Cheeger type inequalities associated with isocapacitary constants on Riemannian manifolds with boundary

TL;DR

This work introduces a sharp Cheeger-type framework for Steklov-type problems on Riemannian manifolds by linking the first Steklov eigenvalue to the isocapacitary constant $Γ_{∂}(M)$ via capacity methods inspired by Maz'ya. It yields two-sided bounds $\frac{1}{4}Γ_{∂}(M) ≤ σ_1(M) ≤ 2Γ_{∂}(M)$ for compact manifolds and analogous bounds for the Dirichlet-to-Neumann spectrum on non-compact manifolds, including Steklov-Dirichlet variants. The paper also demonstrates concrete applications: estimates on compact hyperbolic surfaces and explicit spectral bounds for the Dirichlet-to-Neumann operator on hyperbolic half-spaces, with an explicit DtN formula and dimension-dependent lower bounds. The techniques provide a capacity-based, geometric route to spectral estimates, offering concrete, scalable constants tied to boundary capacity properties. These results advance understanding of Steklov spectra through isocapacitary geometry with practical implications for geometry and analysis on manifolds with boundary.

Abstract

In this paper, we study the Steklov eigenvalue of a Riemannian manifold (M, g) with smooth boundary. For compact M , we establish a Cheeger-type inequality for the first Steklov eigenvalue by the isocapacitary constant. For non-compact M , we estimate the bottom of the spectrum of the Dirichlet-to-Neumann operator by the isocapacitary constant.

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Theorems & Definitions (26)

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