Let $M$ be a Complete simply-connected $n$-dimensional Riemannian manifold with non-positive curvature,$\Omega $ is a open subset of $M$ ,If $n\geq 4$ Is anyone give a estimate of $\frac{Vol_{n}\left ( \Omega \right )}{Vol_{n-1}\left ( \Omega \right )^{\frac{n}{n-1}}}$?,If $\Omega $ replace by $B^{n}\left ( 1 \right )$,it is a unit sphere,estimate?
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5$\begingroup$ I guess you want to say $$\frac{Vol_{n}\left ( \Omega \right )}{Vol_{n-1}\left (\partial \Omega \right )^{\frac{n}{n-1}}}.$$ $\endgroup$– Anton PetruninCommented Aug 31, 2012 at 12:20
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It is an old open problem.
- The case $n=4$ is done by Croke. He shows that $$\frac{Vol_{n}\left ( \Omega \right )}{Vol_{n-1}\left (\partial \Omega \right )^{\frac{n}{n-1}}}\le C_n$$ for some constant $C_n$ which is optimal for $n=4$.
- The case $n=3$ is done by Kleiner.
- The cases $n=1$ and $2$ are trivial.
answered Aug 31, 2012 at 12:29
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1$\begingroup$ I would not say that $n=2$ is trivial, even if it is not difficult. It can be proved in several ways, but the first proof by Weil in 1926 can be considered to have launched the subject, except that many people seem to have forgotten this work. $\endgroup$ Commented Sep 11, 2012 at 21:17