Holomorphic convexity and polynomial-growth holomorphic functions on complete Kähler manifolds with nonpositive bisectional curvature and Euclidean volume growth (Question 1.10 of Wu–Yang)

In a 2025 paper (published in the Transactions of the American Mathematical Society, 2026), Hanyu Wu and Bo Yang constructed an abundance of complete Kähler metrics with negative holomorphic bisectional curvature on total spaces of certain vector bundles over compact Kähler bases, proved dimension estimates for holomorphic functions of polynomial growth on such total spaces, and posed as their headline open question (Question 1.10) whether a complete noncompact Kähler manifold with nonpositive holomorphic bisectional curvature and Euclidean volume growth, in the sense that the limsup of Vol(B(p,r))/r^{2n} lies in (0,+∞), is necessarily holomorphically convex and necessarily admits nonconstant holomorphic functions of polynomial growth. Every known example satisfying the hypotheses is holomorphically convex with polynomial-growth functions, the authors' counterexamples without the volume-growth hypothesis fail it. The problem is open in both directions.

Background

A Kähler manifold is a complex manifold with a Riemannian metric compatible with its complex structure; its holomorphic bisectional curvature (BI) is the Kähler-geometric analogue of sectional curvature, assigning to two complex tangent lines the quantity $R(v,\bar v,w,\bar w)$ normalized by $|v|^2|w|^2$, and $BI\le 0$ means this mixed curvature is nonpositive for all pairs of tangent directions. Since the 1970s, Greene–Wu, Siu, and Yau have advocated studying how much complex function theory the curvature of a complete noncompact Kähler metric forces: whether such manifolds are holomorphically convex, admit separating or slowly growing holomorphic functions, or are Stein. A complex space is holomorphically convex if for every infinite discrete subset of it some holomorphic function is unbounded on that subset - the weakest standard notion of having enough holomorphic functions at infinity - and a Stein manifold is one biholomorphic to a closed submanifold of some $\mathbb{C}^N$, the strongest such notion. The classical rigidity theorem in this direction, due to Siu–Yau and extended by Mok–Siu–Yau, states that a complete simply-connected Kähler manifold of nonpositive sectional curvature whose scalar curvature decays pointwise faster than quadratically is holomorphically isometric to complex Euclidean space.

Wu and Yang (the source of this problem) recently constructed many complete Kähler metrics with $BI<0$ on total spaces of vector bundles $E=L^{\oplus r}$, where $L$ is a negative line bundle (a line bundle with a Hermitian metric of negative curvature form) over a compact Kähler base with $BI<0$. Their construction uses Calabi's ansatz, which builds complete metrics on such total spaces in the form $\pi^*\omega_g+\sqrt{-1}\,\partial\bar\partial u(h(v))$, where $h(v)$ is the fiber norm of a point $v$ and $u$ is a function of it. For a wider class of complete metrics on total spaces of negative line bundles they proved dimension estimates for holomorphic functions of polynomial growth: a holomorphic function $f$ has polynomial growth of order at most $d$ if $|f(q)|\le C(d(q,p)+1)^d$ for a fixed base point $p$, and $\mathcal{O}_d(X,\widetilde{g})$ denotes the vector space of such functions. Using a three-circle theorem of Gang Liu (a Kähler analogue of Hadamard's classical growth estimate on annuli), they showed $\dim\mathcal{O}_d$ is finite, bounded by sums of dimensions of spaces of sections $H^0(M,L^{-k})$, whenever the holomorphic sectional curvature in the radial fiber direction satisfies an integral decay condition $\int_0^\infty s\,k(s)\,ds<\infty$.

What the geometry alone, without the vector-bundle model, forces remained open. The authors show by example that $BI\le 0$ by itself is not enough: on $X=M\times\mathbb{C}$ with a suitable complete metric of nonpositive Gauss curvature on the $\mathbb{C}$ factor, every nonconstant holomorphic function grows at least exponentially in the geodesic distance - but that example has exponential volume growth. The natural remaining hypothesis is Euclidean volume growth, meaning $\limsup_{r\to\infty}\operatorname{Vol}(B(p,r))/r^{2n}\in(0,+\infty)$ where $B(p,r)$ are geodesic balls, $n=\dim_{\mathbb{C}}X$, and $r^{2n}$ is the growth rate of $\mathbb{C}^n$. All known examples with $BI\le 0$ and this volume growth are holomorphically convex and carry nonconstant polynomial-growth holomorphic functions: flat $\mathbb{C}^n$; the Calabi-type total spaces above (which are strongly pseudoconvex, hence holomorphically convex but not Stein, and carry nonconstant polynomial-growth functions by the source's Corollary 1.5); and the smooth affine quadric $\{z_1^2+z_2^2+z_3^2=t\}\subset\mathbb{C}^3$, $t\ne 0$, with the induced Euclidean metric, which Yang–Zheng showed satisfies $BI\le 0$ with Euclidean volume growth, its coordinate functions growing linearly. Whether this is forced or an accident of the examples is unknown. The contrast with the opposite sign is sharp: for nonnegative bisectional curvature with maximal volume growth there is an intensive uniformization program (recent contributions include the rigidity of dimension estimates by Chu–Deng–Hao–Li, the minimal-degree and volume-growth relations of Shi, and the uniformisation of positively curved complete Kähler surfaces by Datar–Pingali–Seshadri), while no comparable structure theory exists on the nonpositive side. Wu–Yang therefore posed, as the headline question of their concluding discussion, the question stated below.

Problem Statement

Let $(X,\widetilde{g})$ be a complete noncompact Kähler manifold of complex dimension $n$ with nonpositive holomorphic bisectional curvature, $BI(\widetilde{g})\le 0$. Assume $\widetilde{g}$ is of at most Euclidean volume growth in the sense that

$\limsup_{r\to\infty}\frac{\operatorname{Vol}(B(p,r))}{r^{2n}}\in(0,+\infty),$

where $B(p,r)$ denotes the geodesic ball with radius $r$ centered at some point $p\in X$.

(1) Is $X$ necessarily holomorphically convex?

(2) Does $X$ necessarily admit nonconstant holomorphic functions of polynomial growth, that is, a nonconstant $f\in\mathcal{O}(X)$ with $|f(q)|\le C(d_{\widetilde{g}}(q,p)+1)^d$ for some constants $C>0$ and $d\ge 0$?

This is Question 1.10 of Wu–Yang, stated in its original generality: no restriction to total spaces of vector bundles, bounded curvature, simply-connectedness, or any decay condition on the curvature is assumed. Both parts are universal claims over all such $(X,\widetilde{g})$, and they are logically independent: failure of holomorphic convexity does not preclude the existence of nonconstant polynomial-growth holomorphic functions, and absence of such functions does not preclude holomorphic convexity.

References

1. Hanyu Wu, Bo Yang, Negatively curved Kähler metrics on total spaces of a class of vector bundles, Transactions of the American Mathematical Society (2026), DOI: 10.1090/tran/9788, https://doi.org/10.1090/tran/9788; preprint arXiv:2508.16354, https://arxiv.org/abs/2508.16354
2. Hanyu Wu, Bo Yang, Kähler geometry on total spaces of vector bundles over elliptic curves, preprint (2025), arXiv:2511.08906, https://arxiv.org/abs/2511.08906
3. Yum-Tong Siu, Shing-Tung Yau, Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay, Annals of Mathematics (2) 105 (1977), 225–264, DOI: 10.2307/1970998, https://doi.org/10.2307/1970998
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5. Bo Yang, Fangyang Zheng, Examples of complete Kähler metrics with nonnegative holomorphic sectional curvature, Journal of Geometric Analysis 33 (2023), Paper No. 47, DOI: 10.1007/s12220-022-01081-0, https://doi.org/10.1007/s12220-022-01081-0
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