Enumeration problems for minimal surfaces
Enumerative geometry is traditionally a branch of algebraic geometry: A classic result is the theorem of Cayley-Salmon (1849), which states that every smooth cubic surface (a degree 3 hypersurface in CP3) contains exactly 27 lines. There are, however, enumerative problems in differential geometry as well. This page records results and conjectures regarding enumeration of minimal submanifolds in spheres, and also some of the tools involved.
In the following, all ambient Riemannian manifolds are smooth and closed, and all submanifolds are smooth, closed, and embedded unless specified otherwise.
(Last update: Aug 2024)
Conjectures
1905 Poincaré: 3 geodesics in every 2-sphere [Poi05]
(Resolved by Grayson [Gra89])
1982 Yau: 4 minimal 2-spheres in every 3-sphere [Yau82]
1989 White: 5 minimal tori in every 3-sphere [Whi89]
Minimal submanifolds in spheres
1917 Birkhoff: 1 immersed geodesic in every 2-sphere [Bir17]
(New tool: min-max theory for curves)
1981 Sacks-Uhlenbeck: 1 immersed minimal 2-sphere in every n-sphere. [SU81]
(New tool: a perturbed energy functional)
1983 Simon-Smith: 1 minimal 2-sphere in every 3-sphere [Smi83]
(New tool: min-max theory for surfaces)
1989 Grayson: 3 geodesics in every 2-sphere [Gra89] (built on Lyusternik–Schnirelmann [LS29])
(New tool: curve shortening flow)
1991 White: 2 minimal 2-spheres in every 3-sphere with positive Ricci curvature, 1 minimal torus in every 3-sphere with positive Ricci curvature [Whi91]
(New tools: degree theory, Ricci flow)
2019 Haslhofer-Ketover: 2 minimal 2-spheres in every 3-sphere with bumpy metric [HK19]
(New tool: mean curvature flow with surgery)
2023 Wang-Zhou: 4 minimal 2-spheres in every 3-sphere with bumpy metric or positive Ricci curvature [WZ23]
(New tool: multiplicity one theorem in min-max)
Bibliography
[Bir17] Poincaré, Henri. "Sur les lignes géodésiques des surfaces convexes." Transactions of the American Mathematical Society 6.3 (1905): 237-274.
[Gra89] Grayson, Matthew A. "Shortening embedded curves." Annals of Mathematics 129.1 (1989): 71-111.
[HK19] Haslhofer, Robert, and Daniel Ketover. "Minimal 2-spheres in 3-spheres." (2019): 1929-1975.
[LS29] Lusternik, Lazar, and Lev Schnirelmann. "Sur le problème de trois géodésiques fermées sur les surfaces de genre 0." CR Acad. Sci. Paris 189 (1929): 269-271.
[Poi05] Poincaré, Henri. "Sur les lignes géodésiques des surfaces convexes." Transactions of the American Mathematical Society 6.3 (1905): 237-274.
[SU81] Sacks, Jonathan, and Karen Uhlenbeck. "The existence of minimal immersions of 2-spheres." Annals of mathematics 113.1 (1981): 1-24.
[Smi83] Smith, Francis R. "On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric." Bulletin of the Australian Mathematical Society 28.1 (1983): 159-160.
[WZ23] Wang, Zhichao, and Xin Zhou. "Min-max minimal hypersurfaces with higher multiplicity." arXiv preprint arXiv:2201.06154 (2022).
[Whi89] White, Brian. "Every three-sphere of positive Ricci curvature contains a minimal embedded torus." (1989): 71-75.
[Whi91] White, Brian. "The space of minimal submanifolds for varying Riemannian metrics." Indiana University Mathematics Journal (1991): 161-200.
[Yau82] Shing-Tung Yau. Seminar in Differential Geometry, Problem Section. pages 669-706, Annals of Mathematics Studies (1982): 102, Princeton University Press, Princeton, N.J.