On the Rosenberg-Stolz Conjecture for $ X \times \mathbb{R}^{2} $ and Its Application in Complex Geometry

Abstract

Let $ X $ be an oriented, closed manifold with $ \dim X \geqslant 2 $. In this article, we give both Riemannian geoemtry and complex geometry results on (sub)manifolds of the type $ X \times \mathbb{C}^{k} $ or $ X \times \mathbb{R}^{k} $. For Riemannian geometry side, we show that if $ X \times \mathbb{C} = X \times \mathbb{R}^{2} $ admits a Riemannian metric $ g $ with uniformly positive scalar curvature and bounded curvature, such that some novel conformally invariant $ g $-angle condition is satisfied, then there exists a complete metric $ \tilde{g} $ conformal to $ g $ such that $ \tilde{g} |_{X} $ has positive scalar curvature. This Riemannian path implies a complex geometry result: we show that if the complex manifold $ X \times \mathbb{C} $ admits a Hermitian metric $ \omega $ whose associated Riemannian metric $ g $ has uniformly positive scalar curvature and is of bounded curvature, then $ X \times \mathbb{C} $ admits a Hermitian metric $ \tilde{\omega} $ with positive Chern scalar curvature, provided that some $ g $-angle condition is satisfied. The Riemannian geometry result partially answers a 1994 Rosenberg-Stolz conjecture in all dimensions. The complex geometry result extends a result of XiaoKui Yang from compact Hermitian manifolds to noncompact Hermitian manifolds of type $ X \times \mathbb{C} $. We further generalize both the Riemannian and complex geometry results to $ X \times \mathbb{R}^{k} $ or $ X \times \mathbb{C}^{k} $ for any $ k \geqslant 1 $ by imposing a generalized conformally invariant angle condition.