Uniqueness of embedded minimal surfaces of genus 2 in round spherical space forms of Heegaard genus 2 (Ketover–Liokumovich Conjecture 1.5)

The Smale conjecture for lens spaces was recently re-proved by Ketover and Liokumovich using minimal surfaces and min-max theory, and the load-bearing rigidity input in their argument is Brendle's resolution of the Lawson conjecture: in a round lens space, the minimal surfaces realizing the Heegaard genus are exactly the quotients of the Clifford torus, unique up to ambient isometry. For the remaining spherical space forms — those of Heegaard genus 2, such as the prism manifolds and the Poincaré homology sphere — the corresponding classification is unknown, and the authors posed it as their Conjecture 1.5: each round spherical space form of Heegaard genus 2 admits exactly one embedded minimal surface of genus 2 up to ambient isometry. The problem is to prove or refute the conjecture, equivalently to classify the embedded minimal genus-2 surfaces in every round genus-2 space form up to ambient isometry.

Background

Let $S^3=\{(z,w)\in\mathbb{C}^2:\ |z|^2+|w|^2=1\}$ be the round 3-sphere. A spherical space form is a quotient $M=S^3/G$ endowed with the quotient metric of constant sectional curvature $+1$, where $G$ is a finite group of isometries of $S^3$ acting freely. When $G$ is cyclic one obtains the lens spaces $L(p,q)$; when $G$ is non-cyclic — for example the binary dihedral, binary tetrahedral, binary octahedral or binary icosahedral groups, i.e. the double covers in $SO(4)$ of the dihedral and Platonic rotation groups, and related products — one obtains manifolds such as the prism manifolds and the Poincaré homology sphere (the quotient of $S^3$ by the binary icosahedral group of order 120, a homology 3-sphere). A Heegaard splitting of a closed 3-manifold is a decomposition into two handlebodies meeting along their common boundary, and the Heegaard genus is the smallest genus of such a splitting surface; the lens spaces are exactly the spherical space forms of Heegaard genus 1, and every non-lens spherical space form has Heegaard genus 2. An embedded minimal surface is a smoothly embedded closed surface whose mean curvature vanishes identically, i.e. a critical point of the area functional under deformations of its embedding, and two such surfaces are identified up to ambient isometry when an isometry of the ambient manifold carries one onto the other.

The classical Smale conjecture, proved by Hatcher, asserts that the inclusion of the isometry group $O(4)$ into the diffeomorphism group of $S^3$ is a homotopy equivalence; its generalization predicts the same homotopy equivalence $\mathrm{Isom}(M)\to\mathrm{Diff}(M)$ for closed 3-manifolds of constant sectional curvature $\pm1$. Bamler and Kleiner established the spherical case for all space forms using Ricci flow through singularities. Ketover and Liokumovich then gave a new proof of the generalized Smale conjecture for all lens spaces by minimal-surface methods: using min-max theory (the variational construction of area-critical surfaces from families of sweepouts), they show that the space of embedded Heegaard tori deformation retracts onto the space of minimal tori. The linchpin of their scheme is a rigidity theorem: Brendle's resolution of the Lawson conjecture states that the Clifford torus $\{(z,w):\ |z|^2=|w|^2=1/2\}$ is, up to ambient isometry, the unique embedded minimal torus in the round $S^3$; combined with a covering argument (Ketover–Liokumovich, Theorem 2.2), this classifies the embedded minimal tori in every round lens space as quotients of Clifford tori.

For the genus-2 space forms this input is missing. If $\Sigma\subset M=S^3/G$ is a genus-2 surface, its preimage in $S^3$ is a connected embedded minimal surface of genus $1+d\geq2$, where $d$ is the covering degree (a divisor of the order of $G$); classifying embedded minimal surfaces of genus at least 2 in the round $S^3$ is a wide-open problem, and uniqueness in fact fails in $S^3$ itself, where Lawson's constructions produce embedded minimal surfaces of every genus. Existence of minimal surfaces realizing the Heegaard genus is nevertheless within reach of min-max theory: Ketover, Liokumovich and Song proved that a strongly irreducible Heegaard surface in a closed Riemannian 3-manifold is isotopic either to a minimal surface of Morse index at most one or to the boundary of a tubular neighborhood of a non-orientable minimal surface with a vertical handle attached, and genus-2 splittings of spherical space forms are strongly irreducible since these manifolds are irreducible and contain no incompressible surfaces. Ketover and Liokumovich therefore posed, as the genus-2 analogue of the Lawson conjecture, their Conjecture 1.5, and identified it as the missing classification that would let their min-max proof of the Smale conjecture extend from lens spaces to the spherical space forms of Heegaard genus 2.

Problem Statement

Prove or refute Conjecture 1.5 of Ketover–Liokumovich: each round spherical space form of Heegaard genus 2 — that is, each quotient $M=S^3/G$ with its constant sectional curvature $+1$ quotient metric, where $G$ is a finite non-cyclic group of orientation-preserving isometries acting freely and linearly on $S^3$ (the prism manifolds and the tetrahedral, octahedral and icosahedral space forms, including the Poincaré homology sphere) — admits exactly one embedded minimal surface of genus 2 up to ambient isometry. Equivalently: classify, for every such $M$, the embedded closed minimal surfaces of genus 2 in $M$, and determine whether they form a single orbit under the isometry group $\mathrm{Isom}(M)$. The conjecture fails for a round genus-2 space form $M$ exactly when $M$ contains two embedded minimal genus-2 surfaces not related by any ambient isometry, or contains none at all. The metric is the round quotient metric, the surfaces are smooth closed embedded orientable surfaces of topological genus 2 with identically vanishing mean curvature, and two solutions are considered the same when they differ by an isometry of $M$.

References

1. Ketover, Daniel & Liokumovich, Yevgeny. The Smale conjecture and min-max theory. Inventiones mathematicae 241 (2025), 1–34. DOI 10.1007/s00222-025-01334-z. arXiv:2310.05756. https://arxiv.org/abs/2310.05756
2. Brendle, Simon. Embedded minimal tori in $S^3$ and the Lawson conjecture. Acta Mathematica 211 (2013), 177–190. DOI 10.1007/s11511-013-0101-2. https://doi.org/10.1007/s11511-013-0101-2
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7. Ketover, Daniel, Liokumovich, Yevgeny & Song, Antoine. On the existence of minimal Heegaard surfaces. arXiv:1911.07161 [math.DG] (2019). https://arxiv.org/abs/1911.07161
8. Li, Xingzhe, Wang, Tongrui & Yao, Xuan. Minimal surfaces with low genus in lens spaces. Journal für die reine und angewandte Mathematik (2025). DOI 10.1515/crelle-2025-0061. arXiv:2406.12584. https://arxiv.org/abs/2406.12584