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

TL;DR

This paper investigates the threshold behavior of the Escobar functional on compact manifolds with boundary, revealing geometric invariants that determine compactness, blow-up, and bifurcation phenomena, especially near the hemisphere case.

Contribution

It provides a detailed analysis of boundary invariants and their role in the threshold landscape, including exact evaluations and conditions for compactness and blow-up.

Findings

Mass reduces to a boundary invariant on zero mean curvature boundary.

Non-umbilic boundaries are automatically subcritical.

Threshold concentration occurs only at umbilic points with specific invariants.

Abstract

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 $\rho_n^{\mathrm{conf}}H_g$ vanishes identically, so the leading obstruction is a renormalized boundary mass $\mathfrak R_g$ (second order, $n\ge5$), followed by a cubic invariant $\Theta_g$ (third order, $n\ge6$), with a Green kernel interaction $\mathsf G_\partial$ in the multi-bubble regime. Exact evaluation of weighted profile moments yields $\kappa_1=\kappa_2=0$: the coefficients of $\operatorname{Ric}_g(\nu,\nu)$ and $\mathrm{Scal}_{\bar g}$ in the bare mass vanish. On $\{H_g=0\}$ the mass reduces to $\mathfrak R_g^{\mathrm{bare}}=\frac{6-n}{2(n-1)(n-3)(n-4)}|\mathring{\mathrm{II}}|^2$. The Lyapunov--Schmidt correction gives $\mathfrak R_g^{\mathrm{red}}\le\mathfrak R_g^{\mathrm{bare}}\le0$ for $n\ge6$; for $n=5$ the nonlocal back-reaction overcomes the positive bare coefficient. In every dimension $n\ge5$, non-umbilic boundaries are automatically subcritical: $C^*_{\mathrm{Esc}}<S_\ast$ whenever $\mathring{\mathrm{II}}\neq0$. 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 $\mathfrak R_g=0$, $\nabla_\partial\mathfrak R_g=0$. Since $\mathfrak R_g^{\mathrm{red}}<0$ at every non-umbilic point, threshold concentration occurs only on the umbilic stratum $\{\mathring{\mathrm{II}}=0\}$. There $\Theta_g$ governs the next bifurcation: $\Theta_g<0$ forces subcriticality; for $n\ge7$, $\Theta_g>0$ 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.