MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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… – Mikhail Katz Jul 3 '13 at 15:36
up vote 5 down vote accepted

There is no Lipschitz or even Gromov-Hausdorff stability - just consider a round metric with long hairy tails of small area.

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.

share|cite|improve this answer
Thanks Sergei. I was wondering about "sea urchins" myself and didn't know how to rule them out. Is that what the Sormani-Wenger metric can do? What about something terribly naive like a lipschitz stability for a nearly-full measure open subset? – alvarezpaiva Jun 18 '13 at 12:06
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.