# Existence of metrics maximizing the first eigenvalue on non-orientable surfaces

## TL;DR

This paper proves the existence of specific metrics that maximize the first eigenvalue normalized by area on non-orientable surfaces, under certain spectral gap conditions.

## Contribution

It establishes the existence of extremal metrics on non-orientable surfaces, building on previous spectral gap results by the authors.

## Key findings

- Existence of extremal metrics under spectral gap conditions
- Extension of eigenvalue maximization theory to non-orientable surfaces
- Utilization of spectral gap conditions from prior work

## Abstract

We prove the existence of metrics maximizing the first eigenvalue normalized by area on closed, non-orientable surfaces assuming two spectral gap conditions. These spectral gap conditions are proved by the authors in \cite{MS3}.