Strict inequality for the Bär-Hijazi-Lott invariant on non-spherical closed spin manifolds

Determine, for closed Riemannian spin manifolds (M^n, g, σ) that are not conformally equivalent to the round sphere (S^n, g0, σ0), whether the strict inequality λ_min^+(M, [g], σ) < (n/2) ω_n^{1/n} holds, where λ_min^+(M, [g], σ) := inf_{g' ∈ [g]} λ_1^+(D_{g'}) · vol(M, g')^{1/n} and ω_n is the volume of (S^n, g0).

Background

The paper studies the spinorial Yamabe-type equation and connects its normalized solutions to the conformal Bär-Hijazi-Lott invariant λ_min^+(M, [g], σ), defined as the infimum over a conformal class [g] of the first positive Dirac eigenvalue scaled by the n-th root of the volume. It is known that λ_min^+(M, [g], σ) ≤ (n/2) ω_n^{1/n} for all closed spin manifolds, with equality attained by the round sphere (S^n, g0, σ0).

The authors highlight that while specific sufficient conditions are known to ensure the strict inequality λ_min^+(M, [g], σ) < (n/2) ω_n^{1/n} (e.g., certain locally conformally flat settings and perturbations of the round metric on the sphere), a general characterization for closed spin manifolds not conformally equivalent to the sphere remains unresolved. This motivates determining precisely which non-spherical closed spin manifolds satisfy or fail the strict inequality.