Relationship between the spectral invariant ρ^∘ and scalar-curvature invariants such as the Yamabe invariant

Investigate and characterize the relationship between the spectral invariant ρ^∘(Y, m_𝔰, g) defined in Definition 5.2 from S^1-equivariant monopole Floer homology and scalar-curvature-based invariants of the same 3-manifold Y, such as the Yamabe invariant; in particular, determine whether there exist inequalities, bounds, or equalities linking ρ^∘ to the Yamabe invariant and identify geometric or topological conditions under which such relationships hold.

Background

The spectral invariant ρ^∘(Y, m_𝔰, g) depends on the metric and arises from an ℝ-filtration of monopole Floer chain complexes induced by the Chern–Simons–Dirac functional. The main application shows ρ^∘ provides lower bounds for the scalar curvature functional on 4-dimensional ribbon rational homology cobordisms and can obstruct the existence of positive scalar curvature metrics.

Motivated by these scalar-curvature interactions, the author explicitly raises the question of how ρ^∘ relates to other scalar-curvature invariants of Y, notably the Yamabe invariant, and defers a systematic paper to future work.