Free boundary minimal disks in convex balls

References

[1] A. Aché, D. Maximo and H. Wu, Metrics with nonnegative Ricci curvature on convex three-manifolds, Geom. Topol. 20 (2016), no. 5, 2905–2922. 10.2140/gt.2016.20.2905Search in Google Scholar

[2] S. Alexakis, T. Balehowsky and A. Nachman, Determining a Riemannian metric from minimal areas, Adv. Math. 366 (2020), Article ID 107025. 10.1016/j.aim.2020.107025Search in Google Scholar

[3] L. Ambrozio, F. C. Marques and A. Neves, Riemannian metrics on the sphere with Zoll families of minimal hypersurfaces, J. Differential Geom. 130 (2025), no. 2, 269–341. 10.4310/jdg/1747156792Search in Google Scholar

[4] G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199–300. 10.1090/S0002-9947-1917-1501070-3Search in Google Scholar

[5] R. Buzano, R. Haslhofer and O. Hershkovits, The moduli space of two-convex embedded spheres, J. Differential Geom. 118 (2021), no. 2, 189–221. 10.4310/jdg/1622743139Search in Google Scholar

[6] O. Chodosh and C. Mantoulidis, Minimal surfaces and the Allen–Cahn equation on 3-manifolds: Index, multiplicity, and curvature estimates, Ann. of Math. (2) 191 (2020), no. 1, 213–328. 10.4007/annals.2020.191.1.4Search in Google Scholar

[7] F. Da Lio, L. Martinazzi and T. Rivière, Blow-up analysis of a nonlocal Liouville-type equation, Anal. PDE 8 (2015), no. 7, 1757–1805. 10.2140/apde.2015.8.1757Search in Google Scholar

[8] C. De Lellis and J. De Lellis-Ramic, Min-max theory for minimal hypersurfaces with boundary, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 1909–1986. 10.5802/aif.3200Search in Google Scholar

[9] U. Dierkes, S. Hildebrandt, A. Küster and O. Wohlrab, Minimal surfaces. I, Grundlehren Math. Wiss. 295, Springer, Berlin 1992. 10.1007/978-3-662-08776-3Search in Google Scholar

[10] N. Edelen, R. Haslhofer, M. N. Ivaki and J. J. Zhu, Mean convex mean curvature flow with free boundary, Comm. Pure Appl. Math. 75 (2022), no. 4, 767–817. 10.1002/cpa.22009Search in Google Scholar

[11] G. Franz, Equivariant index bound for min-max free boundary minimal surfaces, Calc. Var. Partial Differential Equations 62 (2023), no. 7, Paper No. 201. Search in Google Scholar

[12] G. Franz and M. Schulz, Topological control for min-max free boundary minimal surfaces, preprint (2023), https://arxiv.org/abs/2307.00941. 10.1007/s00526-023-02514-6Search in Google Scholar

[13] A. Fraser and M. M.-C. Li, Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary, J. Differential Geom. 96 (2014), no. 2, 183–200. 10.4310/jdg/1393424916Search in Google Scholar

[14] A. M. Fraser, On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math. 53 (2000), no. 8, 931–971. 10.1002/1097-0312(200008)53:8<931::AID-CPA1>3.3.CO;2-0Search in Google Scholar

[15] M. A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71–111. 10.2307/1971486Search in Google Scholar

[16] M. Grüter and J. Jost, Allard type regularity results for varifolds with free boundaries, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 13 (1986), no. 1, 129–169. Search in Google Scholar

[17] M. Grüter and J. Jost, On embedded minimal disks in convex bodies, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 5, 345–390. 10.1016/s0294-1449(16)30378-xSearch in Google Scholar

[18] M. Guaraco, F. Marques and A. Neves, Multiplicity one and strictly stable Allen–Cahn minimal hypersurfaces, preprint (2019), https://arxiv.org/abs/1912.08997. Search in Google Scholar

[19] V. Guillemin, The Radon transform on Zoll surfaces, Adv. Math. 22 (1976), no. 1, 85–119. 10.1016/0001-8708(76)90139-0Search in Google Scholar

[20] R. Haslhofer, Free boundary flow with surgery, preprint (2023), https://arxiv.org/abs/2306.077140. Search in Google Scholar

[21] R. Haslhofer and D. Ketover, Minimal 2-spheres in 3-spheres, Duke Math. J. 168 (2019), no. 10, 1929–1975. 10.1215/00127094-2019-0009Search in Google Scholar

[22] A. E. Hatcher, A proof of the Smale conjecture, Diff ⁢ ( S 3 ) ≃ O ⁢ ( 4 ) , Ann. of Math. (2) 117 (1983), no. 3, 553–607. 10.2307/2007035Search in Google Scholar

[23] K. Irie, K. Irie, F. C. Marques and A. Neves, Density of minimal hypersurfaces for generic metrics, Ann. of Math. (2) 187 (2018), no. 3, 963–972. 10.4007/annals.2018.187.3.8Search in Google Scholar

[24] J. Jost, Existence results for embedded minimal surfaces of controlled topological type. II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 13 (1986), no. 3, 401–426. Search in Google Scholar

[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. 10.4310/jdg/1214443602Search in Google Scholar

[26] D. Ketover, Flipping Heegaard splittings and minimal surfaces, preprint (2022), https://arxiv.org/abs/2211.03745. Search in Google Scholar

[27] D. Ketover, F. C. Marques and A. Neves, The catenoid estimate and its geometric applications, J. Differential Geom. 115 (2020), no. 1, 1–26. 10.4310/jdg/1586224840Search in Google Scholar

[28] P. Laurain and R. Petrides, Existence of min-max free boundary disks realizing the width of a manifold, Adv. Math. 352 (2019), 326–371. 10.1016/j.aim.2019.06.001Search in Google Scholar

[29] M. M.-C. Li, A general existence theorem for embedded minimal surfaces with free boundary, Comm. Pure Appl. Math. 68 (2015), no. 2, 286–331. 10.1002/cpa.21513Search in Google Scholar

[30] M. M.-C. Li and X. Zhou, Min-max theory for free boundary minimal hypersurfaces I—Regularity theory, J. Differential Geom. 118 (2021), no. 3, 487–553. 10.4310/jdg/1625860624Search in Google Scholar

[31] L. Lin, A. Sun and X. Zhou, Min-max minimal disks with free boundary in Riemannian manifolds, Geom. Topol. 24 (2020), no. 1, 471–532. 10.2140/gt.2020.24.471Search in Google Scholar

[32] Y. Liokumovich, F. C. Marques and A. Neves, Weyl law for the volume spectrum, Ann. of Math. (2) 187 (2018), no. 3, 933–961. 10.4007/annals.2018.187.3.7Search in Google Scholar

[33] L. Lyusternik and L. Šnirel’man, Topological methods in variational problems and their application to the differential geometry of surfaces, Uspehi Matem. Nauk (N. S.) 2 (1947), no. 1(17), 166–217. Search in Google Scholar

[34] F. C. Marques, Deforming three-manifolds with positive scalar curvature, Ann. of Math. (2) 176 (2012), no. 2, 815–863. 10.4007/annals.2012.176.2.3Search in Google Scholar

[35] F. C. Marques and A. Neves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683–782. 10.4007/annals.2014.179.2.6Search in Google Scholar

[36] F. C. Marques and A. Neves, Morse index and multiplicity of min-max minimal hypersurfaces, Camb. J. Math. 4 (2016), no. 4, 463–511. 10.4310/CJM.2016.v4.n4.a2Search in Google Scholar

[37] F. C. Marques and A. Neves, Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent. Math. 209 (2017), no. 2, 577–616. 10.1007/s00222-017-0716-6Search in Google Scholar

[38] F. C. Marques and A. Neves, Morse index of multiplicity one min-max minimal hypersurfaces, Adv. Math. 378 (2021), Article ID 107527. 10.1016/j.aim.2020.107527Search in Google Scholar

[39] F. C. Marques, A. Neves and A. Song, Equidistribution of minimal hypersurfaces for generic metrics, Invent. Math. 216 (2019), no. 2, 421–443. 10.1007/s00222-018-00850-5Search in Google Scholar

[40] D. Maximo, I. Nunes and G. Smith, Free boundary minimal annuli in convex three-manifolds, J. Differential Geom. 106 (2017), no. 1, 139–186. 10.4310/jdg/1493172096Search in Google Scholar

[41] W. Meeks, III, L. Simon and S. T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. 10.2307/2007026Search in Google Scholar

[42] V. Millot and Y. Sire, On a fractional Ginzburg–Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal. 215 (2015), no. 1, 125–210. 10.1007/s00205-014-0776-3Search in Google Scholar

[43] J. C. C. Nitsche, Stationary partitioning of convex bodies, Arch. Ration. Mech. Anal. 89 (1985), no. 1, 1–19. 10.1007/BF00281743Search in Google Scholar

[44] R. Petrides, Non planar free boundary minimal disks into ellipsoids, preprint (2023), https://arxiv.org/abs/2304.12111. Search in Google Scholar

[45] A. Pigati and T. Rivière, A proof of the multiplicity 1 conjecture for min-max minimal surfaces in arbitrary codimension, Duke Math. J. 169 (2020), no. 11, 2005–2044. 10.1215/00127094-2020-0002Search in Google Scholar

[46] J. T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes 27, Princeton University, Princeton 1981. 10.1515/9781400856459Search in Google Scholar

[47] H. Poincaré, Sur les lignes géodésiques des surfaces convexes, Trans. Amer. Math. Soc. 6 (1905), no. 3, 237–274. 10.1090/S0002-9947-1905-1500710-4Search in Google Scholar

[48] T. Riviere, Minmax hierarchies, minimal fibrations and a PDE based proof of the Willmore conjecture, preprint (2020), https://arxiv.org/abs/2007.05467. Search in Google Scholar

[49] F. Smith, On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary Riemannian metric, Ph.D. Thesis, University of Melbourne, 1982. 10.1017/S0004972700026216Search in Google Scholar

[50] A. Song, Existence of infinitely many minimal hypersurfaces in closed manifolds, Ann. of Math. (2) 197 (2023), no. 3, 859–895. 10.4007/annals.2023.197.3.1Search in Google Scholar

[51] M. Struwe, On a free boundary problem for minimal surfaces, Invent. Math. 75 (1984), no. 3, 547–560. 10.1007/BF01388643Search in Google Scholar

[52] M. Struwe, Plateau flow or the heat flow for half-harmonic maps, Anal. PDE 17 (2024), no. 4, 1397–1438. 10.2140/apde.2024.17.1397Search in Google Scholar

[53] A. Sun, Z. Wang and X. Zhou, Multiplicity one for min-max theory in compact manifolds with boundary and its applications, Calc. Var. Partial Differential Equations 63 (2024), no. 3, Paper No. 70. 10.1007/s00526-024-02669-wSearch in Google Scholar

[54] Z. Wang, Min-max minimal hypersurface in manifolds with convex boundary and Ric ≥ 0 , Math. Ann. 371 (2018), no. 3–4, 1545–1574. 10.1007/s00208-018-1682-zSearch in Google Scholar

[55] Z. Wang, Existence of infinitely many free boundary minimal hypersurfaces, J. Differential Geom. 126 (2024), no. 1, 363–399. 10.4310/jdg/1707767341Search in Google Scholar

[56] Z. Wang and X. Zhou, Existence of four minimal spheres in S 3 with a bumpy metric, preprint (2023), https://arxiv.org/abs/2305.08755. Search in Google Scholar

[57] J. Wettstein, Uniqueness and regularity of the fractional harmonic gradient flow in S n − 1 , Nonlinear Anal. 214 (2022), Article ID 112592. 10.1016/j.na.2021.112592Search in Google Scholar

[58] J. D. Wettstein, Half-harmonic gradient flow: aspects of a non-local geometric PDE, Math. Eng. 5 (2023), no. 3, Paper No. 058. 10.3934/mine.2023058Search in Google Scholar

[59] B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200. 10.1512/iumj.1991.40.40008Search in Google Scholar

[60] S. T. Yau, Problem section, Seminar on differential geometry, Ann. of Math. Stud. 102, Princeton University, Princeton (1982), 669–706. 10.1515/9781400881918-035Search in Google Scholar

[61] X. Zhou, On the multiplicity one conjecture in min-max theory, Ann. of Math. (2) 192 (2020), no. 3, 767–820. 10.4007/annals.2020.192.3.3Search in Google Scholar

[62] X. Zhou, Mean curvature and variational theory, ICM—International Congress of Mathematicians. Vol. 4. Sections 5–8, European Mathematical Society, Berlin (2023), 2696–2717. 10.4171/icm2022/27Search in Google Scholar

[63] O. Zoll, Ueber Flächen mit Scharen geschlossener geodätischer Linien, Math. Ann. 57 (1903), no. 1, 108–133. 10.1007/BF01449019Search in Google Scholar