In more detail, it is known that the comass norm in H1(M) is monotone under the Ricci flow. Thus, the stable norm in H1(M) is also monotone. It follows that the stable 1-systole of M is monotone under the Ricci flow. In the meantime, the effect on the total volume of M depends on the sign of the total scalar curvature.
A modest proposal would be to prove a stronger version of the Loewner inequality with an error term, similar in spirit to isoperimetric inequalities with an error term (i.e. lower bound for the isoperimetric defect), due to Bonnesen and others.
The paper I have in mind is
Chang, Shu-Cheng: The 2-dimensional Calabi flow. Nagoya Math. J. 181
Here Chang proves that any metric on a compact surface converges to the flat one, without assuming uniformisation.
I was hoping that the systole will be monotone under the Calabi flow on the torus. Evidence toward this is that the analogous Hamilton flow (Ricci flow) on 3-manifolds does have this property. Namely, the stable norm in 1-dimensional homology, or alternatively comass in 1-dimensional cohomology (i.e. on 1-forms) is monotone under the Hamilton flow, as shown in
Ilmanen, Tom; Knopf, Dan: A lower bound for the diameter of solutions
to the Ricci flow with nonzero H1(Mn; R).
Math. Res. Lett. 10 (2003), no. 2-3, 161--168.
Hopefully this will give an error estimate (isosystolic defect) in the Loewner inequality.
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