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

TL;DR

The paper develops a novel conformal-geometry pathway to the Rosenberg–Stolz conjecture for noncompact products $X \times \mathbb{R}^2$ and derives implications for complex geometry on $X \times \mathbb{C}$. By introducing a conformally invariant angle condition and an auxiliary dimension (via $\mathbb{S}^1$), the authors set up an elliptic PDE on a closed manifold to construct a conformal factor that transfers positive scalar curvature from the ambient space to the base $X$. This yields a complete conformal metric $\tilde{g}$ with $R_{\iota_{\zeta}^{*}\iota_{\xi}^{*}\tilde{g}} > 0$ on $X$, providing a partial resolution of Rosenberg–Stolz in all dimensions and enabling a complex-geometric analogue: if $(X \times \mathbb{C},J,\omega)$ has a complete Hermitian metric with $R_g>0$ and the angle condition, then there exists a Hermitian metric with positive Chern scalar curvature on $X \times \mathbb{C}$, extended to the noncompact setting. The framework generalizes naturally to $X \times \mathbb{R}^k$ or $X \times \mathbb{C}^k$ for any $k\ge1$, broadening the impact to cylindrical and line-bundle-type models in complex geometry. Overall, the approach connects ambient positivity, conformal deformation, and extrinsic Gauss–Codazzi data to obtain intrinsic positivity results on base manifolds and their complex-analytic structures.

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 $ ω$ whose associated Riemannian metric $ g $ has uniformly positive scalar curvature and is of bounded curvature, then $ X \times \mathbb{C} $ admits a Hermitian metric $ \tildeω $ 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.

Theorems & Definitions (31)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Example 2.1
• Remark 2.1
• Lemma 3.1
• proof
• Remark 3.1