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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.

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    $\begingroup$ I think you can upvote, but shouldn't accept an answer which does not answer your question. $\endgroup$
    – YCor
    Commented Oct 13, 2016 at 3:26
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    $\begingroup$ Because the conjecture is about simply connected manifolds. $\endgroup$
    – YCor
    Commented Oct 13, 2016 at 8:06
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    $\begingroup$ According to wikipedia: en.wikipedia.org/wiki/Minimal_volume, the conjecture is still open. I do not see at all how RF can help here. There is no positivity assumption on curvature(s) of $M$ in this conjecture and RF in higher dimensions is mostly a mystery without some positivity assumptions. I suggest you ask your teacher for a reference. $\endgroup$
    – Misha
    Commented Aug 4, 2017 at 7:31
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    $\begingroup$ Perhaps the teacher just meant that the Poincare conjecture implies a positive answer in dimension 3? $\endgroup$
    – HJRW
    Commented Aug 4, 2017 at 8:26
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    $\begingroup$ I don't understand. Either your teacher knows the reference or how to prove it. In the latter case they should write a paper. If neither, then your teacher doesn't know and shouldn't claim to. $\endgroup$
    – Deane Yang
    Commented Aug 6, 2017 at 15:23

2 Answers 2

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According to wikipedia, 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)

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    $\begingroup$ Prof. Gromov in his paper noted that if $M$ admit a locally free $\Bbb S^1$-action then $\min {\rm Vol}(M)=0$. In particular $\min {\rm Vol}(\Bbb S^3)=0$. Does this prove the conjecture in dim $3$? $\endgroup$
    – C.F.G
    Commented Sep 3, 2017 at 16:43
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    $\begingroup$ @C.F.G: Perelman proved the Poincare conjecture, implying that the only closed simply-connected 3-manifold is $S^3$. Berger spheres indeed show that $S^3$ satisfies Gromov's Minimal Volume Conjecture. (Yes, you can derive this from the existence of a free $S^1$-action as well, but the result was known before Gromov's paper.) $\endgroup$
    – Misha
    Commented Sep 4, 2017 at 21:52
  • $\begingroup$ the link to Gromov's paper is broken. What paper is it? numdam.org/item/PMIHES_1982__56__5_0 ? $\endgroup$
    – Dima Pasechnik
    Commented Oct 16, 2021 at 8:55
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This response is unfortunately not about simply-connected manifolds, I overlooked that assumption by the author of the question. So, this is not an answer. See my comment below.

Gromov writes in this paper (see Section 0.4) an example of an odd dimensional (not simply-connected) manifold with non-vanishing minimal volume, by constructing a lower bound involving the so-called simplicial volume (and a corresponding metric $g$).

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    $\begingroup$ Can you give a more precise (page) reference in the Gromov's paper? Certainly, closed simply-connected manifolds have zero simplicial volume (see p.14 of the same paper). $\endgroup$
    – Igor Belegradek
    Commented Oct 12, 2016 at 21:00
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    $\begingroup$ I think you overlooked the assumption that the manifold is simply-connected. None of the examples I see on pp 7-10 are simply-connected. Maybe I a missing something. If so, please give a page/line reference. $\endgroup$
    – Igor Belegradek
    Commented Oct 12, 2016 at 22:48
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    $\begingroup$ Ah, yes: I did overlook "simply-connected" in the OP's question. In fact, page 14, 2nd paragraph, lines 4-5, admits lack of knowledge of such manifolds. Yet, this is a marvelous paper with all sorts of examples for other cases. $\endgroup$
    – T. Amdeberhan
    Commented Oct 13, 2016 at 11:04

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