On the Sasakian Structure of Manifolds with Nonnegative Transverse Bisectional Curvature

TL;DR

This paper investigates the geometric structure of noncompact Sasakian manifolds with nonnegative transverse bisectional curvature, confirming a classification result in five dimensions related to the Sasaki analogue of Yau's uniformization conjecture.

Contribution

It proves that 5-dimensional complete noncompact Sasakian manifolds with positive transverse bisectional curvature and maximal volume growth are CR-biholomorphic to the standard Heisenberg group.

Findings

Confirmed the Sasaki analogue of Yau's uniformization conjecture in 5D.

Established CR-biholomorphism to the standard Heisenberg group under given conditions.

Characterized the geometric structure of certain noncompact Sasakian manifolds.

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 $\mathbb{R}^{5}$.