Adrian Chun-Pong Chu - enumerative differential geometry

Adrian Chun-Pong Chu

Enumerative problems for minimal surfaces of prescribed genus

Click here for a lecture note. (2026) (new)

Background

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 (i.e. a degree 3 complex surface in CP3) contains exactly 27 copies of CP1. There are, however, enumerative problems in differential geometry as well. The general theme of the problem is: In any general ambient space, find exactly, or at least, a certain number of "special subspaces".

This page records results and conjectures regarding enumeration of geodesics and minimal surfaces in spheres, and also some of the tools involved. Below, all ambient Riemannian manifolds are smooth and closed, and all submanifolds are smooth, closed, and embedded unless specified otherwise.

Conjectures

1905 Poincaré: Every Riemannian 2-sphere contains at least 3 embedded closed geodesics [Poi05].

(The key difficulty is, when we say “every,” we mean that the statement should hold no matter what the Riemannian metric is!)

1982 S.-T. Yau: Every Riemannian 3-sphere contains at least 4 embedded minimal spheres [Yau82].
1989 B. White: Every Riemannian 3-sphere contains at least 5 embedded minimal tori [Whi89].

Below, we record the known enumerative results for geodesics and minimal surfaces in spheres.

Existence of geodesics in S2

1917 G. D. Birkhoff: Every S2 contains ≥ 1 immersed closed geodesic. [Bir17]

(Tool: min-max theory for curves)

1989 M. Grayson: Every S2 contains ≥ 3 embedded closed geodesics. [Gra89] (Built on Lyusternik-Schnirelmann [LS47]; see also [Lus47; Kli77; Bal78; Jos89; Taĭ92])

(Tool: curve shortening flow)

Existence of minimal surfaces in S3

The table below summarizes the known lower bounds on the number of embedded minimal surfaces of genus g that any 3-sphere must contain, under different assumptions on the ambient metric: (a) positive Ricci curvature (Ric > 0), (b) generic metric, (c) arbitrary metric.

Below is the timeline:

1981 Sacks-Uhlenbeck: Every n-sphere contains ≥ 1 immersed minimal 2-sphere. [SU81]

(Tool: a perturbed energy functional)

1983 Simon-Smith: Every S3 contains ≥ 1 minimal sphere. [Smi83]

(Tool: min-max theory for surfaces)

1991 B. White: Every S3 with Ric > 0 contains ≥ 2 minimal spheres and ≥ 1 minimal torus. [Whi91]

(Tool: degree theory and Ricci flow)

2019 Haslhofer-Ketover: Every S3 with a generic metric contains ≥ 2 minimal spheres. [HK19]

(Tool: mean curvature flow with surgery)

2023 Wang-Zhou: Every S3 with a generic metric, or Ric > 0, contains ≥ 4 minimal spheres. [WZ23]

(Tool: multiplicity one theorem for Simon-Smith min-max)

2024 A. Chu and Yangyang Li: Every S3 with Ric > 0 contains ≥ 5 minimal tori. [CL24]

(Tool: repetitive min-max, interpolation via pinch-off processes)

2024 Xingzhe Li and Zhichao Wang: Every S3 with Ric > 0 contains ≥ 4 minimal tori; every S3 with a generic metric such that Ric > 0 contains ≥ 9 minimal tori. [LW24]

(Tool: a Morse-type inequality)

2025 A. Chu: Every S3 with Ric > 0 contains ≥ 1 minimal surface of genus g, for every g. [Chu25]

(Tool: homology descent in min-max)

2026 A. Chu, Yangyang Li, and Zhihan Wang: Every S3 with Ric > 0 contains ≥ 4 minimal surface of genus 2. [CLW26]

(Tool: enumerative min-max theorem)

Bibliography

[Bal78] Werner Ballmann. “Der Satz von Lusternik und Schnirelmann”. Beitr¨agezur Differentialgeometrie, Heft 1. Vol. 102. Bonner Math. Schriften. Univ. Bonn, Bonn, 1978, pp. 1–25.
[Bir17] Poincaré, Henri. "Sur les lignes géodésiques des surfaces convexes." Transactions of the American Mathematical Society 6.3 (1905): 237-274.
[Chu25] Adrian Chun-Pong Chu. “Minimal surfaces with arbitrary genus in 3-spheres of positive Ricci curvature”. In: arXiv preprint arXiv:2508.06019 (2025).
[CL24] Adrian Chun-Pong Chu and Yangyang Li. “Existence of 5 minimal tori in 3-spheres of positive Ricci curvature”. In: arXiv preprint arXiv:2409.09315 (2024).
[CLW26] Adrian Chun-Pong Chu, Yangyang Li, and Zhihan Wang. “An enumerative min-max theorem for minimal surfaces”. In: arXiv preprint (2026).
[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.
[Jos89] J¨urgen Jost. “A nonparametric proof of the theorem of Lusternik and Schnirelman”. Arch. Math. (Basel) 53.5 (1989), pp. 497–509. issn: 0003-889X,1420-8938. doi: 10.1007/BF01324725.
[L47] L. Lusternik. “Topology of functional spaces and calculus of variations in the large”. Trav. Inst. Math. Stekloff 19 (1947), p. 100.
[LS47] Lazar Aronovich Lyusternik and Lev Genrikhovich Shnirel’man “Topological methods in variational problems and their application to the differential geometry of surfaces”. Uspekhi Matematicheskikh Nauk 2.1 (1947), pp. 166–217.
[LW24] Xingzhe Li and Zhichao Wang. “Existence of embedded minimal tori in three-spheres with positive Ricci curvature”. In: arXiv preprint arXiv:2409.10391(2024).
[Kli77] Wilhelm Klingenberg. Lectures on closed geodesics. Third. Universit¨at Bonn. Mathematisches Institut, Bonn, 1977, 210 pp.
[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.
[Tai92] I. A. Ta˘ımanov. “On the existence of three nonintersecting closed geodesics on manifolds that are homeomorphic to the two-dimensional sphere”. Izv. Ross. Akad. Nauk Ser. Mat. 56.3 (1992), pp. 605–635. issn: 1607-0046,2587-5906. doi: 10.1070/IM1993v040n03ABEH002177.
[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.

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