Milnor conjecture (Ricci curvature) explained
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.[1]
In two dimensions
has finitely generated fundamental group as a consequence that if
for noncompact
, then it is
flat or
diffeomorphic to
, by work of
Cohn-Vossen from 1935.
[2] In three dimensions the conjecture holds due to a noncompact
with
being diffeomorphic to
or having its universal cover
isometrically split. The diffeomorphic part is due to
Schoen-
Yau (1982) while the other part is by Liu (2013).
[3] Another proof of the full statement has been given by Pan (2020).
[4] In 2023 Bruè et al. 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.
Notes and References
- Gromov . M. . 1978-01-01 . Almost flat manifolds . Journal of Differential Geometry . 13 . 2 . 10.4310/jdg/1214434488 . 0022-040X.
- Cohn-Vossen . Stefan . 1935 . Kürzeste Wege und Totalkrümmung auf Flächen . Compositio Mathematica . en . 2 . 69–133 . 1570-5846.
- Liu . Gang . August 2013 . 3-Manifolds with nonnegative Ricci curvature . Inventiones Mathematicae . en . 193 . 2 . 367–375 . 10.1007/s00222-012-0428-x . 0020-9910. 1108.1888 . 2013InMat.193..367L .
- Pan . Jiayin . 2020 . A proof of Milnor conjecture in dimension 3 . Journal für die reine und angewandte Mathematik . en . 2020 . 758 . 253–260 . 10.1515/crelle-2017-0057 . 1435-5345. 1703.08143 .
