Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STATIC AND RELATED CRITICAL SPACES WITH HARMONIC CURVATURE AND THREE RICCI EIGENVALUES

  • Kim, Jongsu (Department of Mathematics Sogang University)
  • Received : 2019.10.10
  • Accepted : 2020.02.11
  • Published : 2020.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this article we make a local classification of n-dimensional Riemannian manifolds (M, g) with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution f to the following equation $$(1)\hspace{20}{\nabla}df=f(r-{\frac{R}{n-1}}g)+x{\cdot} r+y(R)g,$$ where ∇ is the Levi-Civita connection of g, r is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, either (i) or (ii) below holds; (i) (V, g, f + x) is a static space and isometric to a domain in the Riemannian product of an Einstein manifold N and a static space (W, gW, f + x), where gW is a warped product metric of an interval and an Einstein manifold. (ii) (V, g) is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

Keywords

Acknowledgement

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B4004460).

References

  1. L. Ambrozio, On static three-manifolds with positive scalar curvature, J. Differential Geom. 107 (2017), no. 1, 1-45. https://doi.org/10.4310/jdg/1505268028
  2. H. Baltazar, R. Batista, and K. Bezerra, On the volume functional of compact manifolds with boundary with harmonic Weyl tensor, arXiv:1710.06247v1 [math.DG].
  3. H. Baltazar and E. Ribeiro, Jr., Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary, Pacific J. Math. 297 (2018), no. 1, 29-45. https://doi.org/10.2140/pjm.2018.297.29
  4. A. Barros and E. Ribeiro, Jr., Critical point equation on four-dimensional compact manifolds, Math. Nachr. 287 (2014), no. 14-15, 1618-1623. https://doi.org/10.1002/mana.201300149
  5. R. Batista, R. Diogenes, M. Ranieri, and E. Ribeiro Jr., Critical metrics of the volume functional on compact three-manifolds with smooth boundary, J. Geom. Anal. 27 (2017), no. 2, 1530-1547. https://doi.org/10.1007/s12220-016-9730-y
  6. A. L. Besse, Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987. https://doi.org/10.1007/978-3-540-74311-8
  7. J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137-189. https://doi.org/10.1007/PL00005533
  8. J. Corvino, M. Eichmair, and P. Miao, Deformation of scalar curvature and volume, Math. Ann. 357 (2013), no. 2, 551-584. https://doi.org/10.1007/s00208-013-0903-8
  9. A. Derdzinski, Classification of certain compact Riemannian manifolds with harmonic curvature and nonparallel Ricci tensor, Math. Z. 172 (1980), no. 3, 273-280. https://doi.org/10.1007/BF01215090
  10. D. DeTurck and H. Goldschmidt, Regularity theorems in Riemannian geometry. II. Harmonic curvature and the Weyl tensor, Forum Math. 1 (1989), no. 4, 377-394. https://doi.org/10.1515/form.1989.1.377
  11. J. Kim, On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature, J. Geom. Anal. 27 (2017), no. 2, 986-1012. https://doi.org/10.1007/s12220-016-9707-x
  12. J. Kim and J. Shin, Four-dimensional static and related critical spaces with harmonic curvature, Pacific J. Math. 295 (2018), no. 2, 429-462. https://doi.org/10.2140/pjm.2018.295.429
  13. O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), no. 4, 665-675. https://doi.org/10.2969/jmsj/03440665
  14. P. Miao and L.-F. Tam, Einstein and conformally flat critical metrics of the volume functional, Trans. Amer. Math. Soc. 363 (2011), no. 6, 2907-2937. https://doi.org/10.1090/S0002-9947-2011-05195-0
  15. J. Qing and W. Yuan, On scalar curvature rigidity of vacuum static spaces, Math. Ann. 365 (2016), no. 3-4, 1257-1277. https://doi.org/10.1007/s00208-015-1302-0
  16. J. Shin, On the classification of 4-dimensional (m; )-quasi-Einstein manifolds with harmonic Weyl curvature, Ann. Global Anal. Geom. 51 (2017), no. 4, 379-399. https://doi.org/10.1007/s10455-017-9542-8
  17. G. Yun, J. Chang, and S. Hwang, Total scalar curvature and harmonic curvature, Tai-wanese J. Math. 18 (2014), no. 5, 1439-1458. https://doi.org/10.11650/tjm.18.2014.1489
  18. G. Yun and S. Hwang, Rigidity of generalized Bach-flat vacuum static spaces, J. Geom. Phys. 121 (2017), 195-205. https://doi.org/10.1016/j.geomphys.2017.07.016