I'm currently learning some stuffs about systolic inequalities. While reading the relevant sections (p329 to 340) in Berger's Panoramic View of Riemannian Geometry, I noticed a gap in one of the proofs (starting at page 331). The goal is to prove :



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> For any non simply connected compact surface $(M,g)$, $Area(M,g)\geq Sys(M,g)^2/2$.
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Berger considers a periodic geodesic $c$ realizing the systole $L=Sys(M,g)$, pick a point $m$ on $c$ and claims that $Vol B(m,L/2)\geq L^2/2$, which is enough to show the result.


However in the proof of the claim, he invokes the following fact : for $r<L/2$, $B(m,r)$ is a topological disk. He says that this comes from the "very definition of the systole". But this claim is false, it is enough to consider a torus with a long thin finger glued at some point to see it (fig 7.26 on the page where the claim is made shows exactly this). In the paper "Systolic and inter-systolic inequalities", Gromov, facing the same kind of situation, just says "chop the fingers", while this is intuitively convincing I don't see a way to make it rigorous.


My question is the following : is this fixable ?


Thinking a little bit about it, it seems enough for the rest of the argument to show that all but one of the connected component of $M\backslash B(m,r)$ are disks, and that the systole $c$ doesn't meet the components which are disks. But I am not able to prove this at the moment.


I'm aware that another proof is available, through estimating the homological systole. But I like the proof on the next page of Berger that the systolic ration grows at least like the square root of the genus, which uses the same argument.


I should also say that I don't have access the article of Hebda to which Berger refers.



Fixing a proof of the systolic inequality for higher genus surfaces
You don't really need to reattach chopped off fingers to get the estimate. The point is that for almost all (in the sense of Sard) $r$, the points $\gamma(-r)$ and $\gamma(r)$ are in the same connected component of the $r$-level curve of the distance function. Applying the coarea formula and the triangle inequality then gives the estimate.


