Conjugate radius and parabolicity along the Ricci flow
We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if is a complete Ricci flow with nonnegative scalar curvature at , the initial metric is parabolic and not Ricci-flat, and the Einstein tensor is bounded above, then parabolicity is preserved along the flow and the conjugate radius of is finite for every . As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that , has finite conjugate radius.