The Kähler-Ricci flow and the $\bar\partial$ operator on vector fields

Phong, D. H.; Song, Jian; Sturm, Jacob; Weinkove, Ben

doi:10.48550/arxiv.0705.4048

Cited by 4 publications

(8 citation statements)

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“…Proof of Theorem 4: This follows from the arguments of Lemma 5 and Lemma 6 of [PSSW2]. Indeed, one can easily check that the argument of Lemma 5 of [PSSW2] shows that under the assumptions of Theorem 4, R(t) − n C 0 converges exponentially fast to zero.…”

Section: Proof Of Theoremmentioning

confidence: 90%

“…A key step in the proofs of Theorems 1, 2 and 3 is to obtain a uniform lower bound for the first positive eigenvalue λ of the ∂ † ∂ operator on T 1,0 vector fields. The idea of considering this eigenvalue along the Kähler-Ricci flow was introduced in [PS3] and examined further in [PSSW2]. In Section 2 we show that certain curvature conditions imply the desired bound for λ.…”

Section: Conditions (A) and (A'mentioning

confidence: 98%

“…Indeed, one can easily check that the argument of Lemma 5 of [PSSW2] shows that under the assumptions of Theorem 4, R(t) − n C 0 converges exponentially fast to zero. Now apply Lemma 6 of [PSSW2] which states that if ∞ 0 R(t)−n C 0 dt < ∞ then the Kähler-Ricci flow converges exponentially fast in C ∞ to a Kähler-Einstein metric. Q.E.D.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…From Lemmas 5, 6 and 2 we see that the eigenvalue λ is uniformly bounded away from zero. Since the Mabuchi K-energy is bounded below it follows from Theorem 2 of [PSSW2] (or, since the curvature is bounded, the results of [PS3]) that the Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric. Q.E.D.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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The Kähler–Ricci flow with positive bisectional curvature

et al. 2008

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We show that the Kähler-Ricci flow on a manifold with positive first Chern class converges to a Kähler-Einstein metric assuming positive bisectional curvature and certain stability conditions.(A) The Mabuchi K-energy is bounded below on πc 1 (X);(A') The Futaki invariant of X is zero;(B) Let J be the complex structure of X, viewed as a tensor. Then the C ∞ closure of the orbit of J under the diffeomorphism group of X does not contain any complex structure J ∞ with the property that the space of holomorphic vector fields with respect to J ∞ has dimension strictly higher than the dimension of the space of holomorphic vector fields with respect to J. Conditions (A) and (A') and their relations to stability have been studied intensely in the last two decades, and for the definitions we refer the reader to the literature (see [PS1], for example). Condition (B) was introduced in [PS3]. It was shown there that if the curvatures along the Kähler-Ricci flow are uniformly bounded, and if (A) and (B) hold then the Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric. Note that the Riemann curvature tensor is bounded along the flow if the bisectional curvature is nonnegative or, in the case of two complex dimensions, if we have the weaker condition of nonnegative Ricci curvature with traceless curvature operator 2-nonnegative [PS2].Our first result is as follows:Theorem 1 Suppose there exists a Kähler metric g 0 on X with nonnegative bisectional curvature which is positive at one point. Assume condition (A) holds. Then the Kähler-Ricci flow starting at g 0 converges exponentially fast in C ∞ to a Kähler-Einstein metric. Now, at least a priori, the algebraic condition (A') is much weaker than (A). Here, we strengthen the result of [PS3] by replacing (A) by condition (A').Theorem 2 Suppose that the Riemann curvature tensor is uniformly bounded along the Kähler-Ricci flow and that conditions (A') and (B) hold. Then the Kähler-Ricci flow converges exponentially fast in C ∞ to a Kähler-Einstein metric.If n ≤ 2 we have:Theorem 3 Assume X has complex dimension 1 or 2, g 0 has nonnegative bisectional curvature and condition (A') holds. Then the Kähler-Ricci flow starting at g 0 converges exponentially fast in C ∞ to a Kähler-Einstein metric.

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Section: Proof Of Theoremmentioning

confidence: 90%

Section: Conditions (A) and (A'mentioning

confidence: 98%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

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The Kähler–Ricci flow with positive bisectional curvature

et al. 2008

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“…Under the pre-stable condition on complex structures, Phong-Sturm [26] and Phong-Song-Sturm-Weinkove [28] proved some convergence results of Kähler-Ricci flow with extra curvature conditions. We refer the readers to [27] [23] [14] for more recent results on Kähler-Ricci flow.…”

Section: On the Convergence Of Kähler-ricci Flowmentioning

confidence: 99%