Optimality of the boundary mean-curvature defect constant

Determine whether the constant \(4/(3\sqrt3)\) in the boundary mean-curvature defect estimate for conformal metrics on the round hemisphere is optimal.

Background

Theorem \ref{main:boundary} derives a quantitative estimate controlling the squared deviation of the boundary mean curvature $H_g$ from the reference value $h$ by the interior $\sigma_2$-curvature. The estimate is sufficient for the rigidity applications, but the paper does not establish whether its numerical constant can be improved.