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Deformation of the \(\sigma _2\)-curvature

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Abstract

Our main goal in this work is to deal with results concern to the \(\sigma _2\)-curvature. First we find a symmetric 2-tensor canonically associated to the \(\sigma _2\)-curvature and we present an almost-Schur-type lemma. Using this tensor, we introduce the notion of \(\sigma _2\)-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and \(\sigma _2\)-curvature. With a suitable condition on the \(\sigma _2\)-curvature, we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the three-dimensional torus does not admit a metric with constant scalar curvature and nonnegative \(\sigma _2\)-curvature unless it is flat.

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References

  1. Aronszajn, N.: Sur l’unicité du prolongement des solutions des équations aux dérivées partielles elliptiques du second ordre. C. R. Acad. Sci. Paris 242, 723–725 (1956)

    MathSciNet  MATH  Google Scholar 

  2. Ballmann, W., Brin, M., Spatzier, R.: Structure of manifolds of nonpositive curvature. II. Ann. Math. (2) 122(2), 205–235 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berger, M., Ebin, D.: Some decompositions of the space of symmetric tensors on a Riemannian manifold. J. Differ. Geom. 3, 379–392 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  4. Catino, G.: Some rigidity results on critical metrics for quadratic functionals. Calc. Var. Partial Differ. Equ. 54(3), 2921–2937 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang, S.-Y. A., Gursky, M., Yang, P.: Remarks on a fourth order invariant in conformal geometry. Aspects of Mathematics. In: Mok, N. (ed.) Proceedings of a conference in Algebra, Geometry and Several Complex Variable, pp. 353–372. Hong Kong University (1996)

  6. Cheng, X.: A generalization of almost-Schur lemma for closed Riemannian manifolds. Ann. Glob. Anal. Geom. 43(2), 153–160 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, X.: An almost-Schur type lemma for symmetric $(2,0)$ tensors and applications. Pac. J. Math. 267(2), 325–340 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, xxxvi+608. American Mathematical Society; Science Press, Providence; New York (2006)

    Google Scholar 

  9. Cordes, H.O.: Über die eindeutige Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben. Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa. 1956, 239–258 (1956)

    MathSciNet  MATH  Google Scholar 

  10. Fischer, A.E., Marsden, J.E.: Deformations of the scalar curvature. Duke Math. J. 42(3), 519–547 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gursky, M.J., Malchiodi, A.: A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature. J. Eur. Math. Soc. (JEMS) 17(9), 2137–2173 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gursky, M.J., Viaclovsky, J.A.: Rigidity and stability of Einstein metrics for quadratic curvature functionals. J. Reine Angew. Math. 700, 37–91 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Kröncke, K.: Stability of Einstein manifolds. Dissertation. Universität Potsdam (2013)

  14. Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. (N.S.) 17(1), 37–91 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lellis, C.D., Topping, P.M.: Almost-Schur lemma. Calc. Var. Partial Differ. Equ. 43(3–4), 347–354 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, Y.-J., Yuan, W.: Deformations of Q-curvature I. Calc. Var. Partial Differ. Equ. 55(4), 101 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lin, Y.-J., Yuan, W.: A symmetric $2$-tensor canonically associated to $Q$-curvature and its applications (2016). arXiv:1602.01212v2

  18. Schoen, R., Yau, S.T.: Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature. Ann. Math. (2) 110(1), 127–142 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schoen, R., Yau, S.T.: Lectures on Differential Geometry, v+235. International Press, Cambridge (1994)

    MATH  Google Scholar 

  20. Silva Santos, A.: Solutions to the singular $\sigma _2$-Yamabe problem with isolated singularities. Indiana Univ. Math. J. 66(3), 741–790 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251–275 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  22. Viaclovsky, J.A.: Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. 101(2), 283–316 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Federal University of Sergipe, Av. Marechal Rondon s/n, São Cristóvão, SE, 49100-000, Brazil

    Almir Silva Santos & Maria Andrade

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  1. Almir Silva Santos
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  2. Maria Andrade
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Correspondence to Almir Silva Santos.

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This research was partly supported by CAPES/FAPITEC-Brazil.

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Santos, A.S., Andrade, M. Deformation of the \(\sigma _2\)-curvature. Ann Glob Anal Geom 54, 71–85 (2018). https://doi.org/10.1007/s10455-018-9593-5

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  • DOI: https://doi.org/10.1007/s10455-018-9593-5

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