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Milnor conjecture (Ricci curvature)

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(Redirected from Milnor conjecture (Ricci)) For other Milnor conjectures, see Milnor conjecture (disambiguation).

In 1968 John Milnor conjectured 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.

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

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

In 2023 Bruè, Naber and Semola disproved in two preprints the conjecture for six or more dimensions by constructing counterexamples that they described as "smooth fractal snowflakes". The status of the conjecture for four or five dimensions remains open.

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 .
  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 .
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