# Higher order Cheeger inequalities for Steklov eigenvalues

## TL;DR

This paper establishes higher order Cheeger inequalities for Steklov eigenvalues across finite, measurable, and Riemannian spaces, extending previous bounds for the first eigenvalue to higher orders using novel analytical techniques.

## Contribution

It introduces new lower bounds for higher order Steklov eigenvalues based on Cheeger-type constants, generalizing earlier results and developing a unified approach across different mathematical settings.

## Key findings

- Derived lower bounds for $k$-th Steklov eigenvalues.
- Extended Cheeger inequalities to higher eigenvalues.
- Connected Steklov eigenvalues with Dirichlet-Steklov connectivity spectrum.

## Abstract

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considering a family of accelerated Markov operators in the finite and mesurable situations and of mass concentration deformations of the Laplace-Beltrami operator in the manifold setting which converges uniformly to the Steklov operator. As an intermediary step in the proof of the higher order Cheeger type inequality, we define the Dirichlet-Steklov connectivity spectrum and show that the Dirichlet connectivity spectra of this family of operators converges to (or bounded by) the Dirichlet-Steklov spectrum uniformly. Moreover, we obtain bounds for the Steklov eigenvalues in terms of its Dirichlet-Steklov connectivity spectrum which is interesting in its own right and is more robust than the higher order Cheeger type inequalities. The Dirichlet-Steklov spectrum is closely related to the Cheeger-Steklov constants.