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
(2006), 63--73.
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.