##
**Minimal 2-spheres in 3-spheres.**
*(English)*
Zbl 1426.49042

In [Trans. Am. Math. Soc. 18, 199–300 (1917; JFM 46.1174.01)], G. D. Birkhoff used the min-max method to show that any closed Riemannian 2-sphere contains at least one closed geodesic. In [“Topological methods in variational problems and their application to the differential geometry of surfaces”, Usp. Mat. Nauk 2, No. 1, 166–217 (1947)], L. Lusternik and L. Schnirelmann proved that any closed Riemannian 2-sphere contains at least three simple closed geodesics. Later, in [Bull. Aust. Math. Soc. 28, 159–160 (1983; Zbl 0511.49021)], F. R. Smith proved that if \(M\) is a Riemannian 3-manifold diffeomorphic to \(\mathbb{S}^3\), then \(M\) contains an embedded minimal 2-sphere.

In this paper, using combined min-max theory and mean curvature flow, the authors prove that if \(M\) is a 3-manifold diffeomorphic to \(\mathbb{S}^3\) and endowed with a generic metric, then \(M\) contains at least two embedded minimal 2-spheres, that is, they show that exactly one of the following statements holds: (i) \(M\) contains at least one stable embedded minimal 2-sphere, and at least two embedded minimal 2-spheres of index 1, (ii) \(M\) contains no stable embedded minimal 2-spheres, at least one embedded minimal 2-sphere \(\Sigma_1\) of index 1, and at least one embedded minimal 2-sphere \(\Sigma_2\) of index 1 or 2. In this case \(|\Sigma_1|<|\Sigma_2|<2|\Sigma_1|\). A natural family of examples of 3-spheres are ellipsoids \(\{E(a,b,c,d)\}\), where \(E(a,b,c,d)=\left\{\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}+\frac{x_3^2}{c^2}+\frac{x_4^2}{d^2}\right\}\subset\mathbb{R}^4\) with \(a>b>c>d\). In [Enseign. Math., II. Sér. 33, 109–158 (1987; Zbl 0631.53002)], S. T. Yau asked the question whether the only minimal 2-spheres in an ellipsoid centered about the origin in \(\mathbb{R}^4\) are the planar ones. The present authors answer this question by showing that for fixed \(b, c\), and \(d\), if \(a\) is chosen sufficiently large, then the ellipsoid \(E(a,b,c,d)\) contains a nonplanar embedded minimal 2-sphere. Next, the authors show that if \(M\) is a 3-manifold diffeomorphic to \(\mathbb{RP}^3\) endowed with a metric of positive Ricci curvature, then \(M\) admits at least two embedded minimal projective planes. Finally, the authors prove that if \(D\subset M^3\) is a smooth 3-disk with mean convex boundary, then exactly one of the following statements holds: (i) there exists an embedded stable minimal 2-sphere \(\Sigma\subset\text{Int}(D)\), (ii) there exists a smooth foliation \(\{\Sigma_t\}_{t\in[0,1]}\) of \(D\) by mean convex embedded 2-spheres.

In this paper, using combined min-max theory and mean curvature flow, the authors prove that if \(M\) is a 3-manifold diffeomorphic to \(\mathbb{S}^3\) and endowed with a generic metric, then \(M\) contains at least two embedded minimal 2-spheres, that is, they show that exactly one of the following statements holds: (i) \(M\) contains at least one stable embedded minimal 2-sphere, and at least two embedded minimal 2-spheres of index 1, (ii) \(M\) contains no stable embedded minimal 2-spheres, at least one embedded minimal 2-sphere \(\Sigma_1\) of index 1, and at least one embedded minimal 2-sphere \(\Sigma_2\) of index 1 or 2. In this case \(|\Sigma_1|<|\Sigma_2|<2|\Sigma_1|\). A natural family of examples of 3-spheres are ellipsoids \(\{E(a,b,c,d)\}\), where \(E(a,b,c,d)=\left\{\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}+\frac{x_3^2}{c^2}+\frac{x_4^2}{d^2}\right\}\subset\mathbb{R}^4\) with \(a>b>c>d\). In [Enseign. Math., II. Sér. 33, 109–158 (1987; Zbl 0631.53002)], S. T. Yau asked the question whether the only minimal 2-spheres in an ellipsoid centered about the origin in \(\mathbb{R}^4\) are the planar ones. The present authors answer this question by showing that for fixed \(b, c\), and \(d\), if \(a\) is chosen sufficiently large, then the ellipsoid \(E(a,b,c,d)\) contains a nonplanar embedded minimal 2-sphere. Next, the authors show that if \(M\) is a 3-manifold diffeomorphic to \(\mathbb{RP}^3\) endowed with a metric of positive Ricci curvature, then \(M\) admits at least two embedded minimal projective planes. Finally, the authors prove that if \(D\subset M^3\) is a smooth 3-disk with mean convex boundary, then exactly one of the following statements holds: (i) there exists an embedded stable minimal 2-sphere \(\Sigma\subset\text{Int}(D)\), (ii) there exists a smooth foliation \(\{\Sigma_t\}_{t\in[0,1]}\) of \(D\) by mean convex embedded 2-spheres.

Reviewer: Andrew Bucki (Edmond)

### MSC:

49Q05 | Minimal surfaces and optimization |

53E10 | Flows related to mean curvature |

58E12 | Variational problems concerning minimal surfaces (problems in two independent variables) |

49J35 | Existence of solutions for minimax problems |

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\textit{R. Haslhofer} and \textit{D. Ketover}, Duke Math. J. 168, No. 10, 1929--1975 (2019; Zbl 1426.49042)

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