Extending the sharp dimension-estimate rigidity for polynomial-growth holomorphic functions to nonnegative holomorphic sectional curvature

On a complete n-dimensional Kähler manifold with nonnegative holomorphic bisectional curvature, Chu–Deng–Hao–Li determined the full sharp gap structure of the space of holomorphic functions of degree-d polynomial growth: the dimension is at most $\binom{n+d}{d}$ with equality only for flat $\mathbb{C}^{n}$; at least $N_{2}+1$, where $N_{2}=\binom{n+d}{d}-\binom{n+d-2}{d-1}$, it forces the manifold to be biholomorphically isometric to flat $\mathbb{C}^{n}$; and at least $N_{3}+1$, where $N_{3}=\binom{n-1+d}{d}$, it forces maximal volume growth and biholomorphism to $\mathbb{C}^{n}$. The paper asks whether this three-level rigidity picture survives when the bisectional-curvature hypothesis is weakened to nonnegative holomorphic sectional curvature: whether the dimension thresholds $N_{2}+1$ and $N_{3}+1$ still imply, respectively, isometry with flat $\mathbb{C}^{n}$, maximal volume growth, and biholomorphism with $\mathbb{C}^{n}$. The window version of the bisectional statement posed in the first arXiv version was resolved affirmatively by the authors' own March 2026 revision; the holomorphic-sectional-curvature questions (v2 Questions 6.1–6.3) are the open core of the problem.

Background

A Kähler manifold $(M,\omega)$ is a complex $n$-dimensional manifold carrying a compatible Hermitian metric $\omega$. Its holomorphic bisectional curvature (the Kähler analogue of Riemannian sectional curvature) is nonnegative if $R(u,\bar u,v,\bar v)\ge 0$ for all pairs of $(1,0)$-tangent vectors $u,v$ (the complex-linear tangent directions); the weaker condition of nonnegative holomorphic sectional curvature requires this only for $u=v$, i.e. $R(u,\bar u,u,\bar u)\ge 0$ for every $(1,0)$-vector $u$, so non-negativity is demanded on complex lines rather than on pairs of lines. For a base point $p_{0}$, $\mathcal{O}_{d}(M)$ denotes the vector space of holomorphic functions of degree-$d$ polynomial growth, $|f(p)|\le C(1+r(p_{0},p))^{d}$; on $\mathbb{C}^{n}$ these are exactly the polynomials of degree at most $d$, so $\dim\mathcal{O}_{d}(\mathbb{C}^{n})=\binom{n+d}{d}$. A manifold has maximal volume growth if $\mathrm{Vol}(B_{r}(p_{0}))\ge c\,r^{2n}$ for some $c>0$ and all $r>0$, the Euclidean volume rate; a map is biholomorphically isometric if it is simultaneously biholomorphic and an isometry. A central theme going back to Yau's uniformization conjecture - every complete noncompact Kähler manifold with positive bisectional curvature is biholomorphic to $\mathbb{C}^{n}$ - is how strongly the function spaces $\mathcal{O}_{d}(M)$ constrain the complex geometry of $M$. Ni, Chen–Fu–Yin–Zhu, and Gang Liu proved the sharp estimate $\dim\mathcal{O}_{d}(M)\le\binom{n+d}{d}$ under nonnegative bisectional curvature, with equality if and only if $(M,\omega)$ is biholomorphically isometric to flat $(\mathbb{C}^{n},\omega_{\mathbb{C}^{n}})$, where $\omega_{\mathbb{C}^{n}}$ is the standard flat Kähler form; Liu also established this equality rigidity under the weaker hypothesis of nonnegative holomorphic sectional curvature, via his three-circle theorem, a Kähler analogue of Hadamard's classical three-circles theorem on the convexity of $\log\max_{|z|=r}|f|$ in $\log r$. Chu–Deng–Hao–Li then completed the sharp gap structure under nonnegative bisectional curvature. Besides the top level, explicit examples pin the two thresholds: complete metrics $\omega_{\lambda}$ of nonnegative bisectional curvature on $\mathbb{C}^{n}$ itself have $\dim\mathcal{O}_{d}=N_{2}=\binom{n+d}{d}-\binom{n+d-2}{d-1}$ (so no metric rigidity can hold there), and the product $\mathbb{C}^{n-1}\times\mathbb{CP}^{1}$ has $\dim\mathcal{O}_{d}=\binom{n-1+d}{d}=N_{3}$ while not being biholomorphic to $\mathbb{C}^{n}$. Their theorems: first rigidity, $\dim\mathcal{O}_{d}(M)\ge N_{2}+1$ forces $(M,\omega)$ to be biholomorphically isometric to flat $\mathbb{C}^{n}$ (Theorem 1.6); and second rigidity, $\dim\mathcal{O}_{d}(M)\ge N_{3}+1$ forces maximal volume growth (Theorem 1.8) and $M$ biholomorphic to $\mathbb{C}^{n}$ (Theorem 1.9), the last step using the resolution of Yau's uniformization conjecture under maximal volume growth by Liu and by Lee–Tam. Under bisectional curvature the picture is thus complete, and the natural remaining question - posed by the same authors - is whether it survives under nonnegative holomorphic sectional curvature alone, where no comparable structure theory is available; see Gang Liu's survey for the broader dimension-estimate problem.

Problem Statement

Let $(M,\omega)$ be a complete $n$-dimensional Kähler manifold whose holomorphic sectional curvature is nonnegative, i.e. $R(u,\bar u,u,\bar u)\ge 0$ for every $(1,0)$-tangent vector $u$. Let $\mathcal{O}_{d}(M)$ be the space of holomorphic functions of degree-$d$ polynomial growth, i.e. holomorphic functions $f$ satisfying $|f(p)|\le C(1+r(p_{0},p))^{d}$ for some constant $C$, where $r(p_{0},p)$ is the geodesic distance from a fixed base point $p_{0}$; and set $N_{2}=\binom{n+d}{d}-\binom{n+d-2}{d-1}$ and $N_{3}=\binom{n-1+d}{d}$. Determine which of the following implications - all theorems under the stronger hypothesis of nonnegative holomorphic bisectional curvature (Questions 6.1–6.3 of arXiv:2510.14371v2) - remain true under this weaker hypothesis: (Q1) if $\dim\mathcal{O}_{d}(M)\ge N_{2}+1$ for some integer $d\ge 1$, must $(M,\omega)$ be biholomorphically isometric to $(\mathbb{C}^{n},\omega_{\mathbb{C}^{n}})$, i.e. simultaneously biholomorphic and distance-preserving, with $\omega_{\mathbb{C}^{n}}$ the standard flat Kähler form? (Q2) if $\dim\mathcal{O}_{d}(M)\ge N_{3}+1$ for some integer $d\ge 1$, must $(M,\omega)$ have maximal volume growth, i.e. $\mathrm{Vol}(B_{r}(p_{0}))\ge c\,r^{2n}$ for some $c>0$ and all $r>0$, the Euclidean volume rate? (Q3) if $\dim\mathcal{O}_{d}(M)\ge N_{3}+1$ for some integer $d\ge 1$, must $M$ be biholomorphic to $\mathbb{C}^{n}$? (For $d=1$ all three reduce to Liu's known equality-rigidity theorem; the open content is $d\ge 2$.) The single overarching objective is to decide whether the sharp three-level rigidity structure of $\dim\mathcal{O}_{d}(M)$ - isometry with flat $\mathbb{C}^{n}$ at the top level, biholomorphism with $\mathbb{C}^{n}$ throughout the near-maximal window down to $N_{3}+1$, and no conclusion at or below $N_{3}$ - persists when nonnegative bisectional curvature is weakened to nonnegative holomorphic sectional curvature. The thresholds $N_{2}$ and $N_{3}$ are optimal in the bisectional case, and the example $\mathbb{C}^{n-1}\times\mathbb{CP}^{1}$ shows that $N_{3}$ cannot be lowered under holomorphic sectional curvature either.

References

1. Jianchun Chu, Jie Deng, Zihang Hao, Jian Li, “The rigidity of dimension estimate for holomorphic functions on Kähler manifolds”, arXiv:2510.14371 (v1: 2025-10-16, v2: 2026-03-25), DOI: 10.48550/arXiv.2510.14371, https://arxiv.org/abs/2510.14371
2. Gang Liu, “Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds”, Duke Mathematical Journal 165 (2016), no. 15, 2899–2919, DOI: 10.1215/00127094-3645009, https://doi.org/10.1215/00127094-3645009
3. Gang Liu, “On the volume growth of Kähler manifolds with nonnegative bisectional curvature”, Journal of Differential Geometry 102 (2016), no. 3, 485–500, DOI: 10.4310/jdg/1456754016, https://doi.org/10.4310/jdg/1456754016
4. Gang Liu, “On Yau’s uniformization conjecture”, Cambridge Journal of Mathematics 7 (2019), no. 1–2, 33–70, DOI: 10.4310/cjm.2019.v7.n1.a2, https://doi.org/10.4310/cjm.2019.v7.n1.a2
5. Man-Chun Lee, Luen-Fai Tam, “Chern–Ricci flows on noncompact complex manifolds”, Journal of Differential Geometry 115 (2020), no. 3, 529–564, DOI: 10.4310/jdg/1594260018, https://doi.org/10.4310/jdg/1594260018
6. Lei Ni, “A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature”, Journal of the American Mathematical Society 17 (2004), no. 4, 909–946, DOI: 10.1090/s0894-0347-04-00465-5, https://doi.org/10.1090/s0894-0347-04-00465-5
7. Bing-Long Chen, Xiao-Yong Fu, Le Yin, Xi-Ping Zhu, “Sharp dimension estimates of holomorphic functions and rigidity”, Transactions of the American Mathematical Society 358 (2006), no. 4, 1435–1454, DOI: 10.1090/s0002-9947-05-04105-x, https://doi.org/10.1090/s0002-9947-05-04105-x
8. Yuang Shi, “Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds”, arXiv:2511.01263 (2025), DOI: 10.48550/arXiv.2511.01263, https://arxiv.org/abs/2511.01263
9. Zeqing Miao, Hanyu Wu, Bo Yang, “Holomorphic functions on complete Hermitian manifolds with flat Chern connection”, arXiv:2608.01594 (2026), DOI: 10.48550/arXiv.2608.01594, https://arxiv.org/abs/2608.01594
10. Chengjie Yu, Chuangyuan Zhang, “Three circle theorem on almost Hermitian manifolds and applications”, arXiv:2110.12566 (2021), DOI: 10.48550/arXiv.2110.12566, https://arxiv.org/abs/2110.12566
11. Gang Liu, “A Brief Survey on Dimension Estimate of Holomorphic Functions on Kähler Manifolds”, The Journal of Geometric Analysis 33 (2023), no. 4, DOI: 10.1007/s12220-022-01160-2, https://doi.org/10.1007/s12220-022-01160-2
12. Shing-Tung Yau, “Open problems in geometry”, Proceedings of Symposia in Pure Mathematics 54, Part 1 (1993), 1–28, DOI: 10.1090/pspum/054.1/1216573, https://doi.org/10.1090/pspum/054.1/1216573