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August, 2019 Gap Theorems on Critical Point Equation of the Total Scalar Curvature with Divergence-free Bach Tensor
Gabjin Yun, Seungsu Hwang
Taiwanese J. Math. 23(4): 841-855 (August, 2019). DOI: 10.11650/tjm/181102

Abstract

On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture, proposed in 1987 by Besse, has not been resolved except when $M$ has harmonic curvature or the metric is Bach flat. In this paper, we prove some gap properties under divergence-free Bach tensor condition for $n \geq 5$, and a similar condition for $n = 4$.

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Gabjin Yun. Seungsu Hwang. "Gap Theorems on Critical Point Equation of the Total Scalar Curvature with Divergence-free Bach Tensor." Taiwanese J. Math. 23 (4) 841 - 855, August, 2019. https://doi.org/10.11650/tjm/181102

Information

Received: 13 September 2018; Revised: 5 November 2018; Accepted: 6 November 2018; Published: August, 2019
First available in Project Euclid: 18 July 2019

zbMATH: 07088950
MathSciNet: MR3982064
Digital Object Identifier: 10.11650/tjm/181102

Subjects:
Primary: 53C25 , 58E11

Keywords: Bach tensor , Besse conjecture , critical point equation , Einstein metric , total scalar curvature

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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Vol.23 • No. 4 • August, 2019
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