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Rigidity of complete Riemannian manifolds with vanishing Bach tensor

Guangyue Huang|Bingqing Ma
대한수학회 (The Korean Mathematical Society)|2019.09|대한수학회보 (Bulletin of the KMS)|제56권 제5호|pp. 1341-1353|DOI 10.4134/BKMS.b181193

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참고문헌

키워드

rigiditySobolev constantvanishing Bach tensor
주제분류수학

초록

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.

Abstract

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.

참고문헌 (14)

  1. Rudolf Bach, "Zur Weylschen Relativit�tstheorie und der Weylschen Erweiterung des Kr�mmungstensorbegriffs", Mathematische Zeitschrift, 9(1-2), pp.110~135 (1921), Springer Science and Business Media LLC[DOI]
  2. Giovanni Catino, "Integral pinched shrinking Ricci solitons", Advances in Mathematics, 303, pp.279~294 (2016), Elsevier BV[DOI]
  3. Yawei Chu, "A rigidity theorem for complete noncompact Bach-flat manifolds", Journal of Geometry and Physics, 61(2), pp.516~521 (2011), Elsevier BV[DOI]
  4. Yawei Chu, "Rigidity of complete noncompact bach-flat n -manifolds", Journal of Geometry and Physics, 62(11), pp.2227~2233 (2012), Elsevier BV[DOI]
  5. Yi Fang, "A sphere theorem for Bach-flat manifolds with positive constant scalar curvature", Differential Geometry and its Applications, 64, pp.80~91 (2019), Elsevier BV[DOI]
  6. Hai-Ping Fu, "Rigidity theorem for integral pinched shrinking Ricci solitons", Monatshefte für Mathematik, 183(3), pp.487~494 (2017), Springer Science and Business Media LLC[DOI]
  7. Guangyue Huang, "Integral pinched gradient shrinking ρ-Einstein solitons", Journal of Mathematical Analysis and Applications, 451(2), pp.1045~1055 (2017), Elsevier BV[DOI]
  8. Guangyue Huang, "Rigidity of Riemannian manifolds with positive scalar curvature", Annals of Global Analysis and Geometry, 54(2), pp.257~272 (2018), Springer Science and Business Media LLC[DOI]
  9. G. Huang, "Rigidity of Einstein metrics as critical points of some quadratic curvature functionals on complete manifolds", arXiv:1804.10748, ~
  10. Seongtag Kim, "Rigidity of noncompact complete Bach-flat manifolds", Journal of Geometry and Physics, 60(4), pp.637~642 (2010), Elsevier BV[DOI]
  11. Seongtag Kim, "Rigidity of noncompact complete manifolds with harmonic curvature", Manuscripta Mathematica, 135(1-2), pp.107~116 (2011), Springer Science and Business Media LLC[DOI]
  12. Bingqing Ma, "Rigidity of complete noncompact Riemannian manifolds with harmonic curvature", Journal of Geometry and Physics, 124, pp.233~240 (2018), Elsevier BV[DOI]
  13. Bingqing Ma, "Rigidity of Einstein metrics as critical points of quadratic curvature functionals on closed manifolds", Nonlinear Analysis, 175, pp.237~248 (2018), Elsevier BV[DOI]
  14. R. Schoen, "Conformally flat manifolds, Kleinian groups and scalar curvature", Inventiones Mathematicae, 92(1), pp.47~71 (1988), Springer Science and Business Media LLC[DOI]

관련 논문AI추천

RIGIDITY CHARACTERIZATIONS OF COMPLETE RIEMANNIAN MANIFOLDS WITH α-BACH-FLATGuangyue Huang, Qianyu Zeng대한수학회 · 대한수학회지2021 · 제58권 제2호Rigidity of gradient shrinking and expanding Ricci solitonsFei Yang, Liangdi Zhang대한수학회 · 대한수학회보2017 · 제54권 제3호Gradient almost Ricci solitons with vanishing conditions on Weyl tensor and Bach tensor고진석, 황승수대한수학회 · 대한수학회지2020 · 제57권 제2호Some rigidity characterizations of Einstein metrics as critical points for quadratic curvature functionalsBingqing Ma, Guangyue Huang, Jie Yang대한수학회 · 대한수학회보2020 · 제57권 제6호Rigidity characterization of compact Ricci solitonsFengjiang Li, Jian Zhou대한수학회 · 대한수학회지2019 · 제56권 제6호Curvature estimates for gradient expanding Ricci solitonsLiangdi Zhang대한수학회 · 대한수학회보2021 · 제58권 제3호

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