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<p>Let (Ω<sup><i>n</i>+1</sup>, <i>g</i>) be an (<i>n</i> + 1)-dimensional smooth compact connected Riemannian manifold with connected boundary Σ = ∂Ω, whose principal curvatures are bounded from below by a positive constant. In this paper, we provide some sharp lower bounds of the first eigenvalue of the Laplacian on the boundary Σ. In the two dimensional case, we establish our estimates for manifolds whose Gaussian curvature <i>K</i> satisfies <i>K</i> ≥ ±1; in the higher dimensional case, we give our estimates for manifolds with the so-called Ric<Stack> <sub><i>k</i></sub> <sup>Σ</sup> </Stack> condition (see Section 2).</p>

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Lower Bound Estimates of the First Eigenvalue for Boundary of Compact Manifolds

  • 期刊论文

  • 发表日期：

    2025年1月24日

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  • Yiwei Liu,
  • Yihu Yang

摘要

Let (Ωn+1, g) be an (n + 1)-dimensional smooth compact connected Riemannian manifold with connected boundary Σ = ∂Ω, whose principal curvatures are bounded from below by a positive constant. In this paper, we provide some sharp lower bounds of the first eigenvalue of the Laplacian on the boundary Σ. In the two dimensional case, we establish our estimates for manifolds whose Gaussian curvature K satisfies K ≥ ±1; in the higher dimensional case, we give our estimates for manifolds with the so-called Ric k Σ condition (see Section 2).