Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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GRADIENT RICCI SOLITONS WITH HALF HARMONIC WEYL CURVATURE AND TWO RICCI EIGENVALUES

  • Kang, Yutae (Department of Mathematics Sogang University) ;
  • Kim, Jongsu (Department of Mathematics Sogang University)
  • Received : 2020.11.08
  • Accepted : 2020.11.23
  • Published : 2022.04.30
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Abstract

In this article we classify four dimensional gradient Ricci solitons (M, g, f) with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, (V, g) is isometric to one of the following: (i) an Einstein manifold. (ii) a domain in the Riemannian product (ℝ2, g0) × (N, ${\tilde{g}}$), where g0 is the flat metric on ℝ2 and (N, ${\tilde{g}}$) is a two dimensional Riemannian manifold of constant curvature λ ≠ 0. (iii) a domain in ℝ × W with the warped product metric $ds^2+h(s)^2{\tilde{g}}$, where ${\tilde{g}}$ is a constant curved metric on a three dimensional manifold W.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2020R1A2B5B01001862).

References

  1. 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
  2. H.-D. Cao and Q. Chen, On locally conformally flat gradient steady Ricci solitons, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2377-2391. https://doi.org/10.1090/S0002-9947-2011-05446-2
  3. H.-D. Cao and Q. Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), no. 6, 1149-1169. https://doi.org/10.1215/00127094-2147649
  4. X. Chen and Y. Wang, On four-dimensional anti-self-dual gradient Ricci solitons, J. Geom. Anal. 25 (2015), no. 2, 1335-1343. https://doi.org/10.1007/s12220-014-9471-8
  5. B. Chow, S. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, 135, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/135
  6. A. Derdzi'nski, 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
  7. 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
  8. J.-Y. Wu, P. Wu, and W. Wylie, Gradient shrinking Ricci solitons of half harmonic Weyl curvature, Calc. Var. Partial Differential Equations 57 (2018), no. 5, Paper No. 141, 15 pp. https://doi.org/10.1007/s00526-018-1415-x