Steady Ricci solitons with horizontally ϵ-pinched Ricci curvature

Yuxing Deng; Xiaohua Zhu

doi:10.1007/s11425-020-1685-3

Steady Ricci solitons with horizontally ϵ-pinched Ricci curvature

Yuxing Deng, Xiaohua Zhu^*

^*Corresponding author for this work

School of Mathematics and Statistics

Peking University

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In this paper, we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally ϵ-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any κ-noncollapsed gradient steady Ricci soliton (Mⁿ, g, f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature R(x) lim_r(x)→∞R(x)f(x) = C₀ sup_x∈MR(x) with satisfies C0>n−22.

Original language	English
Pages (from-to)	1411-1428
Number of pages	18
Journal	Science China Mathematics
Volume	64
Issue number	7
DOIs	https://doi.org/10.1007/s11425-020-1685-3
Publication status	Published - Jul 2021

Keywords

53C25
53C55
58J05
Ricci flow
Ricci soliton
κ-solutions
ϵ-pinched curvature

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10.1007/s11425-020-1685-3

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title = "Steady Ricci solitons with horizontally ϵ-pinched Ricci curvature",

abstract = "In this paper, we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally ϵ-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any κ-noncollapsed gradient steady Ricci soliton (Mn, g, f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature R(x) limr(x)→∞R(x)f(x) = C0 supx∈MR(x) with satisfies C0>n−22.",

keywords = "53C25, 53C55, 58J05, Ricci flow, Ricci soliton, κ-solutions, ϵ-pinched curvature",

author = "Yuxing Deng and Xiaohua Zhu",

note = "Publisher Copyright: {\textcopyright} 2020, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.",

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T1 - Steady Ricci solitons with horizontally ϵ-pinched Ricci curvature

AU - Deng, Yuxing

AU - Zhu, Xiaohua

PY - 2021/7

Y1 - 2021/7

N2 - In this paper, we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally ϵ-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any κ-noncollapsed gradient steady Ricci soliton (Mn, g, f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature R(x) limr(x)→∞R(x)f(x) = C0 supx∈MR(x) with satisfies C0>n−22.

AB - In this paper, we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally ϵ-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any κ-noncollapsed gradient steady Ricci soliton (Mn, g, f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature R(x) limr(x)→∞R(x)f(x) = C0 supx∈MR(x) with satisfies C0>n−22.

KW - 53C25

KW - 53C55

KW - 58J05

KW - Ricci flow

KW - Ricci soliton

KW - κ-solutions

KW - ϵ-pinched curvature

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DO - 10.1007/s11425-020-1685-3

M3 - Article

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SN - 1674-7283

VL - 64

SP - 1411

EP - 1428

JO - Science China Mathematics

JF - Science China Mathematics

IS - 7

ER -

Steady Ricci solitons with horizontally ϵ-pinched Ricci curvature

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Keywords

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