Rotationally Symmetric Shrinking and Expanding Gradient Kähler-Ricci Solitons

Abstract

We construct new families of Kähler-Ricci solitons on complex line bundles over ℂℙⁿ⁻¹, n ≥ 2. Among these are examples whose initial or final condition is equal to a metric cone ℂⁿ/ℤ_k. We exhibit a noncompact Ricci flow that shrinks smoothly and self-similarly for t < 0, becomes a cone at t = 0, and then expands smoothly and self-similarly for t > 0; this evolution is smooth in space-time except at a single point, at which there is a blowdown of a ℂℙⁿ⁻¹. We also construct certain shrinking solitons with orbifold point singularities.