Levy-Gromov Isoperimetric Inequality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:12:26Z http://mathoverflow.net/feeds/question/96456 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96456/levy-gromov-isoperimetric-inequality Levy-Gromov Isoperimetric Inequality T-' 2012-05-09T14:54:27Z 2012-05-09T22:25:51Z <p>In his paper "Paul Levy's Isoperimetric Inequality", Gromov gives the following isoperimetric inequality:</p> <blockquote> <p>Let $V$ be a closed $(n+1)$-dimensional Riemannian Manifold with $\mathrm{Ric}(V) \geq n \space (= \mathrm{Ric}(S^{n+1}))$. Let $V_0 \subset V$ be a domain with smooth boundary and let $B$ be a round ball in $S^{n+1}$ such that <code>$$\frac{ Vol(V_0)}{Vol(V)}= \frac{Vol(B)}{Vol(S^{n+1})}.$$</code> Then it follows that $$\frac{Vol( \partial V_0)}{Vol(V)} \geq \frac{Vol(\partial B)}{Vol(S^{n+1})}.$$</p> </blockquote> <p>Now my question: in a (slightly earlier) article 'Isoperimetric Inequalities In Riemannian Manifolds', Gromov states that the above inequality will still be true even if $V$ only admits a <strong>negative</strong> lower bound on its Ricci curvature. Does anyone have a reference for a proof of this, or is the statement obvious? It just seems to me that the hypothesis compares the curvature of $V$ to that of $S^{n+1}$, so allowing $\mathrm{Ric}(V)$ to be negative will obscure this. </p> http://mathoverflow.net/questions/96456/levy-gromov-isoperimetric-inequality/96472#96472 Answer by Anton Petrunin for Levy-Gromov Isoperimetric Inequality Anton Petrunin 2012-05-09T16:30:18Z 2012-05-09T16:30:18Z <p>I guess Gromov wanted to say that there is a lower bound for $\mathop{\rm vol}\partial V_0$ in terms of $\mathop{\rm vol} V_0/\mathop{\rm Vol} V$, $\mathop{\rm diam}V$ and lower bound for Ricci curvature. The same proof as in "Paul Levy's Isoperimetric Inequality", gives such a bound, but it is not longer sharp.</p> <p>BTW, there is an analog of Levy--Gromov for open manifolds with $\mathop{\rm Ricc}\ge 0$. It is sharp and gives a lower bound for $\mathop{\rm vol}\partial V_0$ in terms of $\mathop{\rm vol} V_0$ and the volume growth of $V$, BUT as far as I know it is not written. (Please correct me if I am wrong.)</p> http://mathoverflow.net/questions/96456/levy-gromov-isoperimetric-inequality/96512#96512 Answer by Deane Yang for Levy-Gromov Isoperimetric Inequality Deane Yang 2012-05-09T22:25:51Z 2012-05-09T22:25:51Z <p>I'm not sure about who did it first and how sharp the results are, but you can find isoperimetric inequalities for a negative lower bound on Ricci in a paper of Croke (http://www.math.poly.edu/people/faculty.phtml). My recollection is that Gallot also proved a similar isoperimetric inequality, but I can't seem to find the paper.</p>