Volume asymptotics and Einstein moduli on spheres

The moduli spaces of positive Einstein metrics of volume one on S⁶ and S⁸, modulo all diffeomorphisms, each have countably infinitely many connected components. More generally, this holds on Sp+q+1 and Sᵖ⁺¹× Q whenever p,q≥2, d=p+q≤8, and Q is a closed connected positive Einstein q-manifold. For sequences obtained from Böhm’s cap construction and normalized by Ric=dg, we determine the leading volume difference from the limiting sine-cone volume. It is a positive definite quadratic form in the data on a minimal hypersurface. For every such pair, suitable spherical sequences and reflection doubles have volumes converging from above to a positive limit, with successive volume excesses in limiting ratio exp(-2π√(d-1)/(9-d)).