Conjecture: Upper bound on the second Laplacian eigenvalue on RP^n (n ≥ 3)

Prove that for each n ≥ 3, for any metric w g conformally equivalent to the round metric g on RP^n and normalized so that Vol_n(w) = Vol_n(1), the second nonzero Laplace–Beltrami eigenvalue λ_2(w) of −Δ_{w g} satisfies the inequality λ_2(w) < ((2n + 2)^{n/2} + 2n^{n/2})^{2/n}. The conjecture further specifies that the upper bound can be approached by a sequence of metrics degenerating to the disjoint union of a round RP^n and a round sphere whose radii have ratio √(2n + 2) : √n.

Background

The paper studies isoperimetric-type upper bounds for Laplace–Beltrami eigenvalues on real projective spaces RP^n within a fixed conformal class, normalized to preserve volume. Sharp results are known in two dimensions (n = 2), where the second eigenvalue achieves its maximal value via a degenerating sequence to a disjoint union of a round RP^2 and a round sphere.

Extending such sharp upper bounds to higher dimensions is conjectured. Conjecture 3 states a specific inequality for λ_2(w) in dimensions n ≥ 3 and describes the geometric degeneration (to RP^n ∪ S^n with a prescribed ratio of radii) expected to realize the bound. The present paper proves a dimension-dependent upper bound that approaches the conjectured value asymptotically as n → ∞ but does not resolve the conjecture exactly.