A two-piece property for free boundary minimal surfaces in the ball

Lima, Vanderson; Menezes, Ana

doi:10.1090/tran/8223

A two-piece property for free boundary minimal surfaces in the ball
HTML articles powered by AMS MathViewer

by Vanderson Lima and Ana Menezes PDF

Trans. Amer. Math. Soc. 374 (2021), 1661-1686 Request permission

Abstract:

We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean $3$-ball in exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains a nullhomologous diameter, then this region is a closed halfball. Moreover, we prove the regularity at the corners of currents minimizing a partially free boundary problem by following ideas by Grüter and Simon. Our first result gives evidence to a conjecture by Fraser and Li.

References

Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp, Index estimates for free boundary minimal hypersurfaces, Math. Ann. 370 (2018), no. 3-4, 1063–1078. MR 3770163, DOI 10.1007/s00208-017-1549-8
Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp, Compactness analysis for free boundary minimal hypersurfaces, Calc. Var. Partial Differential Equations 57 (2018), no. 1, Paper No. 22, 39. MR 3740402, DOI 10.1007/s00526-017-1281-y
L. Ambrozio and I. Nunes, A gap theorem for free boundary minimal surfaces in the three-ball, Preprint in arXiv:1608.05689, to appear in Comm. Anal. Geom.
Theodora Bourni, Allard-type boundary regularity for $C^{1,\alpha }$ boundaries, Adv. Calc. Var. 9 (2016), no. 2, 143–161. MR 3483600, DOI 10.1515/acv-2014-0032
A. Carlotto, G. Franz, and M. Schulz, Free boundary minimal surfaces with connected boundary and arbitrary genus, Preprint at arXiv:2001.04920 (2020).
Jingyi Chen, Ailana Fraser, and Chao Pang, Minimal immersions of compact bordered Riemann surfaces with free boundary, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2487–2507. MR 3301871, DOI 10.1090/S0002-9947-2014-05990-4
R. Courant, The existence of minimal surfaces of given topological structure under prescribed boundary conditions, Acta Math. 72 (1940), 51–98. MR 2478, DOI 10.1007/BF02546328
Baptiste Devyver, Index of the critical catenoid, Geom. Dedicata 199 (2019), 355–371. MR 3928806, DOI 10.1007/s10711-018-0353-2
Ulrich Dierkes, Stefan Hildebrandt, and Anthony J. Tromba, Regularity of minimal surfaces, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 340, Springer, Heidelberg, 2010. With assistance and contributions by A. Küster. MR 2760441
Frank Duzaar and Klaus Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. 546 (2002), 73–138. MR 1900994, DOI 10.1515/crll.2002.046
Nick Edelen, A note on the singular set of area-minimizing hypersurfaces, Calc. Var. Partial Differential Equations 59 (2020), no. 1, Paper No. 18, 9. MR 4040627, DOI 10.1007/s00526-019-1660-7
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Herbert Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767–771. MR 260981, DOI 10.1090/S0002-9904-1970-12542-3
Abigail Folha, Frank Pacard, and Tatiana Zolotareva, Free boundary minimal surfaces in the unit 3-ball, Manuscripta Math. 154 (2017), no. 3-4, 359–409. MR 3713919, DOI 10.1007/s00229-017-0924-9
Ailana Fraser and Martin Man-chun 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. MR 3178438
Ailana Fraser and Richard Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math. 226 (2011), no. 5, 4011–4030. MR 2770439, DOI 10.1016/j.aim.2010.11.007
Ailana Fraser and Richard Schoen, Minimal surfaces and eigenvalue problems, Geometric analysis, mathematical relativity, and nonlinear partial differential equations, Contemp. Math., vol. 599, Amer. Math. Soc., Providence, RI, 2013, pp. 105–121. MR 3202476, DOI 10.1090/conm/599/11927
Ailana Fraser and Richard Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math. 203 (2016), no. 3, 823–890. MR 3461367, DOI 10.1007/s00222-015-0604-x
Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
Brian Freidin, Mamikon Gulian, and Peter McGrath, Free boundary minimal surfaces in the unit ball with low cohomogeneity, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1671–1683. MR 3601558, DOI 10.1090/proc/13424
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer, Berlin (1998).
A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, Preprint at arXiv:1411.6567v1 (2014).
M. Grüter, Regularitt von minimierenden Strömen bei einer freien Randbedingung, Universitat Düsseldorf, 1985.
Michael Grüter, Optimal regularity for codimension one minimal surfaces with a free boundary, Manuscripta Math. 58 (1987), no. 3, 295–343. MR 893158, DOI 10.1007/BF01165891
Michael Grüter, A remark on minimal surfaces with corners, Variational methods (Paris, 1988) Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 273–282. MR 1205159
Robert Hardt and Leon Simon, Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. of Math. (2) 110 (1979), no. 3, 439–486. MR 554379, DOI 10.2307/1971233
S. Hildebrandt and J. C. C. Nitsche, Minimal surfaces with free boundaries, Acta Math. 143 (1979), no. 3-4, 251–272. MR 549778, DOI 10.1007/BF02392096
N. Kapouleas and M. Li, Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disk, Preprint at arXiv:1709.08556 (2017).
N. Kapouleas and D. Wiygul, Free-boundary minimal surfaces with connected boundary in the 3-ball by tripling the equatorial disc, Preprint at arXiv:1711.00818v2 (2017).
D. Ketover, Free boundary minimal surfaces of unbounded genus, Preprint at arXiv:1612.08692 (2016).
Martin Man-chun Li, A general existence theorem for embedded minimal surfaces with free boundary, Comm. Pure Appl. Math. 68 (2015), no. 2, 286–331. MR 3298664, DOI 10.1002/cpa.21513
M. Li and X. Zhou, Min-max theory for free boundary minimal hypersurfaces I-regularity theory, Preprint at arXiv:1611.02612 (2016).
M. Li and X. Zhou, A maximum principle for free boundary minimal varieties of arbitrary codimension, Preprint at arXiv:1708.05001 (2017).
V. Lima, Bounds for the Morse index of free boundary minimal surfaces, Preprint at arXiv:1710.10971 (2017).
Davi Maximo, Ivaldo Nunes, and Graham Smith, Free boundary minimal annuli in convex three-manifolds, J. Differential Geom. 106 (2017), no. 1, 139–186. MR 3640009
Peter McGrath, A characterization of the critical catenoid, Indiana Univ. Math. J. 67 (2018), no. 2, 889–897. MR 3798860, DOI 10.1512/iumj.2018.67.7251
Antonio Ros, A two-piece property for compact minimal surfaces in a three-sphere, Indiana Univ. Math. J. 44 (1995), no. 3, 841–849. MR 1375352, DOI 10.1512/iumj.1995.44.2011
A. Ros, Stability of minimal and constant mean curvature surfaces with free boundary, Mat. Contemp. 35 (2008), 221–240. MR 2584186
M. V. Safonov, On the boundary estimates for second-order elliptic equations, Complex Var. Elliptic Equ. 63 (2018), no. 7-8, 1123–1141. MR 3802819, DOI 10.1080/17476933.2017.1420066
Pam Sargent, Index bounds for free boundary minimal surfaces of convex bodies, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2467–2480. MR 3626504, DOI 10.1090/proc/13442
James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
Graham Smith and Detang Zhou, The Morse index of the critical catenoid, Geom. Dedicata 201 (2019), 13–19. MR 3978532, DOI 10.1007/s10711-018-0412-8
Bruce Solomon, Harmonic maps to spheres, J. Differential Geom. 21 (1985), no. 2, 151–162. MR 816667
Bruce Solomon and Brian White, A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals, Indiana Univ. Math. J. 38 (1989), no. 3, 683–691. MR 1017330, DOI 10.1512/iumj.1989.38.38032
H. Tran, Index characterization for free boundary minimal surfaces, Preprint at arXiv:1609.01651 (2016).
Z. Wang, Existence of infinitely many free boundary minimal hypersurfaces, Preprint at arXiv:2001.04674 (2020).
B. White, Curvature estimates and compactness theorems in $3$-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math. 88 (1987), no. 2, 243–256. MR 880951, DOI 10.1007/BF01388908
Brian White, The maximum principle for minimal varieties of arbitrary codimension, Comm. Anal. Geom. 18 (2010), no. 3, 421–432. MR 2747434, DOI 10.4310/CAG.2010.v18.n3.a1
Rugang Ye, On the existence of area-minimizing surfaces with free boundary, Math. Z. 206 (1991), no. 3, 321–331. MR 1095757, DOI 10.1007/BF02571346