A note on free boundary hypersurfaces in space form balls. (English) Zbl 1519.53010
Summary: In this article, we establish a relationship between geometric quantities of a hypersurface restricted to its boundary, and the geometric quantities of its boundary as a hypersurface of the boundary of the ball. As a first application, we prove that the quantity of umbilical points of a free boundary surface in the unit ball counted with multiplicities depend only on its topology; moreover, we obtain as consequences that free boundary surfaces are annuli if and only if they have no umbilical points, and a new proof of the Nitsche theorem. Secondly, we prove a geometric integral inequality for compact hypersurfaces, and we characterize topologically the equality under the free boundary condition.
MSC:
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 58C35 | Integration on manifolds; measures on manifolds |
References:
| [1] | Alencar, H.; do Carmo, M., Hypersurfaces with constant mean curvature in space forms, An. Acad. Brasil. Cienc., 66, 265-274 (1994) · Zbl 0810.53051 |
| [2] | Alencar, H.; do Carmo, M., Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc., 120, 4, 1223-1229 (1994) · Zbl 0802.53017 · doi:10.1090/S0002-9939-1994-1172943-2 |
| [3] | Alías, LJ; de Lira, JHS; Malacarne, JM, Constant higher-order mean curvature hypersurfaces in Riemannian spaces, J. Inst. Math. Jussieu, 5, 4, 527-562 (2006) · Zbl 1118.53038 · doi:10.1017/S1474748006000077 |
| [4] | Almgren, FJ Jr, Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2), 84, 277-292 (1966) · Zbl 0146.11905 · doi:10.2307/1970520 |
| [5] | Barbosa, E., Viana, C.: Area rigidity for the equatorial disk in the ball arXiv:1807.07408 (2018) |
| [6] | Catino, G., On conformally flat manifolds with constant positive scalar curvature, Proc. Amer. Math. Soc., 144, 6, 2627-2634 (2016) · Zbl 1335.53046 · doi:10.1090/proc/12925 |
| [7] | Catino, G., A remark on compact hypersurfaces with constant mean curvature in space forms, Bull. Sci. Math., 140, 8, 901-907 (2016) · Zbl 1351.53069 · doi:10.1016/j.bulsci.2016.04.002 |
| [8] | Courant, R., The existence of minimal surfaces of given topological structure under prescribed boundary conditions, Acta Math., 72, 51-98 (1940) · Zbl 0023.39901 · doi:10.1007/BF02546328 |
| [9] | Courant, R., On Plateau’s problem with free boundaries, Proc. Nat. Acad. Sci. USA, 31, 242-246 (1945) · Zbl 0063.00986 · doi:10.1073/pnas.31.8.242 |
| [10] | do Carmo, M., Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc., 277, 2, 685-709 (1983) · Zbl 0518.53059 · doi:10.1090/S0002-9947-1983-0694383-X |
| [11] | Eschenburg, J-H; de Tribuzy, RA, Constant mean curvature surfaces in \(4\)-space forms, Rend. Sem. Mat. Univ. Padova, 79, 185-202 (1988) · Zbl 0652.53003 |
| [12] | Folha, A.; Pacard, F.; Zolotareva, T., Free boundary minimal surfaces in the unit 3-ball, Manuscripta Math., 154, 3-4, 359-409 (2017) · Zbl 1381.35040 · doi:10.1007/s00229-017-0924-9 |
| [13] | Fraser, A.; Schoen, R., The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math., 226, 5, 4011-4030 (2011) · Zbl 1215.53052 · doi:10.1016/j.aim.2010.11.007 |
| [14] | Fraser, A., Schoen, R.: Minimal surfaces and eigenvalue problems. In: Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, pp. 105-121. Contemp. Math., vol. 599. Amer. Math. Soc., Providence, RI (2013) · Zbl 1321.35118 |
| [15] | Fraser, A.; Schoen, R., Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math., 203, 3, 823-890 (2016) · Zbl 1337.35099 · doi:10.1007/s00222-015-0604-x |
| [16] | Freidin, B.; Gulian, M.; McGrath, P., Free boundary minimal surfaces in the unit ball with low cohomogeneity, Proc. Amer. Math. Soc., 145, 4, 1671-1683 (2017) · Zbl 1360.49034 · doi:10.1090/proc/13424 |
| [17] | Kapouleas, Ni., Wiygul, D.: Free boundary minimal surfaces with connected boundary in the \(3\)-ball by tripling the equatorial disc. J. Differential Geom. 123(2), 311-362 (2023) · Zbl 1519.53012 |
| [18] | Kapouleas, N.; Li, MM-C, Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc, J. Reine Angew. Math., 776, 201-254 (2021) · Zbl 1478.53108 · doi:10.1515/crelle-2020-0050 |
| [19] | Lewy, H., On minimal surfaces with partially free boundary, Comm. Pure App. Math., 4, 1, 1-13 (1951) · doi:10.1002/cpa.3160040102 |
| [20] | Lewy, H., On the boundary behavior of minimal surfaces, Proc. Nat. Acad. Sci. USA., 37, 103-110 (1951) · Zbl 0042.15702 · doi:10.1073/pnas.37.2.103 |
| [21] | Li, M.M.-C.: Free boundary minimal surfaces in the unit ball: recent advances and open questions. In: Proceedings of the International Consortium of Chinese Mathematicians 2017, pp. 401-435. Int. Press, Boston, MA (2020) · Zbl 1465.53008 |
| [22] | Nitsche, JCC, Stationary partitioning of convex bodies, Arch. Rat. Mech. Anal., 89, 1, 1-19 (1985) · Zbl 0572.52005 · doi:10.1007/BF00281743 |
| [23] | Stahl, A., Convergence of solutions to the mean curvature flow with a Neumann boundary condition, Calc. Var. Partial Differential Equations, 4, 5, 421-441 (1996) · Zbl 0896.35059 · doi:10.1007/BF01246150 |
| [24] | Tam, L-F; Zhou, D., Stability properties for the higher dimensional catenoid in \(\mathbb{R}^{n+1} \), Proc. Amer. Math. Soc., 137, 10, 3451-3461 (2009) · Zbl 1184.53016 · doi:10.1090/S0002-9939-09-09962-6 |
| [25] | Wang, B., Simons’ equation and minimal hypersurfaces in space forms, Proc. Amer. Math. Soc., 146, 1, 369-383 (2018) · Zbl 1411.53046 · doi:10.1090/proc/13781 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
