Einstein orbifolds — orbifolds carrying Einstein metrics (Ricci curvature proportional to the metric) — arise naturally as limits of sequences of smooth Einstein metrics. A fundamental question: which orbifold singularities can appear in such limits? The answer determines what pathological geometries the Einstein equation can produce.
Ozuch proves that certain negative Einstein orbifolds cannot arise as limits of smooth Einstein metrics. The obstruction is analytic: the linearization of the Einstein equation at the orbifold singular point has a cokernel that prevents smooth deformation. The orbifold is rigid in the wrong direction — perturbations that would smooth the singularity are obstructed by the equation itself.
The result extends to arbitrary dimensions, generalizing previous four-dimensional obstructions. The proof uses the analysis of elliptic operators on orbifolds — a technically demanding extension of standard elliptic theory where the singular points require special treatment in the functional-analytic framework.
The obstruction is computable: given an orbifold singularity type, the cokernel dimension can be calculated from the group action defining the singularity. When the cokernel is non-trivial, no smooth Einstein metric can converge to the orbifold metric. The singularity is forever isolated from smooth geometry.
Not every Einstein orbifold connects to the smooth world. The equation itself builds walls around certain singularities, preventing any smooth path from reaching them. The orbifold exists, the Einstein equation is satisfied, but the solution is orphaned — unreachable from smooth metrics by any continuous deformation.