Today I read about Gromov's definition of minimal volume for smooth manifolds.


$$\min {\rm Vol}(M):=\inf\_{|K\_g|\leq1}\{{\rm Vol}(M,g)\}.$$


Gromov's conjecture states that for every closed simply connected odd-dimensional manifold $\min {\rm Vol}(M)=0$. Is the Gromov conjecture still open? Can anybody give an example for this conjecture?


**Update:** My teacher told to me that this conjecture can be solved by the Ricci flow method but I don't know how to use it. Can anybody give me an explanation to this? and how it works?


Thanks.



Is the Gromov conjecture still open?
According to [wikipedia](http://en.wikipedia.org/wiki/Minimal_volume), the conjecture is still open. I do not see at all how RF can help here. (Apart from dimension 3 when the statement is of course a corollary of Perelman's geometrization theorem.) There is no positivity assumption on curvature(s) of $M$ in this conjecture and the Ricci Flow in higher dimensions is mostly a mystery without some positivity assumptions. From what you wrote, it sounds like your teacher has no idea how to approach this problem via Ricci Flow either; I suggest you work on something else, more doable. 


One more thing: The conjecture is only in odd-dimensional case. For even dimensional manifolds Gromov noted in his paper that there is a bound $MinVol(M^n)\ge c\_n|\chi(M)|$, $c\_n>0$. Hence, all even-dimensional spheres have positive minimal volume.


**Edited: (4 September 2017)** 


An Example of vanishing minimal volume which is due to Gromov is three-sphere. (see [Gromove paper,section 0.4](http://www.ihes.fr/~gromov/PDF/4[35].pdf))


