Yamabe constant evolution and monotonicity along the conformal Ricci flow

Yanlin Li; Abimbola Abolarinwa; Shahroud Azami; Akram Ali; Yanlin Li; Abimbola Abolarinwa; Shahroud Azami; Akram Ali

doi:10.3934/math.2022671

AIMS Mathematics

2022, Volume 7, Issue 7: 12077-12090. doi: 10.3934/math.2022671

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Research article

Yamabe constant evolution and monotonicity along the conformal Ricci flow

1.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
2.
Department of Mathematics, University of Lagos, Akoka, Lagos State, Nigeria
3.
Department of pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran
4.
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia

Received: 12 December 2021 Revised: 06 April 2022 Accepted: 07 April 2022 Published: 21 April 2022
MSC : 53C18, 53C21, 58J35, 53E10, 53E20

We investigate the Yamabe constant's behaviour in a conformal Ricci flow. For conformal Ricci flow metric $ g(t) $, $ t \in [0, T) $, the time evolution formula for the Yamabe constant $ Y(g(t)) $ is derived. It is demonstrated that if the beginning metric $ g(0) = g_0 $ is Yamabe metric, then the Yamabe constant is monotonically growing along the conformal Ricci flow under some simple assumptions unless $ g_0 $ is Einstein. As a result, this study adds to the body of knowledge about the Yamabe problem.
- Yamabe constant,
- conformal Ricci flow,
- Einstein metric,
- scalar curvature
Citation: Yanlin Li, Abimbola Abolarinwa, Shahroud Azami, Akram Ali. Yamabe constant evolution and monotonicity along the conformal Ricci flow[J]. AIMS Mathematics, 2022, 7(7): 12077-12090. doi: 10.3934/math.2022671

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Abstract

We investigate the Yamabe constant's behaviour in a conformal Ricci flow. For conformal Ricci flow metric $ g(t) $, $ t \in [0, T) $, the time evolution formula for the Yamabe constant $ Y(g(t)) $ is derived. It is demonstrated that if the beginning metric $ g(0) = g_0 $ is Yamabe metric, then the Yamabe constant is monotonically growing along the conformal Ricci flow under some simple assumptions unless $ g_0 $ is Einstein. As a result, this study adds to the body of knowledge about the Yamabe problem.

References

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