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

Let$M$ be a Complete simply-connected n-dimensional Riemannian manifold with nonpositive 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?

share|cite|improve this question
I guess you want to say $$\frac{Vol_{n}\left ( \Omega \right )}{Vol_{n-1}\left (\partial \Omega \right )^{\frac{n}{n-1}}}.$$ – Anton Petrunin Aug 31 '12 at 12:20

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.
share|cite|improve this answer
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. – Benoît Kloeckner Sep 11 '12 at 21:17

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.