Sharp degree bounds for rational proper maps between balls of maximal geometric rank: sharpness at minimal target dimension and the above-minimal regime (Problem (D))

For a rational proper holomorphic map $F:\mathbb{B}^n\to\mathbb{B}^N$ between complex unit balls, Huang's geometric rank $\kappa_0\in[0,n-1]$ measures the degeneracy of the map's second fundamental form and splits the theory into four regimes. Al Helal (2024) proved that when the geometric rank is maximal, $\kappa_0=n-1$, and the target dimension is minimal, $N=n(n+1)/2$, one has $\deg F\le n+1$. The target problem is the sharp degree bound for the maximal-geometric-rank class: (a) at minimal target dimension, is $n+1$ sharp, or does the D'Angelo degree-conjecture bound $\deg F\le (N-1)/(n-1)=(n+2)/2$ hold for this class (only $n=2$, where the sharp bound is 3, is known); and (b) in the above-minimal regime $N>n(n+1)/2$ — Problem (D) of the standard taxonomy — no rank-aware sharp bound is known at all. For $n=2$ every rational proper ball map not equivalent to a linear embedding has maximal geometric rank, so part (b) contains the classical open D'Angelo conjecture $\deg F\le 2N-3$ for maps $\mathbb{B}^2\to\mathbb{B}^N$. The problem is open.

Background

Let $\mathbb{B}^n$ denote the unit ball in $\mathbb{C}^n$. A holomorphic map $F:\mathbb{B}^n\to\mathbb{B}^N$ is proper if preimages of compact sets are compact, which forces $F$ to map the boundary sphere to the boundary sphere. Proper holomorphic ball maps with $n\ge 2$ that extend smoothly enough to the boundary are rational (Forstnerič, 1989), so the natural invariant is the algebraic degree $\deg F$; two maps are considered the same up to spherical equivalence, i.e. composition with automorphisms of the source and target balls. Alexander's theorem (proper self-maps of a ball are automorphisms) settled the equidimensional case, and the study of the positive-codimension case has run for fifty years: Webster and Faran handled $N=n+1$, Faran and Huang proved linearity for $N\le 2n-2$, and Huang–Ji classified $N=2n-1$; D'Angelo classified all polynomial proper ball maps, while the classification of rational ones up to spherical equivalence remains a central open program.

A key complexity measure of a ball map $F$ is its geometric rank $\kappa_0\in[0,n-1]$, introduced by Huang: roughly, the rank of the induced quadratic correction (the degeneracy of the second fundamental form) of the map at a general boundary point, computable from a Chern–Moser-type normal form of the map's defining equation (the Chern–Moser theory is the standard local normal-form theory for real hypersurfaces such as the unit sphere). Huang proved that $\kappa_0=0$ forces the map to be a linear embedding, and more generally that maps with $\kappa_0<n-1$ satisfy a semi-linearity property that makes them tractable; he also proved the dimension lower bound $N\ge n+\kappa_0(2n-\kappa_0-1)/2$, so the minimal target dimension depends on the geometric rank. This splits the study of ball maps into four standard regimes: (A) $\kappa_0<n-1$ with minimal $N$; (B) $\kappa_0<n-1$ with $N$ above minimal; (C) $\kappa_0=n-1$ (maximal geometric rank) with the minimal target dimension $N=n(n+1)/2$; and (D) $\kappa_0=n-1$ with $N>n(n+1)/2$. Regimes (A) and (B) are largely understood through work of Huang, Ji, Xu, Yin, and others (for example, the gap theorems classifying maps for $N\le 3n-4$, and Huang–Ji–Xu's confirmation of the degree conjecture for $\kappa_0=1$).

One way to measure the complexity of a rational proper map is its degree. Forstnerič proved $\deg F$ is bounded by a function of the dimensions, and D'Angelo conjectured the sharp bound $\deg F\le 2N-3$ for $n=2$ and $\deg F\le (N-1)/(n-1)$ for $n>2$; the conjecture is known for monomial maps — maps whose components are single monomials — (D'Angelo–Kos–Riehl for $n=2$; Lebl–Peters for $n\ge 3$) and in some geometric-rank-one regimes, but is open in general, with the best general bounds (Meylan's $\deg F\le N(N-1)/2$ for $n=2$; D'Angelo–Lebl's $\deg F\le N(N-1)/(2(2n-3))$ for $n\ge 2$) far from sharp.

The maximal-rank case $\kappa_0=n-1$ is the least understood, precisely because the semi-linearity tool fails there. Al Helal (2024) proved that a proper holomorphic map that is $C^3$-smooth up to the boundary with geometric rank $\kappa_0$ at the minimal target dimension $N=n+\kappa_0(2n-\kappa_0-1)/2$ is rational with $\deg F\le\kappa_0+2$; the $\kappa_0\le n-2$ case was due to Ji–Xu, so the new contribution is the maximal-rank case, giving $\deg F\le n+1$ for rational proper maps with $\kappa_0=n-1$ and $N=n(n+1)/2$. For $n=2$ this recovers Faran's sharp bound 3; for $n=3$ it reproduces the earlier Huang–Ji–Xu bound 4; for $n\ge 4$ it is the first bound at minimal target dimension. What the degree program for maximal-geometric-rank maps still lacks is sharpness: at minimal $N$, the D'Angelo conjecture predicts $\deg F\le(n+2)/2$, roughly half of $n+1$; and in regime (D), $N>n(n+1)/2$, no rank-aware sharp bound is known. Note that for $n=2$ every rational proper ball map not spherically equivalent to a linear embedding has maximal geometric rank $\kappa_0=1$, so regime (D) at $n=2$ is exactly the classical open degree conjecture $\deg F\le 2N-3$ for rational proper maps $\mathbb{B}^2\to\mathbb{B}^N$ with $N>3$.

Problem Statement

Determine the sharp degree bound for rational proper holomorphic maps between complex unit balls whose geometric rank is maximal. Concretely, for rational proper maps $F:\mathbb{B}^n\to\mathbb{B}^N$ with geometric rank $\kappa_0=n-1$ (where $n\ge 2$), find the optimal function $b(n,N)$ with $\deg F\le b(n,N)$, and exhibit maps realizing it, in the two regimes left open by Al Helal (2024):

(1) Minimal target dimension, $N=n(n+1)/2$: determine the sharp bound $b(n,n(n+1)/2)$, i.e. whether the known bound $\deg F\le n+1$ is sharp for $n\ge 3$ or the D'Angelo degree-conjecture bound $\deg F\le(N-1)/(n-1)=(n+2)/2$ holds for this class (so the candidate values to test are $n+1$ and $(n+2)/2$). (For $n=2$ the sharp bound 3 is known, by Faran's classification.)

(2) Above-minimal target dimension, $N>n(n+1)/2$ (Problem (D) of the standard taxonomy): determine the sharp rank-aware degree bound; no rank-aware bound is currently known in this regime, and the general bounds for rational proper maps are far from sharp. For $n=2$, every rational proper map not spherically equivalent to a linear embedding has maximal geometric rank $\kappa_0=1$, so this regime includes the classical open problem of proving $\deg F\le 2N-3$ for rational proper maps $\mathbb{B}^2\to\mathbb{B}^N$.

The two regimes are the minimal and non-minimal cases of one objective: a sharp degree bound for the maximal-geometric-rank class as a function of $(n,N)$.

References

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