Overview
This study investigates Gromov's exact-lift two-form method in the context of scalar-curvature geometry. The research focuses on closed Riemannian spin manifolds that possess a homologically non-trivial closed two-form, where the lift of this form to the universal cover is exact. The core of the method involves using Gromov's twisted \(L^2\)-index to generate harmonic spinors, which are then employed to analyze the geometric properties of the manifold.
Research Context
The study operates within the theoretical framework of scalar-curvature geometry. Key concepts include Gromov's exact-lift two-form method and the application of Gromov's twisted \(L^2\)-index. The geometric structures under consideration are closed Riemannian spin manifolds. A specific condition for these manifolds is the presence of a homologically non-trivial closed two-form whose lift to the universal cover is exact. The research explores the relationship between the scalar curvature of such manifolds and the bottom of the spectrum of their universal Riemannian covering.
Approach
The research approach involves several methodological steps. Initially, Gromov's exact-lift two-form method is applied to the specified Riemannian spin manifolds. The two-form is incorporated into the analysis through Gromov's twisted \(L^2\)-index. This index is instrumental in producing harmonic spinors, specifically for a family of small unitary twists. The analysis then proceeds to examine the equality case of the scalar-curvature comparison. This examination involves interpreting the refined Kato equality defect using a conformal approach. Subsequently, the generated harmonic spinors are utilized to construct a parallel spinor. This parallel spinor is constructed with respect to a suitable conformally related metric. Finally, this construction is used to deduce properties of the original metric.
Findings
- A sharp hyperbolic scalar-curvature comparison was proven for closed Riemannian spin manifolds carrying a homologically non-trivial closed two-form whose lift to the universal cover is exact. This comparison is made with the bottom of the spectrum of the universal Riemannian covering.
- Gromov's twisted \(L^2\)-index was found to produce harmonic spinors for a family of small unitary twists.
- In the equality case, the refined Kato equality defect was interpreted conformally.
- The harmonic spinors were used to construct a parallel spinor with respect to a suitable conformally related metric.
- This construction implies that the original metric is Einstein in the equality case.
- In the positive-spectrum case, the method suggests that the universal cover is real hyperbolic.