On the Sasakian Structure of Manifolds with Nonnegative Transverse Bisectional Curvature
Shu-Cheng Chang, Yingbo Han, Chien Lin, Chin-Tung Wu
TL;DR
The paper addresses a CR analogue of Yau's uniformization conjecture for complete noncompact Sasakian manifolds with nonnegative transverse bisectional curvature, showing that in dimension five a manifold with positive curvature and maximal volume growth is CR-biholomorphic to the Heisenberg group $\mathbb{H}_{2}$. The authors develop a Sasaki–Ricci flow framework, establish Li–Yau–Hamilton type Harnack inequalities for the transverse geometry, and derive long-time existence and curvature estimates under maximal volume growth. They then construct nonconstant CR holomorphic functions of polynomial growth and use CR tangent cones at infinity to obtain proper CR maps, leading to a CR biholomorphism with affine products and, in dimension five, to $\mathbb{H}_{2}$. The results connect Sasakian geometry, CR analysis, and algebraic geometry to realize a CR Yau-type uniformization in the maximal-volume-growth setting, with Ramanujam’s theorem bridging to affine algebraic surfaces. Overall, the work advances the understanding of CR uniformization via Sasaki–Ricci flow and polynomial-growth CR functions, yielding explicit classification in the five-dimensional case.
Abstract
In this paper, we concern with the Sasaki analogue of Yau uniformization conjecture in a complete noncompact Sasakian manifold with nonnegative transverse bisectional curvature. As a consequence, we confirm that any $5$-dimensional complete noncompact Sasakian manifold with positive transverse bisectional curvature and the maximal volume growth must be CR-biholomorphic to the standard Heisenberg group $\mathbb{H}_{2}$ which can be stated as the standard contact Euclidean $5$-space $\mathbb{R}^{5}$.