Sharp thresholds for Escobar and Gagliardo-Nirenberg functionals: the Escobar-Willmore mass, geometric selection, and compactness trichotomy

Abstract: We develop a unified quantitative framework for sharp threshold phenomena in boundary-critical variational problems on compact Riemannian manifolds, covering the Escobar quotient and Gagliardo-Nirenberg inequalities. Via transfer-stability-reduction, we obtain attainment-versus-bubbling alternatives, $H^1$-compactness, and finite-dimensional reductions. Geometric selection is governed by mean curvature $H_g$ and a Willmore-type anisotropy from $|\mathring{\mathrm{II}}|^2$. At hemisphere threshold $S_\ast=C^*_{\mathrm{Esc}}(\mathbb S^n_+)$ for $n\ge5$ on $H_g\equiv0$, we identify a renormalized boundary mass $\mathfrak R_g=κ1(n)\,\mathrm{Ric}_g(ν,ν)+κ_2(n)\,\mathrm{Scal}{g|\partial M}+κ3(n)\,|\mathring{\mathrm{II}}|^2$, $κ_3(n)<0$, yielding one-bubble expansions and energy-only estimators. Threshold dichotomy: if the first nonvanishing coefficient among ${ρ_n^{\mathrm{conf}}H_g,\mathfrak R_g,Θ_g}$ is negative somewhere, then $C^*{\mathrm{Esc}}(M,g)<S_\ast$ and sequences are precompact. At threshold, blow-up concentrates where $H_g$ is critical; on $H_g\equiv0$, stationarity forces $\mathfrak R_g(p)=\nabla_\partial\mathfrak R_g(p)=0$. If $H_g$ is Morse and $\mathfrak R_g\>0$ at all critical points, no bubbling occurs. In multi-bubble regime ($n\ge5$), dynamics governed by $\mathcal W_k=\sum_{i=1}^k\mathfrak R_g(x_i)$ produce $k$-bubble critical points at levels $k^{1/(n-1)}S_\ast$. In the degenerate case we obtain conformal hemispherical rigidity. The GN track yields analogous dichotomies and resolves a question of Christianson et al.: the sharp constant with small Dirichlet windows diverges at optimal capacitary rate, relating threshold to spectral/isoperimetric invariants. Applications include entropy inequalities for fast diffusion, curvature-driven NLS ground states, and (in $n=2$) Euler characteristic recovery from GN measurements.