## Taiwanese Journal of Mathematics

### Gap Theorems on Critical Point Equation of the Total Scalar Curvature with Divergence-free Bach Tensor

#### 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 $n = 4$.

#### Article information

Source
Taiwanese J. Math., Volume 23, Number 4 (2019), 841-855.

Dates
Received: 13 September 2018
Revised: 5 November 2018
Accepted: 6 November 2018
First available in Project Euclid: 18 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1563436871

Digital Object Identifier
doi:10.11650/tjm/181102

Mathematical Reviews number (MathSciNet)
MR3982064

Zentralblatt MATH identifier
07088950

#### Citation

Yun, Gabjin; Hwang, Seungsu. Gap Theorems on Critical Point Equation of the Total Scalar Curvature with Divergence-free Bach Tensor. Taiwanese J. Math. 23 (2019), no. 4, 841--855. doi:10.11650/tjm/181102. https://projecteuclid.org/euclid.twjm/1563436871

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