Conjugate radius and parabolicity along the Ricci flow

Abstract

We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if $(M^n,g(t))_{t\in[0,T]}$ is a complete Ricci flow with nonnegative scalar curvature at $t=0$, the initial metric is parabolic and not Ricci-flat, and the Einstein tensor $\mathrm{Ric}-\tfrac12 Rg$ is bounded above, then parabolicity is preserved along the flow and the conjugate radius of $g(t)$ is finite for every $t\in(0,T]$. As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that $\liminf_{r\to\infty}\Vol(B(p,r))/r^2=0$, has finite conjugate radius.