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<p>Let <i>M</i> be an <i>m</i>(≥ 2)-dimensional closed orientable submanifold in an <i>n</i>-dimensional complete simply-connected Riemannian manifold <i>N</i>, where the sectional curvature of <i>N</i> is bounded above by <i>δ</i>. When <i>δ</i> &lt; 0, inspired by Niu–Xu [20], we give new upper bounds for the first nonzero eigenvalues of the <i>p</i>-Laplacian and the <i>L</i><sub><i>T</i></sub> operator, respectively. These generalize Niu–Xu’s work for the Laplacian [20] and improve the estimates due to Chen [4] for the <i>p</i>-Laplacian and Grosjean [14] for the <i>L</i><sub><i>T</i></sub> operator, respectively. We also obtain several Reilly-type inequalities for the weighted manifolds and some boundary value problems.</p>

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Reilly-type inequalities for submanifolds in Cartan–Hadamard manifolds

  • 期刊论文

  • 发表日期：

    2025年1月20日

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  • Hang Chen,
  • Xudong Gui

摘要

Let M be an m(≥ 2)-dimensional closed orientable submanifold in an n-dimensional complete simply-connected Riemannian manifold N, where the sectional curvature of N is bounded above by δ. When δ < 0, inspired by Niu–Xu [20], we give new upper bounds for the first nonzero eigenvalues of the p-Laplacian and the LT operator, respectively. These generalize Niu–Xu’s work for the Laplacian [20] and improve the estimates due to Chen [4] for the p-Laplacian and Grosjean [14] for the LT operator, respectively. We also obtain several Reilly-type inequalities for the weighted manifolds and some boundary value problems.