\Gamma<\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$, $m\ge 2$, into $\mathrm{Isom}(\mathbb{H}^M_\mathbb{C})$ preserves a totally geodesic copy of $\mathbb{H}^m_\mathbb{C}$; Yue (1996), building on Cao–Mok, proved it when $M\le 2m-1$. The hypothesis $m\ge 2$ is essential: in complex dimension one, uniform lattices are closed surface groups and their quasi-Fuchsian deformations (discrete faithful deformations of the standard lattice embedding, the complex-hyperbolic analogue of quasi-Fuchsian Kleinian groups) into $\mathrm{Isom}(\mathbb{H}^2_\mathbb{C})$ form nontrivial open sets (Parker–Platis). Under the additional hypothesis of maximal Toledo invariant (an integer measuring the size of a representation into a group of Hermitian type), Koziarz–Maubon (2017) proved a neighboring rigidity theorem: maximal representations of uniform complex hyperbolic lattices are, up to compact factors, the standard totally geodesic ones. That result does not address convex co-compact representations without a maximality hypothesis.

Problem Statement

Prove, or refute by explicit construction, the rigidity conjecture for symmetry-rich proper ball maps of low boundary regularity, in the following concrete form. Let $2\le m<M$ and let $f:\mathbb{B}^m\to\mathbb{B}^M$ be a proper holomorphic map between the complex Euclidean unit balls such that (i) $f$ extends to an $\alpha$-Hölder continuous map $\overline{\mathbb{B}^m}\to\overline{\mathbb{B}^M}$ for some $\alpha>0$, and (ii) the image of the natural projection to $\mathrm{Aut}(\mathbb{B}^m)$ of the symmetry group $G_f\subset\mathrm{Aut}(\mathbb{B}^m)\times\mathrm{Aut}(\mathbb{B}^M)$ (the group of pairs $(\varphi,\phi)$ with $\phi\circ f=f\circ\varphi$) is non-compact. The conjecture is that there exist automorphisms $\varphi_1\in\mathrm{Aut}(\mathbb{B}^m)$ and $\varphi_2\in\mathrm{Aut}(\mathbb{B}^M)$ such that $\varphi_2\circ f\circ\varphi_1(z)=(z,0)$ for all $z\in\mathbb{B}^m$ (the totally geodesic linear embedding). The geometric core of this target is the lattice-equivariant case, Conjecture 1.5 of the source preprint, which restates its Conjecture 1.4: if the projection of $G_f$ to $\mathrm{Aut}(\mathbb{B}^m)\cong PU(m,1)$ contains a uniform lattice $\Gamma$ — via harmonic-map theory, the case where $f$ is the $\rho$-equivariant proper holomorphic map associated to a convex co-compact representation $\rho$ of a uniform lattice $\Gamma<\mathrm{Isom}(\mathbb{H}^m_\mathbb{C})$, $m\ge 2$, into $\mathrm{Isom}(\mathbb{H}^M_\mathbb{C})$ — then $f$ is the trivial embedding up to automorphisms, equivalently $\rho$ preserves a totally geodesic copy of $\mathbb{H}^m_\mathbb{C}$ in $\mathbb{H}^M_\mathbb{C}$. After the known cases — Yue's theorem for $M\le 2m-1$, the rational and $C^2$ cases of D'Angelo–Xiao and Gevorgyan–Wang–Zimmer, and the $\alpha>1/2$ case of the source preprint's Theorem 1.3 (a lattice being Zariski dense by the Borel density theorem) — the open core is precisely: the lattice-equivariant case in the region reached by neither theorem, namely dimensions $M\ge 2m$ (outside Yue's range) combined with Hölder exponents $0<\alpha\le 1/2$ (outside Theorem 1.3's range); and, beyond the lattice setting, the weakening of Zariski density to mere non-compactness of the projection at Hölder regularity. Resolving the conjecture also settles the companion question of whether the exponent threshold $\alpha=1/2$ is sharp — a positive resolution lowers the threshold to all $\alpha>0$, while a refutation at some $\alpha\le 1/2$ exhibits a non-linear proper map with non-compact symmetry projection and Hölder boundary extension, showing that rigidity already fails at that exponent under the non-compactness hypothesis; such a counterexample settles the sharpness of $\alpha=1/2$ for the Zariski-dense hypothesis of Theorem 1.3, and exhibits a lattice-type example, only if its symmetry projection is itself Zariski dense (respectively contains a uniform lattice), since a merely elementary non-compact projection would leave those stronger questions open. The conjecture and these surrounding questions all serve the one objective of determining the symmetry–regularity trade-off at which proper ball maps are forced to be totally geodesic.

References

1. Kyle Huang, Jinwoo Park, Aleksander Skenderi, Jaan Amla Srimurthy, Rou Wen, Andrew Zimmer: Rigidity of proper holomorphic maps between balls with Hölder boundary regularity, arXiv:2602.05795 (v2, 19 Feb 2026). DOI 10.48550/arXiv.2602.05795, https://arxiv.org/abs/2602.05795
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