Lower Bound Estimates of the First Steklov Eigenvalue for Compact Manifolds

Abstract

Let (Ωⁿ⁺¹, g) be an (n+1)-dimensional smooth compact connected Riemannian manifold with smooth boundary ∂Ω = Σ. Assume that the Ricci curvature of Ω is nonnegative and the principal curvatures of Σ are bounded from below by a positive constant c. In this paper, by constructing a new weight function, the authors obtain a lower bound of the first nonzero Steklov eigenvalue under the assumption that Sec_Ω ≥ −k, where k is a positive constant. The authors also extend this result to the Steklov-type eigenvalue problem of the weighted Laplacian on a metric measure space.

Keywords

Lower bound / Steklov eigenvalue / Weight function / 53C21 / 53C24

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