2008/06/15 by Albert Chau, Chau, Albert, Luen-Fai Tam +1
Mathematics · #35B35 #58J37 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:35B35 #msc:58J37
paper · pdf · doi:10.48550/arxiv.0806.2457
24 pages
arxiv created 2008/06/15 · arxiv updated 2009/12/01
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_\ijb ≥ cRg_\ijb 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 ti → ∞. We will show as an immediate corollary that the injectivity radius of g(t) along ti is uniformly bounded from below along ti, 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 for (M, g(t)). Combining our results with Cao's splitting for Kähler Ricci flow \citeCao04 and techniques of Ni-Tam \citeNiTam03, 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 a special cases, complete Kähler manifolds with non-negative holomorphic bisectional and average quadratic curvature decay as well as the case of steady gradient Kähler Ricci solitons.