Sasaki Analogue of Yau Uniformization Conjecture

• The paper establishes a Sasaki analogue to Yau’s uniformization theorem, showing compact Sasaki manifolds with positive transverse bisectional curvature are diffeomorphic to (weighted) spheres.
• It employs Sasaki–Ricci flow and volume minimization techniques to preserve curvature positivity and achieve uniformization through controlled Reeb field deformations.
• The work extends classical uniformization from Kähler to Sasaki geometry, offering classification insights for both compact and noncompact cases in CR and contact settings.

The Sasaki analogue of Yau’s uniformization conjecture is a central theme in modern differential geometry, elucidating the structure of Sasaki manifolds under natural curvature conditions and their relation to Kähler geometry. This conjecture and its resolution—primarily through the work of He–Sun—establish a Sasakian parallel to the celebrated Frankel–Mori–Siu–Yau (Yau uniformization) theorem, extending the classification and uniformization phenomena from Kähler to Sasakian, contact, and CR settings. It governs both the compact and noncompact cases and encompasses the influence of holomorphic bisectional curvature, Ricci flow techniques, and rigidity of model spaces.

1. Core Concepts and Definitions

A Sasaki manifold of dimension $2n+1$ is described by a quadruple $(\eta, \xi, \Phi, g)$ where $\eta$ is a contact 1-form, $\xi$ is the Reeb vector field ($\iota_\xi\eta=1$, $\iota_\xi d\eta=0$), $\Phi$ is an endomorphism satisfying $\Phi^2=-\mathrm{Id}+\eta\otimes\xi$, and $g$ is the Riemannian metric

$$g = \eta\otimes\eta + \frac{1}{2} d\eta(\cdot, \Phi\cdot).$$

The Reeb flow induces a one-dimensional foliation $\mathcal{F}_\xi$ whose normal bundle (the contact distribution $\mathcal{D} = \ker\eta$) admits a transverse Kähler structure:

• Transverse metric $g^T$ induced from $g$.
• Transverse Kähler form $\omega^T = \frac{1}{2} d\eta$.
• Transverse complex structure inherited from $\Phi$.

The transverse holomorphic bisectional curvature (THBC) is defined for basic complex directions $X, Y$ by

$$R^T(X, \bar X, Y, \bar Y) = g^T\left(\nabla^T_X\nabla^T_{\bar X}Y - \nabla^T_{\bar X}\nabla^T_X Y - [X, \bar X]^T, \bar Y\right),$$

and positivity (resp. nonnegativity) is defined as $R^T(X, \bar X, Y, \bar Y) > 0$ (resp. $\ge 0$) for all nonzero $X, Y$.

A Kähler cone over $M$ is $(C(M), \bar g, J)$ with $C(M) = M \times \mathbb{R}^+$, metric $\bar g = dr^2 + r^2 g$, and complex structure $J$, intertwining the Sasakian geometry of $M$ with the Kähler geometry of the cone.

2. Statements of the Sasaki Uniformization Conjecture

The compact case, as established by He–Sun (He et al., 2012), can be summarized as follows:

• Theorem (He–Sun): If $(M^{2n+1}, \eta, \xi, \Phi, g)$ is a compact, simply connected Sasaki manifold with positive transverse bisectional curvature, then its Kähler cone $(C(M), J)$ is biholomorphic to $\mathbb{C}^{n+1} \setminus \{0\}$ and $M$ is diffeomorphic to $S^{2n+1}$ equipped with a simple Sasaki metric, deformable through transverse Kähler and Reeb deformations to the standard round sphere.

In the orbifold (quasi-regular) case, $M$ is a weighted Sasaki sphere and the quotient $M/U(1)$ is a weighted projective space.

For noncompact settings, recent work (Chang et al., 15 Jan 2026) confirms:

• Any complete noncompact $5$-dimensional Sasakian manifold with positive transverse bisectional curvature and maximal volume growth is CR–biholomorphic to the standard Heisenberg group $\mathbb{H}_2=\mathbb{C}^2 \times \mathbb{R}$ (standard contact Euclidean $\mathbb{R}^5$).

3. Proof Strategies and Analytic Framework

The analytic approach to the uniformization conjecture for Sasaki manifolds involves two primary flows:

Step 1: Sasaki–Ricci Flow.

The transverse Kähler–Ricci flow

$$\frac{\partial}{\partial t} g^T_{i\bar j} = - R^T_{i\bar j} + g^T_{i\bar j}, \quad g^T(0) = g^T$$

preserves positivity of the transverse bisectional curvature, and long-time existence and subconvergence to a Sasaki–Ricci soliton is ensured via Perelman-type entropy and maximum-principle estimates.

Step 2: Volume Minimization and Reeb Field Deformation.

The volume functional on the Sasaki cone,

$$\operatorname{Vol}(\xi) = \int_M \frac{\eta \wedge (d\eta)^n}{n!}$$

is minimized along its negative gradient (the transverse Futaki invariant) to produce a Sasaki–Einstein metric (modulo positivity). This flow leads, in the compact positive curvature case, to the round sphere by rigidity theorems (Tanno/Goldberg–Kobayashi).

For noncompact manifolds with maximal volume growth and $R^T > 0$, the approach uses CR heat flows, $L^2$-estimates to construct polynomial CR-holomorphic functions, Sasaki–Ricci flow with Li–Yau–Hamilton estimates, and Cheeger–Gromov convergence to explicit model spaces (Chang et al., 15 Jan 2026).

4. Classification: Generalized Frankel Conjecture in Sasaki Geometry

The He–Sun classification (He et al., 2012) provides a Sasaki analogue to Mok’s splitting theorem:

• Theorem: Any compact Sasaki manifold with nonnegative transverse bisectional curvature splits (after finite cover and deformation) as a join of:
  1. Flat Sasaki Euclidean pieces $(\mathbb{R}^{2k+1}, \eta_{\text{flat}})$,
  2. Weighted Sasaki spheres $(S^{2m_0+1}, \eta_{\text{wSph}})$ (Kähler quotients are weighted projective spaces),
  3. Regular Sasaki manifolds over irreducible Hermitian symmetric spaces of rank $\geq 2$.

Positive transverse Ricci curvature implies transverse irreducibility, prohibiting irregular join constructions for Sasaki–Einstein manifolds.

5. Extensions, Rigidity, and Obstructions

• Rigidity: The compact positive case yields only (weighted) round spheres, mirroring the uniqueness of $\mathbb{C}P^n$ in Kähler geometry under positive holomorphic bisectional curvature, and extends to certain Sasaki orbifolds.
• Obstructions: The basic first Chern class $c_1^B$ must be positive and the contact bundle's Chern class must vanish for the normalization $\operatorname{Ric}^T=(2n+2)\omega^T$.
• Non-Einstein Solitons: There exist non-Einstein Sasaki–Ricci solitons of positive curvature, e.g., on weighted spheres, highlighting the genuine novelty over the Kähler setting.

6. Parallelism with Kähler and Uniformization Theorems

The Sasaki uniformization framework generalizes and refines classical results:

Setting	Model Uniformization Space	Curvature Assumption
Kähler (Yau/Mori/Siu)	$\mathbb{CP}^n$	$HBC>0$
Sasaki (compact)	$S^{2n+1}$, weighted spheres	$THBC>0$
Kähler (Mok)	Hermitian symmetric factors	$HBC\ge0$
Sasaki (general)	Flat, symmetric, weighted	$THBC\ge0$
Sasaki (noncompact)	$\mathbb{H}_n$ ($\mathbb{C}^n \times \mathbb{R}$)	$THBC>0$, max vol

This parallelism extends to orbifold quotients and isomorphism criteria via deformation, basic Chern class conditions, and, in the compact case, analogues of Simpson’s theory using Higgs bundles and variation of Hodge structure (Kasuya et al., 2022).

7. Open Problems and Future Directions

For noncompact Sasaki manifolds in higher dimension, the conjecture reduces to open questions about affine pseudoconvex domains: classification of $\Omega \subset \mathbb{C}^n$ such that $M^{2n+1} \cong \Omega \times \mathbb{R}$ is CR–biholomorphic to the standard Heisenberg group remains unresolved for $n > 2$. Topological obstructions related to exotic $\mathbb{R}^4$ structures also arise. Possible generalizations concern orbifold singularities, weaker curvature conditions, and the exploration of quasi-Einstein or constant scalar curvature Sasaki metrics (He et al., 2012, Kasuya et al., 2022, Chang et al., 15 Jan 2026).