CR Yau uniformization conjecture: is every complete noncompact Sasakian manifold with positive transverse bisectional curvature CR-biholomorphic to the Heisenberg group?

The CR Yau uniformization conjecture in Sasakian geometry asks whether every complete noncompact Sasakian $(2n+1)$-manifold with positive transverse holomorphic bisectional curvature is CR-biholomorphic to the standard Heisenberg group $\mathbb{H}_n=\mathbb{C}^n\times\mathbb{R}$ — the odd-dimensional, contact/CR counterpart of Yau's uniformization conjecture for complete noncompact Kähler manifolds with positive bisectional curvature. The conjecture was proposed by Chang–Han–Li–Lin and is restated as Conjecture 2 of Chang–Han–Lin–Wu (arXiv:2601.10017, 2026), which proves it only in dimension five ($n=2$) under the additional hypothesis of maximal volume growth of horizontal balls.

Background

Yau's uniformization conjecture is a central open problem in complex differential geometry: it predicts that a complete noncompact Kähler manifold (a complex manifold with a compatible Riemannian metric whose associated 2-form is closed) whose holomorphic bisectional curvature (a Kähler curvature notion that simultaneously generalizes sectional and Ricci curvature) is positive must be biholomorphic to complex Euclidean space $\mathbb{C}^m$. Under the additional assumption of maximal volume growth — geodesic balls having Euclidean-order volume, $\mathrm{Vol}(B(p,r))\geq Cr^{2m}$ — the conjecture was confirmed by Gang Liu (2019) through elliptic methods and by Lee–Tam (2020) through the Chern–Ricci flow (a parabolic deformation of Hermitian metrics), after bounded-curvature versions by Chen–Zhu (2003) and Chau–Tam (2006); without volume-growth hypotheses the Kähler conjecture remains open, with only partial results in the non-maximal case (Chen–Zhu 2018). A recent study of the volume-growth function (Shi, arXiv:2511.01263) still frames these two proofs as the state of the art for the volume-growth case, confirming that the unrestricted Kähler problem is open.

Sasakian geometry is the odd-dimensional analogue of Kähler geometry. A Sasakian manifold is a $(2n+1)$-dimensional Riemannian manifold $(M,g)$ whose metric cone $(\mathbb{R}^{+}\times M, dr^{2}+r^{2}g)$ is Kähler. It carries a distinguished unit-length Killing vector field $\xi$ (the Reeb field) with dual contact form $\eta$; the metric splits as $g=g^{T}+\eta\otimes\eta$, and the one-dimensional foliation generated by $\xi$ (the Reeb foliation) has local leaf spaces that are complex $n$-manifolds carrying a transverse Kähler metric $g^{T}$. The transverse holomorphic bisectional curvature of $M$ is the holomorphic bisectional curvature of $g^{T}$ on these leaf spaces. Every Sasakian manifold is simultaneously a strictly pseudoconvex CR manifold: the contact distribution $D=\ker\eta$ carries a complex structure, the pseudohermitian torsion vanishes, and a CR-biholomorphism is a diffeomorphism whose differential is complex-linear and CR-holomorphic on $D$. The model space is the Heisenberg group $\mathbb{H}_{n}=\mathbb{C}^{n}\times\mathbb{R}$ with its standard contact structure — equivalently, the standard contact Euclidean space $\mathbb{R}^{2n+1}$ — which plays the role that $\mathbb{C}^{n}$ plays on the Kähler side. For compact Sasakian manifolds the picture is complete: He (2013) developed the Sasaki-Ricci flow (the transverse analogue of the Kähler-Ricci flow), and He–Sun (2016) proved the Sasaki version of the Frankel conjecture: a simply connected closed Sasakian manifold with positive transverse bisectional curvature is CR-equivalent to the round sphere $S^{2n+1}$.

The noncompact case is the subject of an ongoing program by Chang and collaborators. Chang–Han–Lin (2021) proved a CR analogue of Riemann's rigidity theorem: for the space $\mathcal{O}^{CR}_{d}(M)$ of CR-holomorphic functions of polynomial growth of degree at most $d$, one has $\dim_{\mathbb{C}}\mathcal{O}^{CR}_{d}(M)\leq\dim_{\mathbb{C}}\mathcal{O}^{CR}_{d}(\mathbb{H}_{n})$, with equality for some $d$ if and only if $M$ is CR-equivalent to $\mathbb{H}_{n}$ (the supporting three-circle theorem is in Chang–Han–Lin 2019). Chang–Han–Li–Lin (2023) then proved that under the CR maximal volume growth condition — $\mathrm{Vol}(B_{\xi}(p,r))\geq Cr^{2n}$ for the horizontal balls $B_{\xi}$ (the balls for the transverse Kähler metric $g^{T}$ on the contact distribution $D=\ker\eta$) — nonconstant CR-holomorphic functions of polynomial growth exist, and consequently $M$ is CR-biholomorphic to $\Sigma\times\mathbb{R}$ for an affine algebraic variety $\Sigma$. The source (Chang–Han–Lin–Wu, 2026) pushes this to a genuine uniformization statement: under maximal volume growth and nonnegative transverse bisectional curvature, $M$ is CR-biholomorphic to $\Omega\times\mathbb{R}$ for a pseudoconvex domain $\Omega\subset\mathbb{C}^{n}$ (Theorem 1.1(1)); and if moreover $n=2$ and the transverse bisectional curvature is positive, then $M$ is CR-biholomorphic to $\mathbb{H}_{2}$ (Theorem 1.1(2)). The proof establishes long-time existence of the Sasaki-Ricci flow with derivative curvature estimates (Theorem 1.2), shows that the averaged transverse scalar curvature decay bound $\int R^{T}\leq C/(1+r^{2})$ follows automatically from maximal volume growth (Theorem 1.4), develops Li–Yau–Hamilton-type Harnack inequalities for the flow, and completes the $n=2$ case using Ramanujam's (1971) theorem that a smooth contractible complex algebraic surface simply connected at infinity is isomorphic to the affine plane $\mathbb{C}^{2}$. The general conjecture — the CR counterpart of Yau's — remains unresolved.

Problem Statement

Conjecture (CR Yau uniformization conjecture, Chang–Han–Li–Lin; restated as Conjecture 2 of Chang–Han–Lin–Wu, arXiv:2601.10017). Let $M$ be a complete noncompact Sasakian $(2n+1)$-manifold whose transverse holomorphic bisectional curvature is positive. Must $M$ be CR-biholomorphic to the standard Heisenberg group $\mathbb{H}_{n}=\mathbb{C}^{n}\times\mathbb{R}$ with its standard CR/contact structure? The conjecture is stated without any volume-growth hypothesis: an affirmative answer establishes CR-biholomorphism to $\mathbb{H}_{n}$ itself under exactly the stated hypotheses — completeness, noncompactness, and positivity of the transverse holomorphic bisectional curvature — and a refutation exhibits a complete noncompact Sasakian $(2n+1)$-manifold with positive transverse holomorphic bisectional curvature that is not CR-biholomorphic to $\mathbb{H}_{n}$. The known partial results are strictly weaker and do not resolve the conjecture: under the additional hypothesis of maximal volume growth of the horizontal balls $B_{\xi}$ (the balls for the transverse Kähler metric $g^{T}$), $\mathrm{Vol}(B_{\xi}(p,r))\geq C_{1}r^{2n}$ for some fixed point $p\in M$, all $r\geq0$, and a constant $C_{1}>0$, the conjecture is currently proven only for $n=2$ and remains open for $n=1$ and for $n\geq3$; without volume-growth hypotheses it is open in every dimension $n\geq1$, including $n=2$. Conclusions such as CR-biholomorphism to a product $\Omega\times\mathbb{R}$ with $\Omega$ a pseudoconvex domain, or to $\Sigma\times\mathbb{R}$ with $\Sigma$ an affine algebraic variety, are likewise known strictly weaker partial results.

References

1. Shu-Cheng Chang, Yingbo Han, Chien Lin, Chin-Tung Wu, On the Sasakian Structure of Manifolds with Nonnegative Transverse Bisectional Curvature, arXiv:2601.10017 (2026), https://arxiv.org/abs/2601.10017
2. Shu-Cheng Chang, Yingbo Han, Nan Li, Chien Lin, Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian manifolds, Journal für die reine und angewandte Mathematik 802 (2023), 223–254, DOI 10.1515/crelle-2023-0046, https://doi.org/10.1515/crelle-2023-0046
3. Shu-Cheng Chang, Yingbo Han, Chien Lin, On the sharp dimension estimate of CR holomorphic functions in Sasakian manifolds, International Mathematics Research Notices 2021 (2021), no. 17, 12888–12924, DOI 10.1093/imrn/rnz164, https://doi.org/10.1093/imrn/rnz164
4. Shu-Cheng Chang, Yingbo Han, Chien Lin, On the three-circle theorem and its applications in Sasakian manifolds, Calculus of Variations and Partial Differential Equations 58 (2019), no. 3, Art. 101, DOI 10.1007/s00526-019-1538-8, https://doi.org/10.1007/s00526-019-1538-8
5. Weiyong He, Song Sun, Frankel conjecture and Sasaki geometry, Advances in Mathematics 291 (2016), 912–960, DOI 10.1016/j.aim.2015.11.053, https://doi.org/10.1016/j.aim.2015.11.053
6. Weiyong He, The Sasaki-Ricci flow and compact Sasakian manifolds of positive transverse holomorphic bisectional curvature, Journal of Geometric Analysis 23 (2013), 1876–1931, DOI 10.1007/s12220-012-9311-7, https://doi.org/10.1007/s12220-012-9311-7
7. 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
8. 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
9. Bing-Long Chen, Xi-Ping Zhu, On complete noncompact Kähler manifolds with positive bisectional curvature, Mathematische Annalen 327 (2003), no. 1, 1–23, DOI 10.1007/s00208-003-0385-1, https://doi.org/10.1007/s00208-003-0385-1
10. Albert Chau, Luen-Fai Tam, On the complex structure of Kähler manifolds with nonnegative curvature, Journal of Differential Geometry 73 (2006), no. 3, 491–530, DOI 10.4310/jdg/1146169936, https://doi.org/10.4310/jdg/1146169936
11. Binglong Chen, Xiping Zhu, Yau's uniformization conjecture for manifolds with non-maximal volume growth, Acta Mathematica Scientia, Series B 38 (2018), no. 5, 1468–1484, DOI 10.1016/s0252-9602(18)30828-2, https://doi.org/10.1016/s0252-9602(18)30828-2
12. C. P. Ramanujam, A topological characterisation of the affine plane as an algebraic variety, Annals of Mathematics 94 (1971), no. 1, 69–88, DOI 10.2307/1970735, https://doi.org/10.2307/1970735
13. Yuang Shi, Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds, arXiv:2511.01263 (2025), https://arxiv.org/abs/2511.01263