Milnor conjecture (Ricci curvature)
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". The conjecture was proven for the non-trivial cases of two dimensions in 1930s and three dimensions in 2013. It also holds true under some additional restrictions on the manifold itself or definition of curvature.[2]
In 2023 Bruè et al.[3] disproved the conjecture for six or more dimensions by constructing counterexamples .[2] The status of the conjecture for four or five dimensions remains open.
References
Sources
- Milnor, J. (1968-01-01). "A note on curvature and fundamental group". Journal of Differential Geometry. 2 (1): 1–7. doi:10.4310/jdg/1214501132. ISSN 0022-040X.
- Cepelewicz, Jordana (2024-05-14). "Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture". Quanta Magazine. Retrieved 2024-05-15.
- Bruè, Elia; Naber, Aaron; Semola, Daniele (2023). "Fundamental Groups and the Milnor Conjecture". arXiv:2303.15347.
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