On the simply connectedness of nonnegatively curved Kähler manifolds and applications

Abstract

Abstract. We study complete noncompact long-time solutions (M, g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. R ij ≥ cRg ij at (p, t) for all t for some c > 0, then there always exists a local gradient Kähler-Ricci soliton limit around p after possibly rescaling g(t) along some sequence t i → ∞. We will show as an immediate corollary that the injectivity radius of g(t) along t i is uniformly bounded from below along t i , and thus M must in fact be simply connected. Additional results concerning the uniformization of M and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci curvature for (M, g(t)). Combining our results with Cao's splitting for Kähler-Ricci flow (2004) and techniques of Ni and Tam (2003), we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point p ∈ M , then M has a special holomorphic fiber bundle structure. We will treat as special cases, complete Kähler manifolds with nonnegative holomorphic bisectional curvature and average quadratic curvature decay as well as the case of steady gradient Kähler-Ricci solitons.

Expanding solitons with non-negative curvature operator coming out of cones

“…We refer the reader to that paper for more details. Similar estimates may also be found in Proposition 4.1 of the paper [8].…”

Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature

“…Note that an expanding Ricci soliton with nonnegative Ricci curvature has maximal volume growth (cf. [7] or [12]). Namely, there exists a uniform constant δ > 0 such that vol(B r (p)) ≥ δr 2n .…”

“…By the convergence of V (t α ), the integral curve of −V (t α ) will also converge to a point q in D(r 1 ) for some r 1 < r when α is large enough (cf. Page 9 of [12]). As a consequence, q is a zero point of V (t α ) in D(r 1 ).…”

Yamabe Flow and Myers Type Theorem on Complete Manifolds

“…We use the arguments due to A.Chau and L.F.Tam [8](see Theorem 2.1) to show the injectivity radius of x i have the uniformly lower bound with respect to g i (0). By our assumptions on the positivity of Ricci curvature, we may then let W (i) ∈ T M be the unique solutions to (5.2) on (M n , g i (t)) for any i.…”