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Stability of total scalar curvature and the critical point equation

황승수|윤갑진
대한수학회 (The Korean Mathematical Society)|2024.01|대한수학회보 (Bulletin of the KMS)|제61권 제1호|pp. 281-290|DOI 10.4134/BKMS.b230111

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참고문헌

키워드

EinsteinStability of the curvature functionalthe total scalar curvature
주제분류수학

초록

We consider the total scalar curvature functional, and show that if the second variation in the transverse traceless tensor direction is negative, then the metric is Einstein. We also find the relation between the second variation and the Lichnerowicz Laplacian.

Abstract

We consider the total scalar curvature functional, and show that if the second variation in the transverse traceless tensor direction is negative, then the metric is Einstein. We also find the relation between the second variation and the Lichnerowicz Laplacian.

참고문헌 (11)

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  3. R. T. Jantzen, "Conformal geometry and spatially homogeneous cosmology", Ann. Inst. H. Poincar´e Sect. A(N. S. ), 33(2), pp.121~146 (1980)
  4. N. Koiso, "On the second derivative of the total scalar curvature", Osaka Math. J, 16(2), pp.413~421 (1979)
  5. A. Lichnerowicz, "Propagateurs et commutateurs en relativite g´enerale", Inst. Hautes Etudes Sci. Publ. Math., 10, pp.56~ (1961)
  6. Y. Muto, "On Einstein metrics", J. Differential Geometry, 9, pp.521~530 (1974)
  7. M. Obata, "Certain conditions for a Riemannian manifold to be isometric with a sphere", J. Math. Soc. Japan, 14, pp.333~340 (1962)[DOI]
  8. W. Sheng, "Variational properties of quadratic curvature functionals", Sci. China Math, 62(9), pp.1765~1778 (2019)[DOI]
  9. S. E. Stepanov, "On the Licherowicz Laplacian", J. Math. Sci, 263, pp.415~422 (2022)
  10. J. A. Viaclovsky, "Critical metrics for Riemannian curvature functionals, in Geometric analysis", IAS, 2016, pp.22, 197~274
  11. G. Yun, "Gap theorems on critical point equation of the total scalar curvature with divergence-free Bach tensor", Taiwanese J. Math, 23(4), pp.841~855 (2019)[DOI]

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