On the simply connectedness of non-negatively curved Kähler manifolds and applications

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dc.creatorChau, Albert
dc.creatorTam, Luen-Fai
dc.date2008-06-15
dc.date.accessioned2026-07-07T09:44:45Z
dc.date.available2026-07-07T09:44:45Z
dc.descriptionWe 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 \ge 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 $t_i \to \infty$. 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 for $(M, g(t))$. Combining our results with Cao's splitting for Kähler Ricci flow \cite{Cao04} and techniques of Ni-Tam \cite{NiTam03}, we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $p \in M$, 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.
dc.description24 pages
dc.identifierhttps://arxiv.org/abs/0806.2457
dc.identifierhttp://arxiv.org/abs/0806.2457
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/162986
dc.subjectDifferential Geometry
dc.subjectAnalysis of PDEs
dc.subject58J37; 35B35
dc.titleOn the simply connectedness of non-negatively curved Kähler manifolds and applications
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