A Poincar\'{e}-type inequality and a related eigenvalue problem

TL;DR

This paper establishes a Poincaré-type inequality on the unit circle using curvature theorems, generalizes it to higher dimensions, and explores the relationship between the best constant and eigenvalues on manifolds.

Contribution

It introduces a new Poincaré-type inequality involving a weighted average, extends the result to higher-dimensional manifolds, and analyzes the eigenvalue problem related to the inequality.

Findings

The inequality holds for smooth positive functions on the circle.

The proof leverages Fenchel's theorem on total curvature.

The best constant may differ from the first eigenvalue on certain manifolds.

Abstract

Given a smooth positive function $f$ defined on the unit circle satisfying a simple condition, we obtain a Poincar\'{e}-type inequality for an arbitrary function $u$ whose weighted average with respect to $f$ is zero. The proof uses Fenchel's theorem about the total curvature of closed space curves in an essential way. Next we consider the generalization of this result to higher dimensional closed Riemannian manifold and reduce it to an eigenvalue problem. Finally, we point out that even though such Poincar\'{e}-type inequality still holds, the best constant $\lambda_1(f)$ might be different from the first eigenvalue $\lambda_1$ by constructing explicit examples on the standard spheres and flat tori.