• Title/Summary/Keyword: $L^p$ harmonic 1-forms

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L2 HARMONIC 1-FORMS ON SUBMANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY

  • Chao, Xiaoli;Lv, Yusha
    • Journal of the Korean Mathematical Society
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    • v.53 no.3
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    • pp.583-595
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    • 2016
  • In the present note, we deal with $L^2$ harmonic 1-forms on complete submanifolds with weighted $Poincar{\acute{e}}$ inequality. By supposing submanifold is stable or has sufficiently small total curvature, we establish two vanishing theorems for $L^2$ harmonic 1-forms, which are some extension of the results of Kim and Yun, Sang and Thanh, Cavalcante Mirandola and $Vit{\acute{o}}rio$.

FINITENESS AND VANISHING RESULTS ON HYPERSURFACES WITH FINITE INDEX IN ℝn+1: A REVISION

  • Van Duc, Nguyen
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.3
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    • pp.709-723
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    • 2022
  • In this note, we revise some vanishing and finiteness results on hypersurfaces with finite index in ℝn+1. When the hypersurface is stable minimal, we show that there is no nontrivial L2p harmonic 1-form for some p. The our range of p is better than those in [7]. With the same range of p, we also give finiteness results on minimal hypersurfaces with finite index.

COMPLETE NONCOMPACT SUBMANIFOLDS OF MANIFOLDS WITH NEGATIVE CURVATURE

  • Ya Gao;Yanling Gao;Jing Mao;Zhiqi Xie
    • Journal of the Korean Mathematical Society
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    • v.61 no.1
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    • pp.183-205
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    • 2024
  • In this paper, for an m-dimensional (m ≥ 5) complete non-compact submanifold M immersed in an n-dimensional (n ≥ 6) simply connected Riemannian manifold N with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of M, the norm of its weighted mean curvature vector |Hf| and the weighted real-valued function f, we can obtain: • several one-end theorems for M; • two Liouville theorems for harmonic maps from M to complete Riemannian manifolds with nonpositive sectional curvature.