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 $\dim X⩾2$. 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 $k⩾1$ by imposing a generalized conformally invariant angle condition.

26 pages, all comments are welcome. V2 fixed minor typos and changed the title to emphasize the Riemannian geometry result