Min–Oo conjecture for metrics on the hemisphere

Let $g$) be a smooth Riemannian metric on the closed hemisphere $\mathbb{S}^3_+$ satisfying the following conditions: its scalar curvature obeys ${\mathrm R}_g\geq 6$; its induced boundary metric is the standard metric, $g|_{\partial \mathbb{S}^3_+}=g_{\mathbb{S}^2}$; and $\partial \mathbb{S}^3_+$ is totally geodesic with respect to $g$. Min–Oo conjecture. Then $(\mathbb{S}^3_+,g)$ is isometric to the standard round hemisphere $(\mathbb{S}^3_+,g_{\mathbb{S}^3})$. The conjecture aimed to characterize the de Sitter solution as the unique zero-mass initial data under a positive cosmological constant. It is refuted in full generality by counterexamples, although positive rigidity results remain available under strengthened hypotheses.