# Yamabe Invariants, Homogeneous Spaces, and Rational Complex Surfaces

## TL;DR

This paper explores the Yamabe invariant, a key geometric quantity, and its connections to static potentials and Laplacian eigenvalues on smooth compact manifolds, shedding light on an open problem in differential geometry.

## Contribution

It establishes a link between the Yamabe invariant and spectral properties of manifolds, providing new insights into an open problem in geometric analysis.

## Key findings

- Connection between Yamabe invariant and Laplacian eigenvalues
- Insights into static potentials on compact manifolds
- Potential implications for open problems in geometry

## Abstract

The Yamabe invariant is a diffeomorphism invariant of smooth compact manifolds that arises from the normalized Einstein-Hilbert functional. This article highlights the manner in which one compelling open problem regarding the Yamabe invariant appears to be closely tied to static potentials and the first eigenvalue of the Laplacian.