In 1968 John Milnor conjectured ^[1] that the fundamental group of a complete manifold is finitely generated if its Ricci curvature stays nonnegative. In an oversimplified interpretation, such a manifold has a finite number of " holes". A version for almost-flat manifolds holds from work of Gromov. ^{[2] [3]}

In two dimensions ${\displaystyle M^{2}}$ has finitely generated fundamental group as a consequence that if ${\displaystyle \operatorname {Ric} >0}$ for noncompact ${\displaystyle M^{2}}$, then it is flat or diffeomorphic to ${\displaystyle \mathbb {R} ^{2}}$, by work of Cohn-Vossen from 1935. ^{[4] [5]}

In three dimensions the conjecture holds due to a noncompact ${\displaystyle M^{3}}$ with ${\displaystyle \operatorname {Ric} >0}$ being diffeomorphic to ${\displaystyle \mathbb {R} ^{3}}$ or having its universal cover isometrically split. The diffeomorphic part is due to Schoen- Yau (1982) ^{[6] [5]} while the other part is by Liu (2013). ^{[7] [5]} Another proof of the full statement has been given by Pan (2020). ^{[8] [5]}

In 2023 Bruè et al. disproved in two preprints the conjecture for six ^[9] or more ^[5] dimensions by constructing counterexamples that they described as "smooth fractal snowflakes". The status of the conjecture for four or five dimensions remains open. ^[3]

References

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