×

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
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] S. Alexakis, T. Balehowsky, and A. Nachman, Determining a Riemannian metric from minimal areas, preprint, arXiv:1711.09379v2 [math.DG].
[2] F. J. Almgren, Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277-292. · Zbl 0146.11905
[3] R. H. Bamler and B. Kleiner, Ricci flow and diffeomorphism groups of 3-manifolds, preprint, arXiv:1712.06197v1 [math.DG].
[4] G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199-300. · JFM 46.1174.01
[5] H. Bray, S. Brendle, M. Eichmair, and A. Neves, Area-minimizing projective planes in 3-manifolds, Comm. Pure Appl. Math. 63 (2010), no. 9, 1237-1247. · Zbl 1200.53053
[6] S. Brendle, An inscribed radius estimate for mean curvature flow in Riemannian manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 4, 1447-1472. · Zbl 1365.53059
[7] S. Brendle and G. Huisken, Mean curvature flow with surgery of mean convex surfaces in three-manifolds, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 9, 2239-2257. · Zbl 1407.53061
[8] R. Buzano, R. Haslhofer, and O. Hershkovits, The moduli space of two-convex embedded spheres, preprint, arXiv:1607.05604v2 [math.DG].
[9] Y. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749-786. · Zbl 0696.35087
[10] O. Chodosh and C. Mantoulidis, Minimal surfaces and the Allen-Cahn equation on 3-manifolds: Index, multiplicity, and curvature estimates, preprint, arXiv:1803.02716v2 [math.DG].
[11] T. H. Colding and C. De Lellis, “The min – max construction of minimal surfaces” in Surveys in Differential Geometry, Vol. VIII (Boston, 2002), Surv. Differ. Geom. 8, International Press, Somerville, 2003, 75-107. · Zbl 1051.53052
[12] T. H. Colding, D. Gabai, and D. Ketover, On the classification of Heegaard splittings, Duke Math. J. 167 (2018), no. 15, 2833-2856. · Zbl 1403.57013
[13] T. H. Colding and W. P. Minicozzi, II, A Course in Minimal Surfaces, Grad. Stud. Math. 121, Amer. Math. Soc., Providence, 2011. · Zbl 1242.53007
[14] O. Cornea, G. Lupton, J. Oprea, and D. Tanré, Lusternick-Schnirelmann Category, Math. Surveys Monogr. 103, Amer. Math. Soc., Providence, 2003.
[15] L. C. Evans and J. Spruck, Motion of level sets by mean curvature, I, J. Differential Geom. 33 (1991), no. 3, 635-681. · Zbl 0726.53029
[16] M. A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71-111. · Zbl 0686.53036
[17] L. Guth, The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture, Acta Math. 205 (2010), no. 2, 263-286. · Zbl 1210.52004
[18] R. S. Hamilton, Monotonicity formulas for parabolic flows on manifolds, Comm. Anal. Geom. 1 (1993), no. 1, 127-137. · Zbl 0779.58037
[19] R. Haslhofer and B. Kleiner, Mean curvature flow of mean convex hypersurfaces, Comm. Pure Appl. Math. 70 (2017), no. 3, 511-546. · Zbl 1360.53069
[20] R. Haslhofer and B. Kleiner, Mean curvature flow with surgery, Duke Math. J. 166 (2017), no. 9, 1591-1626. · Zbl 1370.53046
[21] A. E. Hatcher, A proof of the Smale conjecture, \(\text{Diff}(S^3)\simeq{\text{O}} (4)\), Ann. of Math. (2) 117 (1983), no. 3, 553-607.
[22] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266. · Zbl 0556.53001
[23] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean curvature, Invent. Math. 84 (1986), no. 3, 463-480. · Zbl 0589.53058
[24] T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520. · Zbl 0798.35066
[25] J. Jost, Embedded minimal surfaces in manifolds diffeomorphic to the three-dimensional ball or sphere, J. Differential Geom. 30 (1989), no. 2, 555-577. · Zbl 0682.53056
[26] D. Ketover, Genus bounds for min – max minimal surfaces, to appear in J. Differential Geom., preprint, arXiv:1312.2666v2 [math.DG].
[27] D. Ketover and Y. Liokumovich, On the existence of unstable minimal Heegaard surfaces, preprint, arXiv:1709.09744v1 [math.DG].
[28] D. Ketover, Y. Liokumovich, and A. Song, On the existence of minimal Heegaard surfaces, in preparation.
[29] D. Ketover, F. C. Marques, and A. Neves, The catenoid estimate and its geometric applications, preprint, arXiv:1601.04514v1 [math.DG].
[30] L. Liokumovich, F. C. Marques, and A. Neves, Weyl law for the volume spectrum, Ann. of Math. (2) 187 (2018), no. 3, 933-961. · Zbl 1390.53034
[31] L. Lusternik and L. Schnirelmann, Topological methods in variational problems and their application to the differential geometry of surfaces (in Russian), Uspekhi Mat. Nauk 2 (1947), no. 1, 166-217.
[32] F. C. Marques and A. Neves, Rigidity of min – max minimal spheres in three-manifolds, Duke Math. J. 161 (2012), no. 14, 2725-2752. · Zbl 1260.53079
[33] F. C. Marques and A. Neves, Morse index and multiplicity of min – max minimal hypersurfaces, Camb. J. Math. 4 (2016), no. 4, 463-511. · Zbl 1367.49036
[34] F. C. Marques and A. Neves, Morse index of multiplicity one min – max minimal hypersurfaces, preprint, arXiv:1803.04273v2 [math.DG]. · Zbl 1367.49036
[35] F. C. Marques, A. Neves, and A. Song, Equidistribution of minimal hypersurfaces for generic metrics, preprint, arXiv:1712.06238v2 [math.DG]. · Zbl 1419.53061
[36] W. Meeks, III, L. Simon, and S. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621-659. · Zbl 0521.53007
[37] B. Sharp, Compactness of minimal hypersurfaces with bounded index, J. Differential Geom. 106 (2017), no. 2, 317-339. · Zbl 1390.53065
[38] F. Smith, On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary Riemannian metric, PhD dissertation, supervised by Leon Simon, University of Melbourne, Melbourne, 1982.
[39] A. Song, Local min – max surfaces and strongly irreducible minimal Heegaard splittings, preprint, arXiv:1706.01037v1 [math.DG].
[40] B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161-200. · Zbl 0742.58009
[41] B. White, The topology of hypersurfaces moving by mean curvature, Comm. Anal. Geom. 3 (1995), nos. 1-2, 317-333.
[42] B. White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 (2000), no. 3, 665-695. · Zbl 0961.53039
[43] B. White, On the bumpy metrics theorem for minimal submanifolds, Amer. J. Math. 139 (2107), no. 4, 1149-1155. · Zbl 1379.53084
[44] S.-T. Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), nos. 1-2, 109-158.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.