# Upper bounds for higher-order Poincar'e constants

## TL;DR

This paper introduces higher-order Poincaré constants for weighted manifolds, providing new upper bounds for eigenvalues of various Laplacians and isoperimetric constants, with applications to manifolds with boundary.

## Contribution

It defines higher-order Poincaré constants and derives upper bounds for eigenvalues and isoperimetric constants, extending previous estimates to new settings.

## Key findings

- Upper bounds for weighted Laplacian eigenvalues
- Upper bounds for Dirichlet eigenvalues and isoperimetric constants
- Application to inscribed radii in manifolds with boundary

## Abstract

We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the $p$-Laplacian. In the case of the closed eigenvalue problem and the Neumann eigenvalue problem these are related with the estimates obtained by Chung-Grigor'yan-Yau and Gozlan-Herry. We also obtain similar upper bounds for Dirichlet eigenvalues and multi-way isoperimetric constants. As an application, for manifolds with boundary of non-negative dimensional weighted Ricci curvature, we give upper bounds for inscribed radii in terms of dimension and the first Dirichlet Poincar'e constant.