# Stability of Pu's isosystolic inequality

The volume of a Riemannian metric on the projective plane is $2\pi$ and length of every non-contractible loop is greater than $\pi - \epsilon$ for some small, positive number $\epsilon$. Is this metric close to the canonical metric?

The question is somewhat vague on purpose. I'm mostly interested in the best constant for a bilipschitz equivalence in terms of $\epsilon$, but I also wonder whether for some sufficiently small $\epsilon$ one can conclude that the curvature is close to 1.

Stability of inequalities is a well-trodden research topic in convex geometry and I was wondering what was known about this in systolic geometry.

-
An interesting "converse" would be to ask whether there is a lower bound for the "isosystolic defect" namely the difference between the area and $\tfrac{2}{\pi}\text{sys}^2$, in the assumption that the area is normalized to $2\pi$ and the curvature in a region of area $A$ is bounded away from $1$ by $\epsilon>0$. For a related question see mathoverflow.net/questions/127599/… – Mikhail Katz Jul 3 '13 at 15:36

One can hope for stability with respect to intrinsic flat distance in the sense of Sormani-Wenger or some similar metric. This distance is basically Federer's flat distance between isometric images is $L^\infty$ (just like the Gromov-Hausdorff distance is the Hausdorff distance in $L^\infty$). The stability in this sense probably amounts to uniqueness of the equality case in the class of integral current spaces arising as limits of projective planes.
Uniformization (along the lines of Pu's proof) gives you a conformal factor $L^2$-close to 1. This implies Lipschitz closeness on nearly full measure. – Sergei Ivanov Jun 18 '13 at 16:58