Milnor conjecture (Ricci curvature)

For other Milnor conjectures, see Milnor conjecture (disambiguation).

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, Naber and Semola 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

1. Milnor, J. (1968). "A note on curvature and fundamental group". Journal of Differential Geometry. 2 (1): 1–7. doi:10.4310/jdg/1214501132. ISSN 0022-040X.
2. Gromov, M. (1978-01-01). "Almost flat manifolds". Journal of Differential Geometry. 13 (2). doi:10.4310/jdg/1214434488. ISSN 0022-040X.
3. Cepelewicz, Jordana (2024-05-14). "Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture". Quanta Magazine. Retrieved 2024-05-15.
4. Cohn-Vossen, Stefan (1935). "Kürzeste Wege und Totalkrümmung auf Flächen". Compositio Mathematica. 2: 69–133. ISSN 1570-5846.
5. Bruè, Elia; Naber, Aaron; Semola, Daniele (2023). "Fundamental Groups and the Milnor Conjecture". arXiv:2303.15347 [math.DG].
6. Schoen, Richard; Yau, Shing-Tung (1982-12-31), Yau, Shing-tung (ed.), "Complete Three Dimensional Manifolds with Positive Ricci Curvature and Scalar Curvature", Seminar on Differential Geometry. (AM-102), Princeton University Press, pp. 209–228, doi:10.1515/9781400881918-013, ISBN 978-1-4008-8191-8, retrieved 2024-05-24
7. Liu, Gang (August 2013). "3-Manifolds with nonnegative Ricci curvature". Inventiones Mathematicae. 193 (2): 367–375. arXiv:1108.1888. Bibcode:2013InMat.193..367L. doi:10.1007/s00222-012-0428-x. ISSN 0020-9910.
8. Pan, Jiayin (2020). "A proof of Milnor conjecture in dimension 3". Journal für die reine und angewandte Mathematik. 2020 (758): 253–260. arXiv:1703.08143. doi:10.1515/crelle-2017-0057. ISSN 1435-5345.
9. Bruè, Elia; Naber, Aaron; Semola, Daniele (2023). "Six dimensional counterexample to the Milnor Conjecture". arXiv:2311.12155 [math.DG].