Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature

Pak-Yeung Chan

Paper · Aug 27, 2019

Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature

Pak-Yeung Chan

math.DGarXiv:1908.10445

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Abstract

We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$ , namely $lim_{r \to \infty} r^{- 1} \int_{B_{r}} S = 0$ , and show that it is a quotient of $Σ \times C^{n - 1 - k} \times N^{k}$ , where $Σ$ and $N$ denote the Hamilton's cigar soliton and some compact Kähler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady Kähler Ricci soliton with $R i c \geq 0$ , together with subquadratic volume growth or $lim sup_{r \to \infty} r S < 1$ must have universal covering space isometric to $Σ \times C^{n - 1 - k} \times N^{k}$ .

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Submitted: Aug 27, 2019
Subjects: math.DG

Ricci solitons Kähler–Ricci solitons Ancient solutions Ricci-flat & Calabi–Yau Kähler geometry Scalar curvature

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
11B.2
Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities
Comparing curvature across space and time.
12B.1
κ-SolutionsBook 12B · κ-Solutions and Surgery
Ancient, nonnegatively curved, noncollapsed: the models for singularities.

[1] 12B.2
The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

[2] 11B.2
Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities
Comparing curvature across space and time.

[3] 12B.1
κ-SolutionsBook 12B · κ-Solutions and Surgery
Ancient, nonnegatively curved, noncollapsed: the models for singularities.

Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature

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