A gap theorem on complete shrinking gradient Ricci solitons

Shijin Zhang

doi:10.1090/proc/13689

A gap theorem on complete shrinking gradient Ricci solitons

Shijin Zhang^*

^*Corresponding author for this work

School of Mathematical Sciences

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

3 Scopus citations

Abstract

In this short note, using Günther’s volume comparison theorem and Yokota’s gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton (Mⁿ, g, f) with sectional curvature K(g) < A and Vol_f (M) ≥ v for some uniform constant A, v, there exists a small uniform constant ɛ_n,A,v > 0 depends only on n, A and v, if the scalar curvature R ≤ ɛ_n,A,v, then (M, g, f) is isometric to the Gaussian soliton (Formula Presented).

Original language	English
Pages (from-to)	359-368
Number of pages	10
Journal	Proceedings of the American Mathematical Society
Volume	146
Issue number	1
DOIs	https://doi.org/10.1090/proc/13689
State	Published - 2018

Keywords

Gap theorem
Sectional curvature
Shrinking gradient Ricci solitons

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title = "A gap theorem on complete shrinking gradient Ricci solitons",

abstract = "In this short note, using G{\"u}nther{\textquoteright}s volume comparison theorem and Yokota{\textquoteright}s gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton (Mn, g, f) with sectional curvature K(g) < A and Volf (M) ≥ v for some uniform constant A, v, there exists a small uniform constant ɛn,A,v > 0 depends only on n, A and v, if the scalar curvature R ≤ ɛn,A,v, then (M, g, f) is isometric to the Gaussian soliton (Formula Presented).",

keywords = "Gap theorem, Sectional curvature, Shrinking gradient Ricci solitons",

author = "Shijin Zhang",

note = "Publisher Copyright: {\textcopyright} 2017 American Mathematical Society.",

year = "2018",

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pages = "359--368",

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T1 - A gap theorem on complete shrinking gradient Ricci solitons

AU - Zhang, Shijin

PY - 2018

Y1 - 2018

N2 - In this short note, using Günther’s volume comparison theorem and Yokota’s gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton (Mn, g, f) with sectional curvature K(g) < A and Volf (M) ≥ v for some uniform constant A, v, there exists a small uniform constant ɛn,A,v > 0 depends only on n, A and v, if the scalar curvature R ≤ ɛn,A,v, then (M, g, f) is isometric to the Gaussian soliton (Formula Presented).

AB - In this short note, using Günther’s volume comparison theorem and Yokota’s gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton (Mn, g, f) with sectional curvature K(g) < A and Volf (M) ≥ v for some uniform constant A, v, there exists a small uniform constant ɛn,A,v > 0 depends only on n, A and v, if the scalar curvature R ≤ ɛn,A,v, then (M, g, f) is isometric to the Gaussian soliton (Formula Presented).

KW - Gap theorem

KW - Sectional curvature

KW - Shrinking gradient Ricci solitons

UR - https://www.scopus.com/pages/publications/85034232720

U2 - 10.1090/proc/13689

DO - 10.1090/proc/13689

M3 - 文章

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SN - 0002-9939

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SP - 359

EP - 368

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 1

ER -

A gap theorem on complete shrinking gradient Ricci solitons

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