Bavard's systolically extremal Klein bottles and three dimensional applications

TL;DR

This paper constructs extremal Riemannian metrics on non-orientable Bieberbach 3-manifolds, including Klein bottles, demonstrating their systolic optimality within conformal classes and extending Bavard's extremal metrics to three dimensions.

Contribution

It proves the existence of systolically extremal metrics on all non-orientable Bieberbach 3-manifolds, generalizing Bavard's Klein bottle metrics to three-dimensional applications.

Findings

Existence of extremal metrics on all non-orientable Bieberbach 3-manifolds.

Extension of Bavard's Klein bottle metrics to three-dimensional manifolds.

Demonstration of systolic extremality within conformal classes.

Abstract

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds satisfy an isosystolic inequality by a general and fundamental result of M. Gromov. In dimension 3, there exist four classes of non-orientable Bieberbach manifolds up to an affine diffeomorphism. In this paper, We prove the existence on each diffeomorphism class of non-orientable Bieberbach 3-manifolds of a two-parameter family of singular Riemannian metrics that are systolically extremal in their conformal class. The proof uses a one-parameter family of singular Riemannian metrics on the Klein bottle discovered by C. Bavard (\cite{bavard88}): each one of these metrics is extremal in its conformal class.