## 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.

### 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

### Keywords:

minimal surfaces; embedded minimal 2-spheres

### Citations:

Zbl 0511.49021; Zbl 0631.53002; JFM 46.1174.01
Full Text:

### References:

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