The evolution of a Riemannian metric by the Ricci flow has been found to be very powerful in studying and classifying manifolds of certain curvature conditions. However, the flow typically develops singularities in finite time, which need to be understood. Quantities monotone in time are a common tool to study geometric flows near singularities.;We define a reduced distance function based at a point at the singular time of a Ricci flow on a complete n-dimensional manifold M. Our curvature bound assumption is weaker than the generic type I condition. We show that the corresponding reduced volume based at singular time is monotone along the flow. Since the quantity being constant implies that the flow is a gradient shrinking soliton, type I singularities can be modeled by those special solutions. We also show the monotonicity of the reduced volume arising from the reduced distance to a compact submanifold of M, and we similarly extend that notion to singular time.
展开▼
