DOI QR코드

DOI QR Code

MONOTONICITY OF THE FIRST EIGENVALUE OF THE LAPLACE AND THE p-LAPLACE OPERATORS UNDER A FORCED MEAN CURVATURE FLOW

  • Mao, Jing
  • Received : 2017.11.12
  • Accepted : 2018.02.21
  • Published : 2018.11.01

Abstract

In this paper, we would like to give an answer to Problem 1 below issued firstly in [17]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the p-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicate constraint, we can give lower bounds for the first nonzero closed eigenvalue of the Laplacian provided additionally the second fundamental form of the initial hypersurface satisfies a pinching condition.

Keywords

Ricci-Hamilton flow;mean curvature flow;Laplace operator;p-Laplace operator

References

  1. J. Mao, Forced hyperbolic mean curvature flow, Kodai Math. J. 35 (2012), no. 3, 500-522.
  2. J. Mao, Deforming two-dimensional graphs in $R^4$ by forced mean curvature flow, Kodai Math. J. 35 (2012), no. 3, 523-531. https://doi.org/10.2996/kmj/1352985452
  3. J. Mao, A class of rotationally symmetric quantum layers of dimension 4, J. Math. Anal. Appl. 397 (2013), no. 2, 791-799.
  4. J. Mao, Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel, J. Math. Pures Appl. (9) 101 (2014), no. 3, 372-393.
  5. J. Mao, Eigenvalue estimation and some results on finite topological type, Ph.D. thesis, IST-UTL, 2013.
  6. J. Mao, G. Li, and C. Wu, Entire graphs under a general flow, Demonstratio Math. 42 (2009), no. 3, 631-640.
  7. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
  8. J. Roth, A remark on almost umbilical hypersurfaces, Arch. Math. (Brno) 49 (2013), no. 1, 1-7.
  9. K. Shiohama and H. W. Xu, Rigidity and sphere theorems for submanifolds, Kyushu J. Math. 48 (1994), no. 2, 291-306. https://doi.org/10.2206/kyushujm.48.291
  10. K. Shiohama and H. W. Xu, Rigidity and sphere theorems for submanifolds. II, Kyushu J. Math. 54 (2000), no. 1, 103-109.
  11. L. Zhao, The first eigenvalue of p-Laplace operator under powers of the mth mean curvature flow, Results Math. 63 (2013), no. 3-4, 937-948.
  12. B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151-171. https://doi.org/10.1007/BF01191340
  13. E. Cabezas-Rivas and C. Sinestrari, Volume-preserving flow by powers of the mth mean curvature, Calc. Var. Partial Differential Equations 38 (2010), no. 3-4, 441-469.
  14. X. Cao, Eigenvalues of ($-{\Delta}+{\frac{R}{2}}$) on manifolds with nonnegative curvature operator, Math. Ann. 337 (2007), no. 2, 435-441.
  15. X. Cao, First eigenvalues of geometric operators under the Ricci flow, Proc. Amer. Math. Soc. 136 (2008), no. 11, 4075-4078. https://doi.org/10.1090/S0002-9939-08-09533-6
  16. X. Cao, S. Hou, and J. Ling, Estimate and monotonicity of the first eigenvalue under the Ricci flow, Math. Ann. 354 (2012), no. 2, 451-463.
  17. P. Freitas, J. Mao, and I. Salavessa, Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds, Calc. Var. Partial Differential Equations 51 (2014), no. 3-4, 701-724.
  18. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306.
  19. G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266.
  20. G. Li and I. Salavessa, Forced convex mean curvature flow in Euclidean spaces, Manuscripta Math. 126 (2008), no. 3, 333-351. https://doi.org/10.1007/s00229-008-0181-z
  21. G. H. Li, J. Mao, and C. X. Wu, Convex mean curvature flow with a forcing term in direction of the position vector, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 2, 313-332.
  22. J.-F. Li, Eigenvalues and energy functionals with monotonicity formulae under Ricci flow, Math. Ann. 338 (2007), no. 4, 927-946.
  23. L. Ma, Eigenvalue monotonicity for the Ricci-Hamilton flow, Ann. Global Anal. Geom. 29 (2006), no. 3, 287-292.

Acknowledgement

Supported by : NSF of China, Fok Ying-Tong Education Foundation