DOI QR코드

DOI QR Code

WEAKLY EINSTEIN CRITICAL POINT EQUATION

  • Hwang, Seungsu ;
  • Yun, Gabjin
  • 투고 : 2015.07.01
  • 발행 : 2016.07.31

초록

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when g is weakly Einstein. In higher dimensional case $n{\geq}5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard n-sphere when it is n-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

키워드

total scalar curvature;critical point metric;weakly Einstein;Einstein metric;linearized scalar curvature

참고문헌

  1. M. Berger, Quelques formules de variation pour une structure riemannienne, Ann. Sci. Ecole Norm. Sup. (4) 3 (1970), 285-294. https://doi.org/10.24033/asens.1194
  2. A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3) 10, A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1987.
  3. Y. Euh, J. H. Park, and K. Sekigawa, A curvature identity on a 4-dimensional Riemannian manifold, Results Math. 63 (2013), no. 1-2, 107-114. https://doi.org/10.1007/s00025-011-0164-3
  4. N. Koiso, A decomposition of the space M of Riemannian metrics on a manifold, Osaka J. Math. 16 (1979), no. 2, 423-429.
  5. J. Lafontaine, Remarques sur les varietes conformement plates, Math. Ann. 259 (1982), no. 3, 313-319. https://doi.org/10.1007/BF01456943
  6. M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), no. 3, 333-340. https://doi.org/10.2969/jmsj/01430333
  7. Q. Wang, J. N. Gomes, and C. Xia, h-almost Ricci soliton, arXiv.org: 1411.6416v2, 2015.
  8. G. Yun, J. Chang, and S. Hwang, Total scalar curvature and harmonic curvature, Taiwanese J. Math. 18 (2014), no. 5, 1439-1458. https://doi.org/10.11650/tjm.18.2014.1489
  9. G. Yun, J. Chang, and S. Hwang, On the structure of linearization of the scalar curvature, Tohoku Math. J. (2) 67, (2015), no. 2, 281-295. https://doi.org/10.2748/tmj/1435237044

피인용 문헌

  1. Locally conformally flat weakly-Einstein manifolds vol.111, pp.5, 2018, https://doi.org/10.1007/s00013-018-1221-x

과제정보

연구 과제 주관 기관 : National Research Foundation of Korea(NRF)