Generalized L-geodesic and Monotonicity of the Generalized Reduced Volume in the Ricci Flow

Suppose M is a complete n-dimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T ). For any 0 < p < 1, (p0, t0) ∈ M × (0, T ), q ∈ M , we define the Lp-length between p0 and q, Lp-geodesic, the generalized reduced distance lp and the generalized reduced volume Ṽp(τ), τ = t0 − t, corresponding to the Lp-geodesic at the point p0 at time t0. Under the condition Rij ≥ −c1gij on M × (0, t0) for some constant c1 > 0, we will prove the existence of a Lp-geodesic which minimize the Lp(q, τ)-length between p0 and q for any τ > 0. This result for the case p = 1/2 is conjectured and used many times but no proof of it was given in Perelman’s papers on Ricci flow. Let g(τ) = g(t0 − τ) and let Ṽ τ p (τ) be the rescaled generalized reduced volume. Suppose M also has nonnegative curvature operator with respect to the metric g(t) for any t ∈ (0, T ) and when 1/2 < p < 1, M has uniformly bounded scalar curvature on (0, T ). Let 0 < c < 1 and let τ0 = min((2(1− p))−1/(2p−1), t0). For any 1/2 ≤ p < 1 we prove that there exists a constant A0 ≥ 0 with A0 = 0 for p = 1/2 such that e−A0τ Ṽp(τ) is a monotone decreasing function in (0, τ1) where τ1 = (1 − c)τ0 if 1/2 < p < 1 and τ1 = t0 if p = 1/2. When (M, g) is an ancient κ-solution of the Ricci flow, we will prove a monotonicity property of the rescaled generalized volume Ṽ τ p (τ) with respect to τ for any 1/2 ≤ p < 1. When p = 1/2, the Lp-length, Lp-geodesic, the lp function and Ṽp(τ) are equal to the L-length, L-geodesic, the reduced distance l and the reduced volume Ṽ (τ) introduced by Perelman in his papers on Ricci flow. We will also prove a conjecture on the reduced distance l and the reduced volume Ṽ which was used by Perelman without proof in [P1]. 1991 Mathematics Subject Classification. Primary 58J35, 53C44 Secondary 58C99.