Sharp degree bounds for rational proper holomorphic maps between balls: the D'Angelo conjecture and the classification of valid quadruples (n, r, d, k)

For $n \ge 2$, a proper holomorphic map between unit balls that is sufficiently smooth up to the boundary is automatically rational, and its degree is bounded in terms of the domain and target dimensions alone; the sharp form of this bound is unknown. D'Angelo and Lebl's 2009 study of the complexity of proper ball maps crystallized the problem in two linked forms. First, determine the sharp bound $\Phi(n,N)$ on the degree of rational proper maps $f : B_n \to B_N$; the D'Angelo conjecture asserts $\deg f \le 2N-3$ for $n = 2$ and $\deg f \le (N-1)/(n-1)$ for $n \ge 3$, the values known to be sharp for monomial maps. Second, classify the valid quadruples $(n,r,d,k)$: those for which there exists a $k$-dimensional convex family of proper rational maps from $B_n$ of generic degree $d$ and rank $r$ (the degree problem is exactly the $k = 0$ case). The conjecture is confirmed only for monomial maps, geometric rank one, the low-codimension gap intervals (degree $\le 2$ up to $N = 3n-3$ for $n \ge 4$; degree $\le 3$ up to $N = 4n-6$ for $n \ge 7$), and bounded in the maximal-geometric-rank case; the best general bound, $\deg f \le N(N-1)/(2(2n-3))$ (Meylan; D'Angelo–Lebl), is far from sharp. The problem remains open, with active partial progress through 2025.

Background

Let $B_n$ denote the unit ball in $\mathbb{C}^n$. A holomorphic map $f : B_n \to B_N$ is proper precisely when $\|f(z)\| \to 1$ whenever $\|z\| \to 1$ from inside; proper maps between balls are the central model case of the classification and boundary-regularity theory of holomorphic maps in several complex variables. In one complex variable, proper maps of the disc to itself are finite Blaschke products (finite products of disc automorphisms), which exist in every degree, so no degree bound in terms of the target dimension is possible. For $n \ge 2$ the picture changes drastically: Forstnerič proved that a proper map between balls that is sufficiently smooth up to a boundary patch is necessarily rational, i.e. a quotient $p/q$ of a vector-valued holomorphic polynomial by a scalar polynomial with $q \ne 0$ near the closed ball, and that its degree — the maximum of the degrees of $p$ and $q$ after reduction to lowest terms — is bounded by a function of $n$ and $N$ alone. Determining the sharp form of this bound has been a driving open problem of the area since the 1980s. The known landmarks are of two kinds. First, low-codimension classifications: when $N = n \ge 2$ every proper self-map of the ball is an automorphism; Faran classified proper maps $B_2 \to B_3$ (degree at most 3, four equivalence classes); Huang proved that maps $B_n \to B_N$ with $n < N \le 2n-2$ are linear; Huang–Ji proved degree at most 2 at $N = 2n-1$; Huang–Ji–Xu initiated the theory of gap intervals and proved degree at most 2 up to $N = 3n-4$, extended to $N = 3n-3$ (for $n \ge 4$) by Andrews–Huang–Ji–Yin; and Ji–Yin proved degree at most 3 for $N \le 4n-6$ when $n \ge 7$ (the third gap interval). Second, structured classes: for monomial proper maps (maps whose components are single monomials) the sharp bounds are known — D'Angelo–Kos–Riehl proved $\deg f \le 2N-3$ for $n = 2$ and Lebl–Peters proved $\deg f \le (N-1)/(n-1)$ for $n \ge 3$, both sharp. These monomial sharp bounds and the gap results crystallized into the D'Angelo conjecture: every rational proper map $f : B_n \to B_N$ satisfies $\deg f \le 2N-3$ when $n = 2$ and $\deg f \le (N-1)/(n-1)$ when $n \ge 3$. Related to these classifications is the geometric rank of Huang–Ji–Xu, an integer invariant of a rational proper map, bounded by $n-1$, measuring how degenerate the map's homogeneous expansion is: the conjecture is known for maps of geometric rank one (Huang–Ji–Xu), and Al Helal proved the bound $\deg f \le n+1$ for maps of maximal geometric rank $n-1$ at the minimal target dimension $N = n(n+1)/2$ where that rank occurs. By contrast, the best general bounds — due to Meylan for $n = 2$ and to D'Angelo–Lebl for all $n \ge 2$ — are $\deg f \le N(N-1)/(2(2n-3))$, quadratic in $N$ and far from the conjectured linear behavior. D'Angelo–Lebl's 'On the complexity of proper holomorphic mappings between balls' (2009), placed the degree question inside a broader quantitative complexity theory of proper maps. Since a proper map is determined up to unitary changes of target coordinates by its squared norm $\|f\|^2$, one can form convex combinations $\sum_i \lambda_i \|f_i\|^2$ of squared norms of proper maps; such collections are called convex families. The paper defines the rank of a proper map as the rank of its underlying Hermitian form — equivalently the smallest integer $N_0$ for which there is a norm-equivalent map sending $B_n$ properly to $B_{N_0}$ — defines the rank $r$ of a family as the maximum of the ranks of its members, and defines the generic degree $d$ as the degree of the maps on a dense open subset of the parameter set. A quadruple of integers $(n, r, d, k)$ is called valid when there exists a $k$-dimensional convex family of proper rational maps from $B_n$ of generic degree $d$ and rank $r$. D'Angelo–Lebl prove, for example, that for fixed $n$ and $d$ every convex family of origin-preserving polynomial proper maps has dimension at most $\delta(n, d-1)^2$, where $\delta(n, m)$ is the dimension of the space of polynomials of degree at most $m$ in $n$ variables with vanishing constant term, and they construct many valid quadruples; but they state explicitly that giving necessary and sufficient conditions for validity 'is an extremely difficult problem', of which the degree-estimate problem is exactly the $k = 0$ case. The same paper contributes the improved general bound above, structural results on inverse images (for a polynomial proper map $p$ the zero fiber $p^{-1}(0)$ is empty or the single point $\{0\}$; for a rational map with fixed denominator $q$ the candidates for $f^{-1}(0)$ form an explicitly determined finite set $S(q)$), and a completion construction (Proposition 4) for which the paper asserts, as a proven impossibility, that no bound on the number of completing components or their degree in terms of $n$ and $\deg g$ exists.

Problem Statement

Determine the sharp quantitative complexity of rational proper holomorphic maps between unit balls, as posed by D'Angelo–Lebl (2009) in two linked forms. (1) Sharp degree bounds. For each pair $(n, N)$ with $n \ge 2$ and $N \ge n$, determine $\Phi(n,N) = \max \{ \deg f : f \in \operatorname{Rat}(B_n, B_N) \}$, the maximal degree of a rational proper holomorphic map from the unit ball $B_n \subset \mathbb{C}^n$ to the unit ball $B_N \subset \mathbb{C}^N$ (the maximum exists: Forstnerič proved degree is bounded in terms of $n$ and $N$ alone). In particular, prove or refute the D'Angelo conjecture: $\deg f \le 2N - 3$ for $n = 2$, and $\deg f \le (N-1)/(n-1)$ for $n \ge 3$. Determine $\Phi(n,N)$ explicitly: prove that every rational proper map $B_n \to B_N$ has $\deg f$ at most $\Phi(n,N)$, and exhibit explicit proper maps of degree $\Phi(n,N)$ showing that no smaller bound is possible. No structural hypothesis — monomial, group-invariant, bounded geometric rank, gap-interval codimension, or boundary regularity beyond what rationality already provides — may be imposed. (2) Valid quadruples. Give necessary and sufficient conditions on a quadruple of integers $(n, r, d, k)$ for it to be valid: for there to exist a $k$-dimensional convex family of proper rational mappings from $B_n$ of generic degree $d$ and rank $r$ (the rank of a family is the maximum of the ranks of its members). Part (1) is the $k = 0$ case of part (2): a valid quadruple $(n, r, d, 0)$ certifies a single proper rational map from $B_n$ of degree $d$ and rank $r$, where the rank $r$ is the minimal norm-equivalent target dimension rather than the ambient target dimension $N$, so recovering $\Phi(n,N)$ from the quadruple classification requires translating between the rank parameter $r$ and the target dimension $N$; a solution of (2) therefore contains a solution of (1).

References

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