Group data versus map data for rational proper maps of balls: detection of map invariants from the associated symmetry groups

For a rational proper holomorphic map $f:\mathbb{B}_n\to\mathbb{B}_N$ , D'Angelo and Xiao attached five symmetry groups $A_f,\Gamma_f,G_f,T_f,H_f$ , and Grundmeier and Lebl, using a normal form for rational ball maps, added the denominator-related groups $D_f,\Sigma_f,\Delta_f^{(a,b)}$ and proved $G_f\le\Gamma_f\le D_f\le\Sigma_f$ together with a realization theorem: every subgroup $\Gamma\le U(n)$ defined by finitely many invariant polynomials arises as $\Gamma_f$ of a polynomial proper ball map. The source poses as its open problem the determination of what properties of the map $f$ can be detected using these groups: which invariants of $f$ are determined by the group data, individually or in combination, beyond the known isolated detections. As of late 2025 only Frobenius-type gap results for three special cyclic group families are known; the detection dictionary remains open.

Background

A proper holomorphic map $f:\mathbb{B}_n\to\mathbb{B}_N$ , where $\mathbb{B}_k$ is the unit ball in $\mathbb{C}^k$ , is a holomorphic map with $f(\mathbb{B}_n)\subset\mathbb{B}_N$ that maps the boundary sphere of $\mathbb{B}_n$ to the boundary sphere of $\mathbb{B}_N$ . Forstnerič proved that such a map with sufficiently smooth boundary extension is rational, that is, a ratio $f=p/g$ of vector-valued and scalar holomorphic polynomials, so rational proper maps are the central objects of the subject. Two such maps are spherically equivalent if they differ by automorphisms of the source and target balls; the automorphism group $\operatorname{Aut}(\mathbb{B}_n)$ is the group of biholomorphic self-maps of the ball. A recurring theme since Cartan is to study maps invariant under subgroups of the automorphism group: Rudin and Forstnerič constructed invariant proper maps for general targets, D'Angelo–Lichtblau and Lichtblau gave the decisive characterization of which groups admit invariant proper rational maps between balls, and Lichtblau showed that the invariance group $G_f$ of a non-linear proper rational ball map is always finite, fixed-point-free, and cyclic. To use symmetry as an invariant of the map itself, D'Angelo and Xiao introduced five groups attached to a proper rational map: $A_f$ (pairs $(\varphi,\tau)$ of source and target automorphisms with $\tau\circ f=f\circ\varphi$ ), its projections $\Gamma_f\le\operatorname{Aut}(\mathbb{B}_n)$ and $T_f\le\operatorname{Aut}(\mathbb{B}_N)$ , and the pointwise stabilizers $G_f\le\operatorname{Aut}(\mathbb{B}_n)$ ( $f=f\circ\varphi$ ) and $H_f\le\operatorname{Aut}(\mathbb{B}_N)$ ( $\tau\circ f=f$ ). They proved: $\Gamma_f$ is noncompact if and only if $f$ is spherically equivalent to a linear fractional embedding, that is, a proper map whose components are linear functions of $z$ divided by one common affine function of $z$ (extended to $C^2$ boundary regularity by Gevorgyan–Wang–Zimmer and to Hölder exponent above $1/2$ by Huang–Park–Skenderi–Srimurthy–Wen–Zimmer); $\Gamma_f$ contains the $n$ -torus if and only if $f$ is spherically equivalent to a monomial map, one whose every component is a single monomial in $z$ ; if $\Gamma_f$ contains the central circle of $U(n)$ then $f$ is spherically equivalent to a polynomial map; and every finite subgroup arises as $\Gamma_f$ . Grundmeier and Lebl put $f=p/g$ into a normal form ( $f(0)=p(0)=0$ , denominator $g=1+g_2+\cdots+g_{d-1}$ with the quadratic part $g_2$ diagonalized, available up to spherical equivalence by Lebl's exhaustion-function construction) and showed that then all these groups become closed subgroups of the unitary groups, with $G_f\le\Gamma_f\le D_f\le\Sigma_f\le U(n)$ , where $D_f$ preserves the whole denominator $g$ , $\Sigma_f$ preserves its quadratic part $g_2$ , and $\Delta_f^{(a,b)}$ preserves the bidegree- $(a,b)$ part of the underlying form $|g|^2-\|p\|^2$ — groups that are typically much easier to compute than $\Gamma_f$ . They further characterized $\Gamma_f$ exactly as the subgroup of $U(n)$ defined by the single invariant polynomial $\rho=|g|^2-\|p\|^2$ , that is, the group fixing $\rho$ ( $\rho\circ U=\rho$ ), and proved a converse realization theorem: every subgroup $\Gamma\le U(n)$ defined by finitely many invariant polynomials $\rho_1,\dots,\rho_\ell$ , meaning $\Gamma=\{U\in U(n):\rho_i\circ U=\rho_i\ \text{for all}\ i\}$ (a class containing all finite groups and the compact groups cut out by finitely many polynomial invariance conditions) equals $\Gamma_f$ for some polynomial proper map in normal form. They also constructed degree-4 maps with $G_f=\{I,-I\}$ whose denominator is $1+\sum_j\sigma_j z_j^2$ for small distinct $\sigma_j$ . Conspicuously not determined is which map properties the group data encodes in general; on the realization side the paper notes only that the $N$ produced by its construction 'depends on the invariant polynomials given', and after its degree-4 constructions it remarks that the approach 'easily generalizes to more complicated denominators'. A 2025 follow-up by D'Angelo–Grundmeier–Lichtblau studies the minimal target dimensions of invariant sphere maps for the three cyclic group families that can occur, obtaining Frobenius-type (postage-stamp) gaps in the possible number of terms, and explicitly states that combinatorial questions about the relationship between degree, number of terms, and a given group $\Gamma_f$ remain.

Problem Statement

Let $f=p/g:\mathbb{B}_n\to\mathbb{B}_N$ be a rational proper holomorphic map of degree $d>1$ , in lowest terms with $g(0)=1$ , written in the Grundmeier–Lebl normal form: $f(0)=p(0)=0$ , $g=1+g_2+\cdots+g_{d-1}$ with homogeneous parts $g_k$ of degree $k$ and $g_2(z)=\sum_{\ell=1}^n\sigma_\ell z_\ell^2$ , $0\le\sigma_1\le\cdots\le\sigma_n$ . Attach to $f$ the automorphism-group data of D'Angelo–Xiao — $A_f=\{(\varphi,\tau)\in\operatorname{Aut}(\mathbb{B}_n)\oplus\operatorname{Aut}(\mathbb{B}_N):\tau\circ f=f\circ\varphi\}$ , its projections $\Gamma_f$ and $T_f$ , and the stabilizers $G_f=\{\varphi:f=f\circ\varphi\}$ , $H_f=\{\tau:\tau\circ f=f\}$ — and the denominator-related subgroups of $U(n)$ of Grundmeier–Lebl: $D_f=\{U:g\circ U=g\}$ , $\Sigma_f=\{U:g_2\circ U=g_2\}$ , and $\Delta_f^{(a,b)}=\{U:\bigl(|g|^2-\|p\|^2\bigr)_{(a,b)}$ is $U$ -invariant $\}$ , where $(\cdot)_{(a,b)}$ denotes the bidegree- $(a,b)$ part. The problem is the detection question posed by the source: determine what properties of the map $f$ can be detected using the above groups, individually or in combination. The task is to settle this for each of the following four properties of $f$ , all four determinations required, and each asked beyond the known detections (noncompact $\Gamma_f$ iff linear fractional; torus in $\Gamma_f$ iff monomial up to equivalence; central circle in $\Gamma_f$ implies polynomial up to equivalence; $G_f=\{I,-I\}$ with distinct positive $\sigma_\ell$ forces $f$ even of degree at least 4): (i) the degree of $f$ ; (ii) the spherical-equivalence class of $f$ ; (iii) whether $f$ is polynomial rather than rational, up to spherical equivalence; (iv) the coefficient-level data of $f$ , namely the coefficients of the polynomials $p$ and $g$ and the bidegree parts $\bigl(|g|^2-\|p\|^2\bigr)_{(a,b)}$ of the underlying form. For each of these, determine which of the groups $A_f,\Gamma_f,G_f,T_f,H_f,D_f,\Sigma_f,\Delta_f^{(a,b)}$ detect the property, individually or in combination; the four properties are logically nested (the coefficient data (iv) determines the degree (i) and the spherical-equivalence class (ii), and (ii) determines whether $f$ is polynomial up to spherical equivalence), but each of the four determinations is required in its own right.

References

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