Rigidity of $$\kappa $$-noncollapsed steady Kähler–Ricci solitons

Deng, Yue; Zhu, Xiaohua

doi:10.1007/s00208-019-01807-6

Cited by 11 publications

(7 citation statements)

References 23 publications

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“…By using the result in [16, Theorem 1.3], we conclude that (M ∞ , g ∞ ) is a nonflat, κ-noncollapsed Kähler Ricci steady soliton with nonnegative bisectional curvature. However, such a Kähler steady soliton does not exist by [31,Theorem 1.2].…”

Section: Next We Definementioning

confidence: 99%

Ancient solutions to the Ricci flow with isotropic curvature conditions

Cho¹,

Yu²

2020

Preprint

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We show that every n-dimensional, κ-noncollapsed, noncompact, complete ancient solution to the Ricci flow with uniformly PIC for n = 4 or n ≥ 12 has weakly PIC 2 and bounded curvature. Combining this with the results in [12], we prove that any such solution is isometric to either a family of shrinking cylinders (or a quotient thereof) or the Bryant soliton. Also, we classify all complex 2-dimensional, κ-noncollapsed, complete ancient solutions to the Kähler Ricci flow with weakly PIC.

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Section: Next We Definementioning

confidence: 99%

Ancient solutions to the Ricci flow with isotropic curvature conditions

Cho¹,

Yu²

2020

Preprint

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“…Under the assumption that the soliton is κ noncollapsed with dimension n ≥ 4, Deng-Zhu [29] obtained the estimate (7) if |Rm| ≤ C(r + 1) −1 and Ric ≥ 0 outside a compact subset (see also [31,22,5]). However, there does exist collapsed steady solitons (see [10,28,32]). Munteanu-Sung-Wang [52] and the first named author [16] proved that the upper bound of |Rm| in (4) holds, i.e.…”

Section: Introductionmentioning

confidence: 99%

On a dichotomy of the curvature decay of steady Ricci soliton

Chan,

Zhu

2021

Preprint

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“…When the scalar curvature has no uniform decay, we hope to prove a dimension reduction theorem for steady Ricci solitons. In [23], the second named author and Xiaohua Zhu proved the dimension reduction for steady gradient Kähler-Ricci solitons with nonnegative bisectional curvature. Under (1.3), we can find a geodesic line in the limit of a sequence of steady gradient Ricci solitons.…”

Section: Introductionmentioning

confidence: 99%

On Four-dimensional Steady gradient Ricci solitons that dimension reduce

Chow¹,

Deng²,

Ma³

2020

Preprint

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In this paper, we will study the asymptotic geometry of 4dimensional steady gradient Ricci solitons under the condition that they dimension reduce to 3-manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either dimension reduce to a spherical space form S 3 /Γ or weakly dimension reduce to the 3-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE 4-manifolds or dimension reduce to 3-dimensional manifolds. As an application, we prove that any steady gradient Kähler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperkähler ALE 4-manifolds.

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Rigidity of $$\kappa $$-noncollapsed steady Kähler–Ricci solitons

Cited by 11 publications

References 23 publications

Ancient solutions to the Ricci flow with isotropic curvature conditions

Ancient solutions to the Ricci flow with isotropic curvature conditions

On a dichotomy of the curvature decay of steady Ricci soliton

On Four-dimensional Steady gradient Ricci solitons that dimension reduce

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