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In the paper of Andrews and Bryan Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere http://arxiv.org/abs/0908.3606, the isoperimetric profile is defined by $h(\psi)=\inf ( |\partial\Omega|:\Omega\subset M, |\Omega|=\psi|M| )$

Theorem 2: Let $M$ be a smooth compact Riemannian surface. Then the isoperimetric profile satisfies $$h(\psi)=\sqrt{4\pi|M|\psi} - \frac{|M|^{3/2}\sup_MK}{4\sqrt{\pi}}\psi^{3/2} + O(\psi^2)$$ as $\psi\to 0$.

In the proof of above theorem, they say that "The upper bound follows since $|\partial B_r(p)|\geq h(|B_r(p)|/|M|).$ But I don't know how this inequality implies the upper bound for the equality in Theorem 2.

Does anyone read this paper? Do you know how to achieve this? Thanks.

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I'm not sure it works, but have you tried plugging into $|\partial B_r(p)|\geq h(|B_r(p)|/|M|)$ the expansion of $|\partial B_r(p)|$ and $|B_r(p)|$ in term of $r$ ? –  Thomas Richard Apr 18 '13 at 15:52
    
@Thomas Richard, thats exactly what you do. I think the only issue that might be a tiny bit tricky is getting a good expansion of $r$ in terms of $|B_r(p)|$, but I think that this is not too hard.. –  Otis Chodosh Apr 18 '13 at 16:15
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