The Yamabe problem asks: within a conformal class, can you find a metric with constant scalar curvature? The Q-curvature generalization asks the same for Branson's Q-curvature — a fourth-order curvature invariant with conformal transformation properties. Both have been solved in specific cases.
The Q/R curvature ratio introduces a novel conformal invariant that couples the two curvatures. Instead of prescribing Q-curvature or scalar curvature independently, the problem asks for a metric where their ratio is constant. This coupling creates a richer variational structure than either problem alone.
The ratio Q/R is a conformal invariant of a specific weight — it transforms predictably under conformal changes of metric. The Yamabe-type problem for this ratio produces a fourth-order PDE whose solutions give the constant-ratio metrics. The equation couples the fourth-order Q-curvature operator with the second-order scalar curvature operator, creating a system that is neither purely fourth-order nor purely second-order.
New Sobolev inequalities emerge from the variational structure. The functional whose critical points solve the constant-ratio problem has specific coercivity properties that translate into embedding theorems for Sobolev spaces on the manifold. These inequalities are sharper than the standard Sobolev inequalities because they exploit the conformal structure.
A new conformal invariant, a new Yamabe problem, and new Sobolev inequalities. All from the ratio of two curvature quantities that had been studied independently for decades.