Interior local rigidity theorem. Israel Lima.

The Interior–Boundary Rigidity Theorem states that, for a compact Riemannian three-dimensional manifold with boundary, any small change in the interior scalar curvature necessarily produces a corresponding change in the boundary mean curvature. More precisely, the size of the variation of the scalar curvature inside the domain dominates, in a quantitative way, the square of the variation of the mean curvature along the boundary. This shows that the geometry of the interior cannot deform independently of the geometry induced on the boundary. The result establishes a new coercive rigidity principle connecting bulk geometric energy to boundary geometric energy, with implications for geometric analysis, quasi-local mass in general relativity, holography, and elasticity theory.