Sharp thresholds for the Escobar functional: the Escobar-Willmore mass, geometric selection, and compactness trichotomy

We study the hemisphere threshold for the conformally covariant Escobar functional on compact Riemannian manifolds $(M^n,g)$ with boundary. The near-threshold landscape is organized by boundary invariants: the first-order coefficient ${\rho }_n^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}}{H}_g$ vanishes identically, so the leading obstruction is a renormalized boundary mass ${\mathfrak{R}}_g$ (second order, $n\ge 5$), followed by a cubic invariant ${\mathrm{\Theta }}_g$ (third order, $n\ge 6$), with a Green kernel interaction ${\mathsf{G}}_{\mathrm{\partial }}$ in the multi-bubble regime. Exact evaluation of weighted profile moments yields ${\kappa }_1={\kappa }_2=0$: the coefficients of ${\mathrm{Ric}}_g(\nu ,\nu )$ and ${\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}}_{\bar{g}}$ in the bare mass vanish. On $\{H_g=0\}$ the mass reduces to ${\mathfrak{R}}_g^{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{e}}=\frac{6-n}{2(n-1)(n-3)(n-4)}\mathrm{\mid }\mathring{\mathrm{I}\mathrm{I}}{\mathrm{\mid }}^2$. The Lyapunov–Schmidt correction gives ${\mathfrak{R}}_g^{\mathrm{r}\mathrm{e}\mathrm{d}}\le {\mathfrak{R}}_g^{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{e}}\le 0$ for $n\ge 6$; for $n=5$ the nonlocal back-reaction overcomes the positive bare coefficient. In every dimension $n\ge 5$, non-umbilic boundaries are automatically subcritical: $C_{\mathrm{E}\mathrm{s}\mathrm{c}}^{\ast }<S_{\ast }$ whenever $\mathring{\mathrm{I}\mathrm{I}}\mathrm{\ne }0$. At threshold, on manifolds not conformally diffeomorphic to the hemisphere, every blow-up of positive constrained critical points is one-bubble and concentrates at an umbilic point with ${\mathfrak{R}}_g=0$, ${\mathrm{\nabla }}_{\mathrm{\partial }}{\mathfrak{R}}_g=0$. Since ${\mathfrak{R}}_g^{\mathrm{r}\mathrm{e}\mathrm{d}}<0$ at every non-umbilic point, threshold concentration occurs only on the umbilic stratum $\{\mathring{\mathrm{I}\mathrm{I}}=0\}$. There ${\mathrm{\Theta }}_g$ governs the next bifurcation: ${\mathrm{\Theta }}_g<0$ forces subcriticality; for $n\ge 7$, ${\mathrm{\Theta }}_g>0$ yields compactness and hemispherical rigidity. In the multi-bubble regime we establish global compactness at Escobar multiples with equal-mass quantization and conditional exclusion of pure multi-bubbling.