Quantitative stability and strict inequalities for the Escobar functional

Abstract

For Euclidean domains whose boundaries are sufficiently close to a sphere in $C^2$, we bound the deficit of the Escobar constant from below by the squared $H^{3/2}$ norm of the boundary graph, after fixing its constant and first spherical harmonics. This holds in every dimension $n\ge 3$ and implies a bound in Fraenkel asymmetry with optimal quadratic power. Under fixed $C^{6,\alpha }$ and size bounds, the asymmetry estimate also holds for star-shaped domains in dimensions $n\ge 6$. For compact Riemannian manifolds with minimal boundary and uniform geometric bounds, we prove $S_{\ast }-C_{\text{Esc}}^{\ast }(M,g)\ge c{a}^4/\log (\Lambda /a)$ in dimension six and $S_{\ast }-C_{\text{Esc}}^{\ast }(M,g)\ge c{a}^4$ in higher dimensions, where $a=\Vert \mathring{\mathrm{II}}{\Vert }_{L^{\infty }}>0$. A family satisfying the same bounds has deficit comparable to $a^{8/3}$, excluding uniform lower bounds with smaller powers. We also compute the cubic coefficient of the standard boundary bubble quotient and give uniform remainder estimates.

arXiv comment52 pages. Split from arXiv:2601.22665v2 (Part 1/3), with new results on the geometric stability of the Escobar functional. Other parts forthcoming. Comments welcome!