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This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in absolute value by 1. Kotschick proved that there are distinct smooth structures on $k(S^2 \times S^2) \sharp (1 + k)(S^1 \times S^3)$, $k$ sufficiently large, for which in the standard smooth structure, the minimal volume $=0$ (by finding a fixed-point free circle action), and another smooth structure for which the minimal volume is bounded away from 0.

My question is whether the converse is true: if there is a metric in which the minimal volume $=0$, must the smooth structure be standard? The existence of a polarized F-structure in this case may be relevant.

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Assume it is true. Then any two homeomorphic 4-manifolds which admit $S^1$-action must be diffeomorphic. Is it true? –  Anton Petrunin Mar 31 '10 at 19:11
    
@Anton: I think the answer to your question is yes, if you assume the S1 actions preserve a smooth structure. Do you know any 4-manifold counterexample? –  Ian Agol Mar 31 '10 at 19:22
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Fintushel long ago classified up to diffeomorphism circle actions on closed 4-manifolds by the isomorphism type of their orbit spaces. (I am mildly confused here because in the original paper Fintushel works with locally smooth actions, hence he only derives topological classification, but I have seen elsewhere that this works smoothly). More recently Baldridge showed that SW-invariants vanish for 4-manifolds with circle actions that have fixed points. Anyway, this gives some hope that what Agol asks is true. –  Igor Belegradek Mar 31 '10 at 20:13

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