Bavard's systolically extremal Klein bottles and three dimensional applications (Q2668994)

There are 10 equivalence classes of three-dimensional Bieberbach manifolds up to diffeomorphism. (A Bieberbach manifold is a compact manifold that admits a flat Riemannian metric.) Four of these equivalence classes are non-orientable. In this paper, the author constructs on each of these four types of non-orientable Bieberbach 3-manifolds, a two-parameter family of singular metrics which maximize the isosystolic ratio \( \frac{\mathrm{sys}(g)^3}{\mathrm{Vol}(g)}\) in their conformal class. The proof utilizes \textit{C. Bavard}'s construction [Geom. Dedicata 27, No. 3, 349--355 (1988; Zbl 0667.53033)] of a 1-parameter family of singular metrics on the Klein bottle which maximize the isosystolic ratio \(\frac{\mathrm{sys}(g)^2}{\mathrm{Area}(g)}\) in their conformal class.