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Isoperimetric profile comparisons and Yamabe constants

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Abstract

We estimate from below the isoperimetric profile of \({S^2 \times {\mathbb R}^2}\) and use this information to obtain lower bounds for the Yamabe constant of \({S^2 \times {\mathbb R}^2}\) . This provides a lower bound for the Yamabe invariants of products S 2 ×  M 2 for any closed Riemann surface M. Explicitly we show that Y (S 2 ×  M 2) >  (2/3)Y(S 4).

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References

  1. Ammann, B., Dahl, M., Humbert, E.: Smooth Yamabe invariant and surgery. preprint arXiv.org math-DG. 0804.1418

  2. Akutagawa K., Florit L., Petean J.: On Yamabe constants of Riemannian products. Comm. Anal. Geom. 15, 947–969 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Akutagawa K., Neves A.: 3-manifolds with Yamabe invariant greater than that of RP 3. J. Differential Geom. 75, 359–386 (2007)

    MATH  MathSciNet  Google Scholar 

  4. Aubin T.: Equations differentielles non-lineaires et probleme de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55, 269–296 (1976)

    MATH  MathSciNet  Google Scholar 

  5. Bayle, V.: Propriétés du concavité du profil isopérimétrique et applications. Ph.D. Thesis, p. 52 (2004)

  6. Bohm C., Wang M., Ziller W.: A variational approach for compact homogeneous Einstein manifolds. Geom. Funct. Anal. 14, 681–733 (2004)

    Article  MathSciNet  Google Scholar 

  7. Bray H., Neves A.: Classification of prime 3-manifolds with Yamabe invariant greater than RP 3. Ann. of Math. 159, 407–424 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Druet O.: Sharp local isoperimetric inequalities involving the scalar curvature. Proc. Amer. Math. Soc. 130(8), 2351–2361 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gursky M., LeBrun C.: Yamabe invariants and Spin c structures. Geom. Funct. Anal. 8, 965–977 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ilias S.: Constantes explicites pour les inegalites de Sobolev sur les varietes riemannienes compactes. Ann. Inst. Fourier (Grenoble) 33(2), 151–165 (1983)

    MATH  MathSciNet  Google Scholar 

  11. Kobayashi, O.: On large scalar curvature Research Report 11. Dept. Math. Keio Univ. (1985)

  12. Kobayashi O.: Scalar curvature of a metric with unit volume. Math. Ann. 279, 253–265 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. LeBrun C.: Yamabe constants and the perturbed Seiberg-Witten equations. Comm. Anal. Geom. 5, 535–553 (1997)

    MATH  MathSciNet  Google Scholar 

  14. Morgan F.: Isoperimetric estimates in products. Ann. Global Anal. Geom. 30, 73–79 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Morgan F.: In polytopes, small balls around some vertex minimize perimeter. J. Geom. Anal. 17, 97–106 (2007)

    MATH  MathSciNet  Google Scholar 

  16. Obata M.: The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geom. 6, 247–258 (1971)

    MATH  MathSciNet  Google Scholar 

  17. Pedrosa R.: The isoperimetric problem in spherical cylinders. Ann. Global Anal. Geom. 26, 333–354 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Petean J.: Computations of the Yamabe invariant. Math. Res. Lett. 5, 703–709 (1998)

    MATH  MathSciNet  Google Scholar 

  19. Petean J.: Best Sobolev constants and manifolds with positive scalar curvature metrics. Ann. Global Anal. Geom. 20, 231–242 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Petean J., Yun G.: Surgery and the Yamabe invariant. Geom. Funct. Anal. 9, 1189–1999 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ros, A.: The isoperimetric problem, Global Theory of Minimal Surfaces (Proc. Clay Math. Institute Summer School, 2001). Amer. Math. Soc., Providence. (2005)

  22. Schoen R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20, 479–495 (1984)

    MATH  MathSciNet  Google Scholar 

  23. Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Lectures Notes in Math, vol 1365, pp. 120–154. Springer-Verlag, Berlin (1989)

  24. Trudinger N.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. 22, 265–274 (1968)

    MATH  MathSciNet  Google Scholar 

  25. Yamabe H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12, 21–37 (1960)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CIMAT, A.P. 402, 36000, Guanajuato, GTO, Mexico

    Jimmy Petean & Juan Miguel Ruiz

  2. Departamento de Matemáticas, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina

    Jimmy Petean

Authors
  1. Jimmy Petean
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  2. Juan Miguel Ruiz
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Correspondence to Jimmy Petean.

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Petean, J., Ruiz, J.M. Isoperimetric profile comparisons and Yamabe constants. Ann Glob Anal Geom 40, 177–189 (2011). https://doi.org/10.1007/s10455-011-9252-6

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  • DOI: https://doi.org/10.1007/s10455-011-9252-6

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