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Conformal Rigidity and Non-rigidity of the Scalar Curvature on Riemannian Manifolds

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Abstract

For a compact smooth manifold \((M,g_0)\) with a boundary, we study the conformal rigidity and non-rigidity of the scalar curvature in the conformal class. It is known that the sign of the first eigenvalue for a linearized operator of the scalar curvature by a conformal change determines the rigidity/non-rigidity of the scalar curvature by conformal changes when the scalar curvature \(R_{g_0}\) is positive. In this paper, we show the sign condition of \(R_{g_0}\) is not necessary, and a reversed rigidity of the scalar curvature in the conformal class does not hold if there exists a point \(x_0 \in M\) with \(R_{g_0}(x_0) > 0.\)

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Acknowledgements

The first author thanks Seongtag Kim for his initial presentation and discussion on the rigidity problem. The work of J. Byeon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1A5A1028324).

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Authors and Affiliations

  1. Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Taejon, 305-701, Republic of Korea

    Jaeyoung Byeon

  2. Stochastic Analysis and Application Research Center, KAIST, 291 Daehak-ro, Yuseong-gu, Taejon, 305-701, Republic of Korea

    Sangdon Jin

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  1. Jaeyoung Byeon
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  2. Sangdon Jin
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Correspondence to Jaeyoung Byeon.

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Byeon, J., Jin, S. Conformal Rigidity and Non-rigidity of the Scalar Curvature on Riemannian Manifolds. J Geom Anal 31, 9745–9767 (2021). https://doi.org/10.1007/s12220-021-00626-z

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