Rigidity of positive rupture solutions to a biharmonic equation with critical negative exponent

We establish two rigidity theorems for positive rupture solutions of the conformally invariant equation ${\mathrm{\Delta }}^{2}u=u^{-7}$ in ${\mathbb{R}}^3\setminus \{0\}$, which extend continuously to the origin with $u(0)=0$. First, we prove that if the associated conformal metric $g=u^{-4}\mathrm{\mid }dx{\mathrm{\mid }}^2$ has nonnegative scalar curvature, then every such solution is radially symmetric and has the sharp rupture profile $u(x)\sim (\frac{4}{3}{)}^{\frac{1}{4}}\mathrm{\mid }x{\mathrm{\mid }}^{\frac{1}{2}}$ as $x\to 0$; in particular, the metric is complete at the origin. The principal novelty is a global rigidity theorem requiring neither curvature nor symmetry: the single global condition $u(x)=o(\mathrm{\mid }x\mathrm{\mid })$ at infinity forces $u(x)\equiv (\frac{4}{3}{)}^{\frac{1}{4}}\mathrm{\mid }x{\mathrm{\mid }}^{\frac{1}{2}}$. This result provides a sharp answer to the uniqueness question posed by McKenna and Reichel in a class with no a priori symmetry assumption. Finally, we construct complementary examples demonstrating the essential roles of the curvature and growth hypotheses.