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L2-harmonic p-forms on submanifolds with finite total curvature

Funds:  Supported by the Natural Science Foundation of Anhui Provincia Education Department (KJ2017A341) and the Talent Project of Fuyang Normal University (RCXM201714), the second author is supported by the Natural Science Foundation of Anhui Province of China (1608085MA03) and the Fundamental Research Funds of Tongling Xueyuan Rencai Program (2015TLXYRC09).
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https://doi.org/10.3969/j.issn.0253-2778.2020.03.006
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  • Corresponding author: ZHOU Jundong(Corresponding author), male, born in 1983, PhD/associate professor. Research field: Algebraic coding. E-mail: zhoujundong109@163.com
  • Received Date: 13 February 2019
  • Accepted Date: 10 January 2020
  • Rev Recd Date: 10 January 2020
  • Publish Date: 31 March 2020
    Abstract

  • Abstract

    Let M be an n-dimensional complete submanifold with flat normal bundle in an (n+l)-dimensional sphere Sn+l. Let Hp(L2(M)) be the space of all L2-harmonic p-forms (2≤p≤n-2) on M. Firstly, we show that Hp(L2(M)) is trivial if the total curvature of M is less than a positive constant depending only on n. Secondly, we show that the dimension of Hp(L2(M)) is finite provided the total curvature of M is finite.

    Abstract

    Let M be an n-dimensional complete submanifold with flat normal bundle in an (n+l)-dimensional sphere Sn+l. Let Hp(L2(M)) be the space of all L2-harmonic p-forms (2≤p≤n-2) on M. Firstly, we show that Hp(L2(M)) is trivial if the total curvature of M is less than a positive constant depending only on n. Secondly, we show that the dimension of Hp(L2(M)) is finite provided the total curvature of M is finite.
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    • References(13)
  • [1]
    YUN G. Total scalar curvature andL2-harmonic 1-forms on a minimal hypersurface in Euclidean space[J]. Geom. Dedicata, 2002, 89: 135-141.
    [2]
    NI L. Gap theorems for minimal submanifolds in Rn+1[J]. Commun. Anal. Geom., 2001,9: 641-656.
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    CAVALCANTE M P, MIRANDOLA H, VITRIO F. L2-harmonic 1-forms on submanifolds with ?倕nite total curvature[J]. J. Geom. Anal., 2014, 24: 205-222.
    [7]
    ZHU P, FANG SW. Finiteness of non-parabolic ends on submanifolds in spheres[J]. Ann. Global Anal. Geom., 2014, 46: 187-196.
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    LIN H Z. Vanishing theorems for harmonic forms on complete submanifolds in Euclidean space[J]. J. Math. Anal. Appl., 2015, 425: 774-787.
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    GAN W Z, ZHU P, FANG S W. L2-harmonic 2-forms on minimal hypersurfaces in spheres[J]. Diff. Geom. Appl., 2018,56: 202-210.
    [10]
    LI P, WANG J P. Minimal hypersurfaces with fnite index[J]. Math. Res. Lett., 2002, 9: 95-104.
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    HAN Y B. The topological structure of complete noncompact sumanifolds in spheres[J]. J. Math. Anal. Appl., 2018, 457: 991-1006.
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    LI P. Geometric Analysis[M]. Cambridge Stud. Adv. Math., 2012.
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    LI P. On the Sobolev constant and the p-spectrum of a compact Riemannian manifold[J]. Ann. Sci. c. Norm. Supér., 1980, 13: 451-468.
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    [1]
    YUN G. Total scalar curvature andL2-harmonic 1-forms on a minimal hypersurface in Euclidean space[J]. Geom. Dedicata, 2002, 89: 135-141.
    [2]
    NI L. Gap theorems for minimal submanifolds in Rn+1[J]. Commun. Anal. Geom., 2001,9: 641-656.
    [3]
    SEO K. Minimal submanifolds with small total scalar curvature in Euclidean space[J]. Kodai Math. J., 2008, 31: 113-119.
    [4]
    SEO K. Rigidity of minimal submanifolds in hyperbolic space[J]. Arch. Math. 2010, 94: 173-181.
    [5]
    FU H P, XU H W. Total curvature and L2-harmonic 1-forms on complete submanifolds in space forms[J]. Geom. Dedicata, 2010, 144: 129-140.
    [6]
    CAVALCANTE M P, MIRANDOLA H, VITRIO F. L2-harmonic 1-forms on submanifolds with ?倕nite total curvature[J]. J. Geom. Anal., 2014, 24: 205-222.
    [7]
    ZHU P, FANG SW. Finiteness of non-parabolic ends on submanifolds in spheres[J]. Ann. Global Anal. Geom., 2014, 46: 187-196.
    [8]
    LIN H Z. Vanishing theorems for harmonic forms on complete submanifolds in Euclidean space[J]. J. Math. Anal. Appl., 2015, 425: 774-787.
    [9]
    GAN W Z, ZHU P, FANG S W. L2-harmonic 2-forms on minimal hypersurfaces in spheres[J]. Diff. Geom. Appl., 2018,56: 202-210.
    [10]
    LI P, WANG J P. Minimal hypersurfaces with fnite index[J]. Math. Res. Lett., 2002, 9: 95-104.
    [11]
    HAN Y B. The topological structure of complete noncompact sumanifolds in spheres[J]. J. Math. Anal. Appl., 2018, 457: 991-1006.
    [12]
    LI P. Geometric Analysis[M]. Cambridge Stud. Adv. Math., 2012.
    [13]
    LI P. On the Sobolev constant and the p-spectrum of a compact Riemannian manifold[J]. Ann. Sci. c. Norm. Supér., 1980, 13: 451-468.
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