On the Calabi Flow

By Xiuxiong Chen, Weiyong He. 2006

90 citations recorded by the source corpus.

In this paper, we study the Calabi flow on a polarized Kähler manifold and some related problems. We first give a precise statement on the short time existence of the Calabi flow for any c3,α(M) initial Kähler potential. As an application, we prove a stability result: any metric near a constant scalar curvature Kähler (CscK) metric will flow to a nearby CscK metric exponentially fast. Secondly, we prove that a compactness theorem in the space of the Kähler metrics given uniform Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-singularity result about a weak constant scalar curvature metric in a punctured disc.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e56b0c63d132fdb37def2f33fac181dc691f734c

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature K\ (2010, methodology): “We refer the readers to [11] for detailed proof.”
Kähler–Ricci flow, Kähler–Einstein metric, and K–stability (2015, background): “2, it suffices to deal with the case that none of the limits in C has constant scalar curvature (the other case is easier, and is already treated in [7]), so we will always assume this is case in the remainder of this subsection.”
Geometric pluripotential theory on Kähler manifolds (2019, background): “Streets [99, 100], the CAT(0) property allows to study of the Calabi flow [39] in the context of the metric completion, leading to precise convergence results for this flow in [13].”
Smooth and singular Kahler-Einstein metrics (2014, background): “The most general long time existence result known is due to Streets [230] though the question of regularity of such weak flows is open (assuming curvature bounds along the flow a smooth solution exists for all time by Chen–He [64]).”
On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties (2014, background): “2, which says that any orbit closure of a K-semistable Fano manifold contains only one isomorphic class of K-polystable Q-Fano variety (this is an extension of the result of [12] for the Fano case).”
Space of Kähler metrics III – On the lower bound of the Calabi energy and geodesic distance (2006, methodology): “To prove the metric is equivalent, we follow Yau’s proof of the Calabi conjecture (cf. [ 13 ]). Following the notation of Sect.”
A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics (2010, background): “. The existence and convergence of the Calabi ﬂow on a Riemann surface was shown by Chrusciel [30]. In the general higher dimensional case almost all results are conditional. It was proved by Chen-He [28] that the Calabi ﬂow exists as long as the Ricci curvature stays uniformly bounded. Moreover, th”