Optimal Curvature Estimates for Homogeneous Ricci Flows

Abstract

AbstractWe prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n)({\mathrm{scal}}(g(t)) - {\mathrm{scal}}(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, with constants depending only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on that space. The above curvature estimates follow from a gap theorem for Ricci-flatness on homogeneous spaces. This theorem is proved by contradiction, using a local $W^{2,p}$ convergence result which holds without symmetry assumptions.