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Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons

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Abstract

In this paper, we study the eigenvalue problem of poly-drifting Laplacian and get a general inequality for lower order eigenvalues on compact smooth metric measure spaces with boundary (possibly empty). Applying this general inequality, we obtain some universal inequalities for lower order eigenvalues for the eigenvalue problem of poly-drifting Laplacian on bounded connected domains in Euclidean spaces or unit spheres. Moreover, we separately get some universal inequalities for the eigenvalue problem of poly-drifting Laplacian on bounded connected domains in the Gaussian and cylinder shrinking solitons.

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References

  1. Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues, In: Davies, E.B., Safalov, Y. (eds.) Spectral Theory and Geometry (Edinburgh, 1998). London Math. Soc. Lecture Notes, vol. 273, pp. 95–139. Cambridge University Press, Cambridge (1999)

  2. Cao, H.D.: Recent progress on Ricci solitons. In: Recent Advances in Geometric Analysis. Adv. Lect. Math., vol. 11, pp. 1–38. International Press, Somerville (2010)

  3. Chen, D., Cheng, Q.M.: Extrinsic estimates for eigenvalues of the Laplace operator. J. Math. Soc. Jpn. 60, 325–339 (2008)

    Article  MATH  Google Scholar 

  4. Chen, D., Cheng, Q.M., Wang, Q., Xia, C.: On eigenvalues of a system of elliptic equations and of the biharmonic operator. J. Math. Anal. Appl. 387, 1146–1159 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheng, Q.M., Huang, G., Wei, G.: Estimates for lower order eigenvalues of a clamped plate problem. Calc. Var. PDE. 38, 409–416 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cheng, Q.M., Ichikawa, T., Mametsuka, S.: Estimates for eigenvalues of the poly-Laplacian with any order. Commun. Contemp. Math. 11(4), 639–655 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cheng, Q.M., Ichikawa, T., Mametsuka, S.: Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere. Calc. Var. PDE. 36(4), 507–523 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheng, Q.M., Peng, Y.: Estimates for eigenvalues of \(\mathfrak{L}\) operator on self-shrinkers. Commun. Contemp. Math. 15(6), 23 (2013). (1350011)

  9. Cheng, Q.M., Yang, H.C.: Estimates on eigenvalues of Laplacian. Math. Ann. 331, 445–660 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cheng, Q.M., Yang, H.C.: Inequalities for eigenvalues of a clamped plate problem. Trans. Am. Math. Soc. 358(6), 2625–2635 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cheng, Q.M., Yang, H.C.: Bounds on eigenvalues of Dirchlet Laplacian. Math. Ann. 58(2), 159–175 (2007)

    Google Scholar 

  12. Colding, T.H., Minicozzi II, W.P.: Generic mean curvature flow I; generic singularities. Ann. Math. 175(2), 755–833 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cheng, X., Mejia, T., Zhou, D.: Eigenvalue estimate and compactness for closed \(f\)-minimal surfaces. Pac. J. Math. 271(2), 347–367 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cheng, X., Zhou, D.: Eigenvalues of the drifted Laplacian on complete metric measure spaces (2013) arXiv:1305.4116

  15. Du, F., Wu, C., Li, G., Xia, C.: Eigenvalues of the buckling problem of arbitrary order and of the polyharmonic operator on Ricci flat manifolds. J. Math. Anal. Appl. 417, 601–621 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  16. Du, F., Wu, C., Li, G., Xia, C.: Estimates for eigenvalues of the bi-drifting Laplacian operator. Zeitschrift füer Angewandte Mathematik und Physik (ZAMP). doi:10.1007/s00033-014-0426-5

  17. Futaki, A., Li, H., Li, X.D.: On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons. Ann. Glob. Anal. Geom. 44(2), 105–114 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Futaki, A., Sano, Y.: Lower diameter bounds for compact shrinking Ricci solitons. Asian J. Math. 17(1), 17–32 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. El Soufi, A., Harrell II, E.M., Ilias, S.: Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds. Trans. Am. Math. Soc. 361, 2337–2350 (2009)

    Article  MATH  Google Scholar 

  20. Harrell II, E.M., Stubbe, J.: On trace inequalities and the universal eigenvalue estimates for some partial differential operators. Trans. Am. Math. Soc. 349, 1797–1809 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Harrel II, E.M., Michel, P.L.: Commutator bounds for eigenvalues with applications to spectral geometry. Commun. Partial. Differ. Equ. 19, 2037–2055 (1994)

    Article  Google Scholar 

  22. Hile, G.N., Yeh, R.Z.: Inequalities for eigenvalues of the biharmonic operator. Pac. J. Math. 112, 115–133 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jost, J., Li-Jost, X., Wang, Q., Xia, C.: Universal bounds for eigenvalues of polyharmonic operator. Trans. Am. Math. Soc. 363(4), 1821–1854 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ma, L., Du, S.H.: Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. C. R. Math. Acad. Sci. Paris 348(21–22), 1203–1206 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Ma, L., Liu, B.Y.: Convex eigenfunction of a drifting Laplacian operator and the fundamental gap. Pac. J. Math. 240, 343–361 (2009)

    Article  MATH  Google Scholar 

  26. Ma, L., Liu, B.Y.: Convexity of the first eigenfunction of the drifting Laplacian operator and its applications. N. Y. J. Math. 14, 393–401 (2008)

    MATH  Google Scholar 

  27. Payne, L.E., Pólya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289–298 (1956)

    MATH  Google Scholar 

  28. Pereira, R.G., Adriano, L., Pina, R.: Universal bounds for eigenvalues of the polydrifting Laplacian operator in compact domains in the \(\mathbb{R}^{n}\) and \(\mathbb{S}^{n}\). Ann. Glob. Anal. Geom. 47, 373–397 (2015)

    Article  MathSciNet  Google Scholar 

  29. Wei, G.F., Wylie, W.: Comparison geometry for the Bakry–Émery Ricci tensor. J. Differ. Geom. 83, 377–405 (2009)

    MATH  MathSciNet  Google Scholar 

  30. Sun, H.J., Cheng, Q.M., Yang, H.C.: Lower order eigenvalues of Dirichlet Laplacian. Manuscripta Math. 125(2), 139–156 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Wang, Q., Xia, C.: Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds. J. Funct. Anal. 245, 334–352 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Wang, Q., Xia, C.: Inequalities for eigenvalues of a clamped plate problem. Calc. Var. PDE. 40(1–2), 273–289 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. Xia, C.: Eigenvalues on Riemannian Manifolds. Impresso no Brasil / Printed in Brazil Capa, Noni Geiger / Sérgio R. Vaz (2013)

  34. Xia, C., Xu, H.: Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds. Ann. Glob. Anal. Geom. 45, 155–166 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  35. Yang, H.C.: An estimate of the difference between consecutive eigenvalues. Preprint IC/91/60 of ICTP, Trieste (1991)

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Acknowledgments

F. Du was partially supported by CNPq, Brazil and the NSF of China (Grant No. 11401131). J. Mao was partially supported by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the NSF of China (Grant No. 11401131). Q.-L. Wang was partially supported by CNPq, Brazil. The authors would like to thank the anonymous referee for his or her careful reading and valuable comments.

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Authors and Affiliations

  1. School of Mathematics and Physics Science, Jingchu University of Technology, Jingmen, 448000, China

    Feng Du

  2. Departamento de Matemática, Universidade de Brasilia, Brasília, DF, 70910-900, Brazil

    Feng Du & Qiaoling Wang

  3. Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, 264209, China

    Jing Mao

  4. School of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Chuanxi Wu

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  1. Feng Du
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  2. Jing Mao
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Correspondence to Jing Mao.

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Du, F., Mao, J., Wang, Q. et al. Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons. Ann Glob Anal Geom 48, 255–268 (2015). https://doi.org/10.1007/s10455-015-9469-x

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  • DOI: https://doi.org/10.1007/s10455-015-9469-x

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