D'Angelo's degree conjecture for rational proper holomorphic maps between complex unit balls: unresolved regimes beyond the third gap interval

The D'Angelo conjecture predicts the sharp universal degree bound for rational proper holomorphic maps between complex unit balls: $\deg F\le2N-3$ for maps $\mathbb{B}^2\to\mathbb{B}^N$ and $\deg F\le\frac{N-1}{n-1}$ for $n\ge3$. The source paper (Ji–Yin, Math. Z. 295, 2020) confirmed the cubic bound $\deg F\le3$ at the third-gap endpoint $N=4n-6$ for $n\ge7$. The subsequent literature (2020–2026) leaves the conjecture open in its remaining regimes: the endpoint $N=4n-6$ for $3\le n\le6$ (predicted bound 3, and 2 when $n=3$), which lies in the between-gap strip $3n-3\le N\le3n$ where irreducible maps exist; all other target dimensions outside the resolved gap intervals; and the two-dimensional bound $2N-3$. Recent work — Gao–Ng's gaps at all levels, Gul–Ji–Yin's and Yin–Yuan's 3-jet determination and maximal-geometric-rank classification — advances adjacent classification problems without settling any of these degree bounds. For $n\le6$, the target $N=4n-6$ lies outside the (empty) third gap interval.

Background

Let $\mathbb{B}^n=\{z\in\mathbb{C}^n:\|z\|<1\}$ be the unit ball and let $\mathrm{Rat}(\mathbb{B}^n,\mathbb{B}^N)$ denote the set of proper holomorphic maps $F:\mathbb{B}^n\to\mathbb{B}^N$ that are rational, i.e. of the form $F=(P_1,\dots,P_N)/Q$ with holomorphic polynomials having no common factor; properness means the preimage of every compact set is compact, and it forces $F$ to send the boundary sphere $\partial\mathbb{B}^n$ into $\partial\mathbb{B}^N$. The degree of $F$ is $\deg F=\max\{\deg P_1,\dots,\deg P_N,\deg Q\}$, as in algebraic geometry. Two such maps are holomorphically equivalent if one is obtained from the other by composing with biholomorphic self-maps (automorphisms) of the source and target balls; the degree is invariant under equivalence. For $n\ge2$ and $N=n$, a theorem of Alexander says every proper self-map of the ball is an automorphism, so the central question is how complicated proper maps can become as the target dimension $N$ grows.

The answer exhibits a striking rigidity pattern. Faran showed that proper maps $\mathbb{B}^2\to\mathbb{B}^3$ have degree at most $3$. Huang proved linearity for $n+1\le N\le2n-2$; Huang–Ji settled $N=2n-1$ (degree at most $2$ for $n>2$); Hamada classified rational proper maps into $\mathbb{B}^{2n}$ for $n\ge4$; and Huang–Ji–Xu showed that for $4\le n\le N\le3n-4$ every proper rational map is equivalent to the Whitney map $(z',z_n)\mapsto(z',z_nz')$ or to a quadratic monomial map of D'Angelo, a bound extended to $N\le3n-3$ for $n\ge4$ by Andrews–Huang–Ji–Yin. In these ranges no new maps appear as $N$ grows — the known maps of lower target dimension just acquire extra zero components. This is the gap phenomenon, and the ranges where it holds are the gap intervals $I_k=[kn+1,(k+1)n-\frac{k(k+1)}{2}-1]$ for $1\le k\le K(n)$, where $K(n)$ is the largest integer $k$ with $\frac{k(k+1)}{2}<n$. Huang–Ji–Yin established the third gap interval: for $n>7$ and $3n+1\le N\le4n-7$, every proper rational map is equivalent to $(G,0)$ with $G\in\mathrm{Rat}(\mathbb{B}^n,\mathbb{B}^{3n})$, while genuinely irreducible maps exist for $3n-3\le N\le3n$ and $N\ge4n-6$ (Faran–Huang–Ji–Zhang constructed maps into $\mathbb{B}^{3n-2}$ not equivalent to any polynomial map). The engine of these theorems is the geometric rank: after moving a boundary point to the origin and putting the map into a normal form, the second-order tangential terms define an $(n-1)\times(n-1)$ matrix whose rank measures the failure of linearity at that point; the geometric rank $\kappa_0(F)$ is the maximum of these ranks over the boundary, rank $0$ means linear, and rank $1$ maps were classified by Huang–Ji–Xu. A rationality theorem of Huang–Ji–Xu (any $C^3$-smooth proper map into $\mathbb{B}^N$ with $N\le\frac{n(n+1)}{2}$ is automatically rational) makes the rational class the natural universe for the whole discussion.

Against this background D'Angelo conjectured, in his 1993 monograph on the geometry of real hypersurfaces, a sharp universal degree bound for all rational proper ball maps: $\deg F\le\begin{cases}2N-3,&n=2,\\[2pt]\dfrac{N-1}{n-1},&n\ge3.\end{cases}$ The bound is sharp: monomial maps (maps whose components are monomials) attain it, and the conjecture is proved for monomial maps by D'Angelo–Kos–Riehl ($n=2$) and Lebl–Peters ($n\ge3$), and for maps of geometric rank at most $1$ by Huang–Ji–Xu. The only bound known for all rational proper maps is far weaker: $\deg F\le\frac{N(N-1)}{2(2n-3)}$ (Meylan for $n=2$, D'Angelo–Lebl in general). The source paper concerns the endpoint of the third gap interval: at $N=4n-6$, where $I_3$ ends and irreducible maps appear, the conjecture predicts $\deg F\le4-\frac{3}{n-1}$, i.e. $\deg F\le3$ for $n\ge4$, and Ji–Yin proved exactly this for $n\ge7$. The conjecture is also entangled with the gap conjecture of Huang–Ji–Yin — that reducibility to fewer target components occurs exactly on the gap intervals — for which Ebenfelt showed a sum-of-squares (SOS) conjecture on Hermitian-form decompositions would suffice, and which Gao–Ng recently advanced with hyperplane-restriction (Macaulay-type) machinery.

Problem Statement

D'Angelo's degree conjecture, in its standard formulation (D'Angelo's 1993 monograph; as stated in Ji–Yin 2020): for every rational proper holomorphic map $F\in\mathrm{Rat}(\mathbb{B}^n,\mathbb{B}^N)$ with $n\ge2$ and $N\ge n$, $\deg F\le\begin{cases}2N-3,&n=2,\\[2pt]\dfrac{N-1}{n-1},&n\ge3.\end{cases}$ Settle this conjecture in every regime where it remains unverified — prove the bound, or refute it by exhibiting an explicit rational proper map that violates it. Confirmed to date: monomial maps; maps of geometric rank $\le1$; $N\le3n-3$ with $n\ge4$ (degree $\le2$); and, by the source result of Ji–Yin (2020), the endpoint $N=4n-6$ with $n\ge7$ (degree $\le3$). The concrete unresolved regimes immediately continuing the source work are: (a) the endpoint target $N=4n-6$ in low dimensions $3\le n\le6$, which lies in the between-gap strip $3n-3\le N\le3n$ where irreducible maps exist — predicted bound $\deg F\le3$ for $4\le n\le6$ and $\deg F\le2$ for $n=3$; (b) for $n\ge3$, every remaining target dimension $N\ge3n-2$ other than the settled endpoint $N=4n-6$ with $n\ge7$: the unresolved part of the between-gap strip $3n-2\le N\le3n$ (with $N=3n-3$ settled for $n\ge4$), where genuinely irreducible maps exist, the interior of the third gap interval ($3n+1\le N\le4n-7$, where maps are known to reduce to at most $3n$ target components, so the degree question transfers to the still-open strip), and all $N\ge4n-5$; (c) the higher gap intervals $I_k$ for $k\ge4$; and (d) the two-dimensional case $n=2$, where the conjectural bound $2N-3$ is confirmed at $N=3$ and at $N=4$ (sharp bound $5$), with $N\ge5$ open, while the best general bound is quadratic in $N$. A complete answer either proves the bound in all of these regimes or refutes the conjecture with an explicit counterexample in one of them.

References

1. Shanyu Ji, Wanke Yin: D'Angelo conjecture in the third gap interval. Mathematische Zeitschrift 295 (2020), 1583–1595. DOI: 10.1007/s00209-019-02428-0. URL: https://doi.org/10.1007/s00209-019-02428-0
2. John P. D'Angelo: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics, CRC Press, Boca Raton, 1993. ISBN: 0-8493-8272-6. DOI: 10.1201/9780203739785. URL: https://doi.org/10.1201/9780203739785
3. John P. D'Angelo, Šimon Kos, Emily Riehl: A sharp bound for the degree of proper monomial mappings between balls. The Journal of Geometric Analysis 13 (2003), 581–593. DOI: 10.1007/BF02921879. URL: https://doi.org/10.1007/BF02921879
4. Jiří Lebl, Han Peters: Polynomials constant on a hyperplane and CR maps of spheres. Illinois Journal of Mathematics 56 (2012), 155–175. DOI: 10.1215/ijm/1380287465. URL: https://doi.org/10.1215/ijm/1380287465
5. Francine Meylan: Degree of a holomorphic map between unit balls from $\mathbb{C}^2$ to $\mathbb{C}^n$. Proceedings of the American Mathematical Society 134 (2006), 1023–1030. DOI: 10.1090/S0002-9939-05-08476-5. URL: https://doi.org/10.1090/S0002-9939-05-08476-5
6. John P. D'Angelo, Jiří Lebl: On the complexity of proper holomorphic mappings between balls. Complex Variables and Elliptic Equations 54 (2009), 187–204. DOI: 10.1080/17476930902759403. URL: https://doi.org/10.1080/17476930902759403
7. H. Alexander: Proper holomorphic mappings in $\mathbb{C}^n$. Indiana University Mathematics Journal 26 (1977), 137–146. DOI: 10.1512/iumj.1977.26.26010. zbMATH: Zbl 0391.32015. URL: https://doi.org/10.1512/iumj.1977.26.26010
8. James J. Faran: Maps from the two-ball to the three-ball. Inventiones Mathematicae 68 (1982), 441–475. DOI: 10.1007/BF01389412. URL: https://doi.org/10.1007/BF01389412
9. Xiaojun Huang: On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions. Journal of Differential Geometry 51 (1999), 13–33. DOI: 10.4310/jdg/1214425024. URL: https://doi.org/10.4310/jdg/1214425024
10. Xiaojun Huang, Shanyu Ji: Mapping $\mathbb{B}^n$ into $\mathbb{B}^{2n-1}$. Inventiones Mathematicae 145 (2001), 219–250. DOI: 10.1007/s002220100140. URL: https://doi.org/10.1007/s002220100140
11. Hidetaka Hamada: Rational proper holomorphic maps from $\mathbb{B}^n$ into $\mathbb{B}^{2n}$. Mathematische Annalen 331 (2005), 693–711. DOI: 10.1007/s00208-004-0606-2. URL: https://doi.org/10.1007/s00208-004-0606-2
12. Xiaojun Huang, Shanyu Ji, Dekang Xu: A new gap phenomenon for proper holomorphic mappings from $\mathbb{B}^n$ into $\mathbb{B}^N$. Mathematical Research Letters 13 (2006), 509–523. arXiv: math/0605068. URL: https://arxiv.org/abs/math/0605068
13. James Faran, Xiaojun Huang, Shanyu Ji, Yuan Zhang: Polynomial and Rational Maps between Balls. Pure and Applied Mathematics Quarterly 6 (2010), 829–842. DOI: 10.4310/PAMQ.2010.v6.n3.a10. URL: https://doi.org/10.4310/PAMQ.2010.v6.n3.a10
14. Xiaojun Huang, Shanyu Ji, Wanke Yin: On the Third Gap for Proper Holomorphic Maps between Balls. Mathematische Annalen 358 (2014), 115–142. DOI: 10.1007/s00208-013-0952-z. URL: https://doi.org/10.1007/s00208-013-0952-z
15. Jared Andrews, Xiaojun Huang, Shanyu Ji, Wanke Yin: Mapping $\mathbb{B}^n$ into $\mathbb{B}^{3n-3}$. Communications in Analysis and Geometry 24 (2016), 279–300. DOI: 10.4310/CAG.2016.v24.n2.a2. URL: https://doi.org/10.4310/CAG.2016.v24.n2.a2
16. Shanyu Ji, Wanke Yin: Upper boundary points of the gap intervals for rational maps between balls. Asian Journal of Mathematics 22 (2018), 493–506. DOI: 10.4310/AJM.2018.v22.n3.a5. URL: https://doi.org/10.4310/AJM.2018.v22.n3.a5
17. Peter Ebenfelt: On the HJY Gap Conjecture in CR geometry vs. the SOS Conjecture for polynomials. Contemporary Mathematics 681 (2017), 125–135. DOI: 10.1090/conm/681/13688. URL: https://doi.org/10.1090/conm/681/13688
18. Jennifer Brooks, Dusty Grundmeier: Sum of squares conjecture: the monomial case in $\mathbb{C}^3$. Mathematische Zeitschrift 299 (2021), 919–940. DOI: 10.1007/s00209-021-02725-7. URL: https://doi.org/10.1007/s00209-021-02725-7
19. Yun Gao, Sui-Chung Ng: A hyperplane restriction theorem for holomorphic mappings and its application for the gap conjecture. Mathematische Annalen 388 (2024), 3169–3182. DOI: 10.1007/s00208-023-02604-y. URL: https://doi.org/10.1007/s00208-023-02604-y
20. Jiří Lebl: Exhaustion functions and normal forms for proper maps of balls. Mathematische Annalen 390 (2024), 3133–3154. DOI: 10.1007/s00208-024-02837-5. URL: https://doi.org/10.1007/s00208-024-02837-5
21. Nuray Gul, Shanyu Ji, Wanke Yin: Three-Jets Determinations of Normalized Proper Holomorphic Maps from $\mathbb{H}_n$ into $\mathbb{H}_{3n-2}$. Complex Analysis and Operator Theory 18 (2024), Article 17. DOI: 10.1007/s11785-023-01468-z. URL: https://doi.org/10.1007/s11785-023-01468-z
22. Abdullah Al Helal: Degree of Ball Maps with Maximum Geometric Rank. Preprint (2024). arXiv: 2410.14959. URL: https://arxiv.org/abs/2410.14959
23. Wanke Yin, Pingsan Yuan: Proper holomorphic maps between balls with maximal geometric rank. Mathematische Annalen 394 (2026). DOI: 10.1007/s00208-026-03343-6. URL: https://doi.org/10.1007/s00208-026-03343-6
24. Sui-Chung Ng, Zhenhan Tu, Wanke Yin: Proper Holomorphic Mappings among Bounded Symmetric Domains and Related Hartogs Domains. Acta Mathematica Sinica, English Series 42 (2026), 1213–1229. DOI: 10.1007/s10114-025-3377-1. URL: https://doi.org/10.1007/s10114-025-3377-1