Positive Transverse Bisectional Curvature

• Positive transverse bisectional curvature is a property where the transverse Kähler structure in Sasakian manifolds exhibits strictly positive curvature, ensuring rigidity and topological control.
• The Sasaki–Ricci flow preserves this curvature condition, with analytical tools such as Li–Yau–Hamilton inequalities guaranteeing long-term convergence and scalar curvature bounds.
• This curvature condition underpins uniformization results, leading to the classification of certain Sasakian manifolds and bridging geometric analysis with global CR-structure studies.

Positive transverse bisectional curvature is a geometric property arising in the study of Sasakian manifolds, which are odd-dimensional Riemannian manifolds equipped with a contact structure and a compatible metric making their structure transverse to the Reeb foliation formally Kähler. In the Sasakian context, the notion of bisectional curvature is defined on the transverse Kähler structure, and its positivity becomes a key rigidity and classification condition. This property not only governs the local geometry but also has significant global and topological implications, closely paralleling uniformization phenomena in Kähler geometry.

1. Transverse Curvature in Sasakian Geometry

Given a Sasakian manifold $(M^{2n+1}, \eta, \xi, \Phi, g)$, the contact subbundle is $D = \ker \eta \subset TM$. The transverse Kähler metric is $g^T = g|_D$, with associated Kähler form $\omega = d\eta$. The transverse Levi-Civita connection $\nabla^T$ on $D$ and its curvature tensor are

$R^T(X,Y,Z,W) = g^T(Rm^T(X,Y)Z, W).$

Transverse holomorphic bisectional curvature is defined as follows. For two $\Phi$-invariant real planes $\sigma_1 = \langle X, \Phi X \rangle$, $\sigma_2 = \langle Y, \Phi Y \rangle$ at $D_p$, set

$$U = \frac{1}{\sqrt{2}}(X - i\Phi X), \quad V = \frac{1}{\sqrt{2}}(Y - i\Phi Y),$$

then

$$B^T(\sigma_1, \sigma_2) := R^T(U, \bar U, V, \bar V) = \langle Rm^T(X, Y)Y, X \rangle + \langle Rm^T(X, \Phi Y)\Phi Y, X \rangle.$$

The positivity condition asserts that $B^T(\sigma_1, \sigma_2)>0$ for all choices of such planes. In local holomorphic coordinates $z^i$, the transverse curvature tensor takes the form

$$R^T_{i\bar{j}k\bar{\ell}} = -\partial_k \partial_{\bar \ell}(g^T_{i\bar j}) + g^{p\bar q}\, \partial_k g^T_{i\bar q}\, \partial_{\bar \ell}g^T_{p\bar j},$$

and the transverse Ricci form is $$\rho^T = -i\, \partial\bar\partial \log\det(g^T_{i\bar j})$$ (Chang et al., 15 Jan 2026, He, 2011).

2. Sasaki–Ricci Flow and Curvature Preservation

The Sasaki–Ricci flow evolves the transverse Kähler metric via

$\partial_t g^T = -\Ric^T,$

mirroring the Kähler–Ricci flow on Kähler manifolds. The evolution of the transverse curvature is governed by

$$\partial_t R^T_{i\bar j} = \Delta_B R^T_{i\bar j} + F_{i\bar j}(Rm^T),$$

where $F$ is a quadratic expression in $Rm^T$. The algebraic null-vector condition, originally due to Bando and Mok, ensures that nonnegative (or positive) transverse holomorphic bisectional curvature is preserved under the flow. If the initial $R^T$ is nonnegative everywhere and positive somewhere, strict positivity is obtained for all $t>0$ (He, 2011).

Perelman's $W$-functional, when restricted to basic (foliated) functions, remains monotone along the coupled flow involving $g^T$, auxiliary potential $f$, and time parameter $\tau$. Uniform scalar curvature and diameter bounds, as well as noncollapsing lower volume bounds for transverse metric balls, are established using these monotonicity and maximum principle techniques.

3. Li–Yau–Hamilton Inequalities and Long-Time Behavior

For the Sasaki–Ricci flow with positive transverse bisectional curvature, a Li–Yau–Hamilton matrix estimate holds: $$Z_{\alpha\bar\beta} = \partial_t R^T_{\alpha\bar\beta} + \nabla^T_\gamma R^T_{\alpha\bar\beta} X^\gamma + \nabla^T_{\bar\gamma} R^T_{\alpha\bar\beta} X^{\bar\gamma} + R^T_{\alpha\bar\gamma}R^T_{\gamma\bar\beta} + \frac{R^T_{\alpha\bar\beta}}{t} \ge 0$$ for any holomorphic Legendre tangent vector $X$. Tracing yields the scalar Harnack inequality: $$\partial_t R^T - \frac{|\nabla^T R^T|^2}{R^T} + \frac{R^T}{t} \ge 0,$$ implying that $t R^T$ is nondecreasing and establishing heat type monotonicity for scalar curvature (Chang et al., 15 Jan 2026).

Curvature bounds of the form $\|\nabla^m Rm^T\|(x,t) \le C_m t^{-(m+1)/2}$ and injectivity radius bounds proportional to $t^{1/2}$ arise, ensuring long-time existence and convergence of the flow in the presence of positive transverse bisectional curvature and maximal transverse volume growth (He, 2011).

4. Uniformization Theorem in Dimension Five

A central result is the CR-Yau uniformization for complete noncompact 5-dimensional Sasakian manifolds:

• Let $(M^5,\eta,\xi,\Phi,g)$ satisfy everywhere positive transverse holomorphic bisectional curvature and maximal transverse volume growth:

$\liminf_{r\to\infty} \frac{\Vol_\xi(B_\xi(x_0, r))}{r^4} > 0.$

• Then $M$ is CR-biholomorphic and diffeomorphic to the standard Heisenberg group $\HH_2=\C^2\times\R$ with its flat contact form and standard complex structure (Chang et al., 15 Jan 2026).

The proof proceeds via the following steps:

• Existence of nonconstant CR-holomorphic functions of polynomial growth by Cheeger–Colding theory and Hörmander $L^2$ estimates in the transverse setting.
• Running the Sasaki–Ricci flow to control curvature and volume; maximal volume growth and transverse curvature bounds lead to exhaustion by CR-biholomorphic charts.
• The leaf space modeled by CR-holomorphic functions is shown to be contractible and biholomorphic to $\C^2$ via Ramanujam's theorem, establishing $\C^2\times\R$ as the unique diffeomorphic and CR-equivalent structure (Chang et al., 15 Jan 2026).

5. Geometric and Topological Implications

Positive transverse bisectional curvature has strong implications for the global geometry and topology of Sasakian manifolds:

• In real dimension five, manifolds with this property and maximal volume growth are uniquely modeled by the Heisenberg group, which is flat in the CR sense.
• The injectivity radius lower bound and curvature decay guarantee simple connectivity at infinity and contractibility.
• For compact manifolds, the Sasaki–Ricci flow with positive initial transverse bisectional curvature converges (up to diffeomorphism pull-back) to a Sasaki–Ricci soliton, with all transverse curvature positivity preserved throughout the flow (He, 2011).

The equivalence between maximal volume growth and the existence of nontrivial CR-holomorphic functions of polynomial growth further establishes an interplay between geometric analysis and function theory on Sasakian manifolds.

6. Key Identities and Analytical Tools

Important analytical tools and identities:

• The transverse complex Monge–Ampère equation for the Sasaki–Ricci flow on basic potentials:

$$\partial_t\varphi = \log\frac{\det(g^T_{i\bar j}+\partial_i\partial_{\bar j}\varphi)}{\det(g^T_{i\bar j})},$$

with $\xi\varphi=0$.

• Evolution of the transverse scalar:

$$\partial_t R^T = \Delta_B R^T + |Rc^T|^2 \ge \Delta_B R^T.$$

• Harnack inequality for $0< t_1 < t_2$ and $x, y$:

$$R^T(x, t_1) \le R^T(y, t_2) \exp\left(\frac{d_{t_1}^2(x,y)}{4(t_2-t_1)}\right).$$

• Volume preservation under the Sasaki–Ricci flow: if $\Vol_\xi(B_\xi(p,r))\ge C r^{2n}$ initially, this persists for all $t$ (Chang et al., 15 Jan 2026, He, 2011).

7. Broader Context and Research Directions

Positive transverse bisectional curvature is a Sasakian analog of positive holomorphic bisectional curvature in Kähler geometry, central to modern uniformization and rigidity results. The extension of Yau’s uniformization phenomenon to the CR/Sasakian setting demonstrates a deep relationship among curvature, volume growth, and global structure. Methodologies involving the Cheeger–Colding theory, Perelman-inspired functionals, and sophisticated maximum principle techniques provide a template for future work in odd-dimensional and foliated geometric flows. A plausible implication is that higher-dimensional analogues and variants with weaker curvature bounds remain fertile ground for further study (Chang et al., 15 Jan 2026, He, 2011).