Conformal geometry has multiple curvature quantities — scalar curvature, Q-curvature, sigma_2-curvature — each with its own Yamabe-type problem: given a conformal class, can you find a metric within it that makes the chosen curvature constant? Each problem has its own existence theory, its own uniqueness results, and its own variational structure. They have been studied independently for decades.
Fang, Wei, and Gonzalez unify them. They introduce a variational framework for formally self-adjoint polydifferential operators that encompasses all three curvature prescriptions as special cases. The unified functional has the same structure for each curvature — only the operator changes — and the existence and uniqueness results follow from properties of the operator rather than the specific curvature.
The unification reveals why the three problems share structural features (conformal covariance, variational characterization, bubbling phenomena) while differing in technical details (regularity requirements, dimension restrictions, sign conditions). The shared features come from the operator being formally self-adjoint and conformally covariant. The differences come from the order and symbol of the operator.
New uniqueness results fall out: for sigma_2-curvature on manifolds with positive scalar curvature, the constant-curvature metric is unique within its conformal class under conditions that the individual theory could not establish. The unified proof works because it uses properties common to all three operators.
Three curvature problems, decades of separate development. One variational framework shows they are instances of the same mathematical structure — a polydifferential operator with the right symmetries.