Sharp Yamabe Bounds for Spherical Three-Space Forms via Four-Dimensional Poincaré--Einstein Compactness

Let $\mathrm{\Gamma }<SO(4)$ be a nontrivial finite group acting freely on $S^3$, and let $M=S^3\mathrm{/}\mathrm{\Gamma }$. We prove that every smooth conformal class $C$ on $M$ satisfies $$Y(M,C)\le \mathrm{\mid }\mathrm{\Gamma }{\mathrm{\mid }}^{-2\mathrm{/}3}Y(S^3),$$ with equality for the quotient-round conformal class. Consequently, $$\sigma (M)=\mathrm{\mid }\mathrm{\Gamma }{\mathrm{\mid }}^{-2\mathrm{/}3}Y(S^3)\mathrm{.}$$ Our approach uses four-dimensional Poincaré–Einstein fillings of the ball. We first establish compactness for fillings whose conformal infinities range over a compact family of positive-scalar-curvature metrics. Since compactness alone does not guarantee continuation through degenerate Einstein metrics, we use equivariant Fredholm parity to obtain a filling with the prescribed boundary conformal class and group action. A common interior fixed point then makes the Li–Qing–Shi boundary test function invariant, and its descent to $M$ gives the sharp covering factor.