Attaining the optimal constant for higher-order Sobolev inequalities on manifolds via asymptotic analysis

Let $(M,g)$ be a closed Riemannian manifold of dimension n, and k≥ 1 an integer such that $n>2k$. We show that there exists B₀>0 such that for all u ∈ Hᵏ(M), \[\|u\|_{L2^(M)}^2 ≤ K_0^2 ∫_M |Δ_gk/2 u|^2 \,dv_g + B_0 \|u\|_{Hᵏ⁻¹(M)}^2,\] where 2^ = 2n/n-2k and Δg = -divg(∇·). Here K₀ is the optimal constant for the Euclidean Sobolev inequality (∫Rⁿ |u|2⁾2/2^ ≤ K₀² ∫Rⁿ |∇ᵏ u|² for all u ∈ Cc^∞(Rⁿ). This result is proved as a consequence of the pointwise blow-up analysis for a sequence of positive solutions (u_α)_α to polyharmonic critical non-linear equations of the form (Δg + α)ᵏ u = u2⁻1 in M. We obtain a pointwise description of u_α, with explicit dependence in α as α→ ∞.