Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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RIGIDITY OF IMMERSED SUBMANIFOLDS IN A HYPERBOLIC SPACE

  • Nguyen, Thac Dung (Department of Mathematics, Mechanics and Informatics Hanoi University of Sciences (HUS-VNU))
  • Received : 2015.11.30
  • Published : 2016.11.30
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Abstract

Let $M^n$, $2{\leq}n{\leq}6$ be a complete noncompact hypersurface immersed in ${\mathbb{H}}^{n+1}$. We show that there exist two certain positive constants 0 < ${\delta}{\leq}1$, and ${\beta}$ depending only on ${\delta}$ and the first eigenvalue ${\lambda}_1(M)$ of Laplacian such that if M satisfies a (${\delta}$-SC) condition and ${\lambda}_1(M)$ has a lower bound then $H^1(L^2(M))=0$. Excepting these two conditions, there is no more additional condition on the curvature.

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References

  1. H. D. Cao, Y. Shen, and S. Zhu, The structure of stable minimal hypersurfaces in ${\mathbb{R}}^{n+1}$, Math. Res. Lett. 4 (1997), no. 5, 637-644. https://doi.org/10.4310/MRL.1997.v4.n5.a2
  2. G. Carron, $L^2$ harmonic forms on non compact manifolds, Lectures given at CIMPA's summer school "Recent Topics in Geometric analysis", Arxiv:0704.3194v1.
  3. G. Carron, Une suite exacte en $L^2$-cohomologie, Duke Math. J. 95 (1998), no. 2, 343-372. https://doi.org/10.1215/S0012-7094-98-09510-2
  4. L. F. Cheung and P. F. Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001), no. 3, 525-530. https://doi.org/10.1007/PL00004840
  5. N. T. Dung and K. Seo, Vanishing theorems for $L^2$ harmonic 1-forms on complete submanifolds in a Riemannian manifold, J. Math. Anal. Appl. 423 (2015), no. 2, 1594-1609. https://doi.org/10.1016/j.jmaa.2014.10.076
  6. H. P. Fu and Z. Q. Li, $L^2$ harmonic 1-forms on complete submanifolds in Euclidean space, Kodai Math. J. 32 (2009), no. 3, 432-441. https://doi.org/10.2996/kmj/1257948888
  7. H. P. Fu and D. Y. Yang, Vanishing theorems on complete manifolds with weighted Poincare inequality and applications, Nagoya Math. J. 206 (2012), 25-37. https://doi.org/10.1215/00277630-1548475
  8. D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727.
  9. J. J. Kim and G. Yun, On the structure of complete hypersurfaces in a Riemannian manifold of nonnegative curvature and $L^2$ harmonic forms, Arch. Math. 100 (2013), no. 4, 369-380. https://doi.org/10.1007/s00013-013-0486-3
  10. P. F. Leung, An estimate on the Ricci curvature of a submanifold and some applications, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1051-1063. https://doi.org/10.1090/S0002-9939-1992-1093601-7
  11. P. Li and L. F. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), no. 2, 359-383. https://doi.org/10.4310/jdg/1214448079
  12. P. Li and J. P. Wang, Stable minimal hypersurfaces in a nonnegatively curved manifold, J. Reine Angew. Math. 566 (2004), 215-230.
  13. L. Ni, Gap theorems for minimal submanifolds in ${\mathbb{R}}^{n+1}$, Comm. Anal. Geom. 9 (2001), no. 3, 641-656. https://doi.org/10.4310/CAG.2001.v9.n3.a2
  14. K. Seo, $L^2$ harmonic 1-forms on minimal submanifolds in hyperbolic space, J. Math. Anal. Appl. 371 (2010), no. 2, 546-551. https://doi.org/10.1016/j.jmaa.2010.05.048