Estimates for Eigenvalues of the Dirichlet Laplacian on Riemannian Manifolds

TL;DR

The paper studies Dirichlet Laplacian eigenvalues on bounded domains in complete Riemannian manifolds, deriving universal upper and lower bounds by extending Li–Yau and Yang inequalities to curved settings. The authors develop a unified framework using Riesz means, Harrell–Stubbe-type estimates, and Tauberian arguments, incorporating curvature through mean-curvature terms and projective-space embeddings. They obtain explicit lower bounds for eigenvalues on domains in projective spaces and minimal submanifolds, and establish Weyl-type heat-kernel bounds that connect spectral data to embedding geometry. This work advances spectral geometry by providing quantitative bounds that relate geometry and spectrum in non-Euclidean settings, with potential applications to geometric analysis and mathematical physics.

Abstract

We revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds. By building on classical results like Li-Yau's and Yang's inequalities, we derive upper and lower bounds for eigenvalues. For the projective spaces and their minimal submanifolds, we also give explicit estimates on lower bounds for eigenvalues of the Dirichlet Laplacian.

Theorems & Definitions (31)

• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Theorem 2.2: The recursion formula of Cheng and Yang ChengYang07
• proof : Proof of Theorem \ref{['thm:ChengYang07']}
• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma 3.1: ChenCheng08