Free boundary minimal surfaces: a survey of recent results

 Access to full text

 Full (PDF)

 Full (DjVu)

1. Aiex, N., Hong, Index, H.estimates for surfaces with constant mean curvature in 3-dimensional manifolds, preprint (arXiv: 1901.00944). Zbl07294622MR4176856DOI10.1007/s00526-020-01855-w
2. Aché, A., Maximo, D., Wu, H., Metrics with nonnegative Ricci curvature on convex three-manifolds, Geom. Topol.20 (2016), no. 5, 2905-2922. Zbl1350.53049MR3556351DOI10.2140/gt.2016.20.2905
3. Almgren, F., Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277-292. Zbl0146.11905MR200816DOI10.2307/1970520
4. Ambrozio, L., Buzano, R., Carlotto, A., Sharp, B., Geometric convergence results for closed minimal surfaces via bubbling analysis, preprint (arXiv: 1803.04956). Zbl1426.53013
5. Ambrozio, L., Buzano, R., Carlotto, A., Sharp, B., Bubbling analysis and geometric convergence results for free boundary minimal surfaces, J. Éc. polytech. Math.6 (2019), 621-664. Zbl1426.53013MR4014631DOI10.5802/jep.102
6. Ambrozio, L., Carlotto, A., Sharp, B., Compactness of the space of minimal hypersurfaces with bounded volume and p-th Jacobi eigenvalue, J. Geom. Anal.26 (2016), no. 4, 2591-2601. Zbl1354.53069MR3544933DOI10.1007/s12220-015-9640-4
7. Ambrozio, L., Carlotto, A., Sharp, B., Compactness analysis for free boundary minimal hypersurfaces, Calc. Var. PDE57 (2018), no. 1, 1-39. Zbl1414.53008MR3740402DOI10.1007/s00526-017-1281-y
8. Ambrozio, L., Carlotto, A., Sharp, B., Comparing the Morse index and the first Betti number of minimal hypersurfaces, J. Differential Geom., 108 (2018), no. 3, 379-410. Zbl1385.53051MR3770846DOI10.4310/jdg/1519959621
9. Ambrozio, L., Carlotto, A., Sharp, B., Index estimates for free boundary minimal hypersurfaces, Math. Ann., 370 (2018), no. 3-4, pages 1063-1078. Zbl1391.53007MR3770163DOI10.1007/s00208-017-1549-8
10. Brendle, S., Embedded minimal tori in S3 and the Lawson conjecture, Acta Math.211 (2013), no. 2, 177-190. Zbl1305.53061MR3143888DOI10.1007/s11511-013-0101-2
11. Buzano, R., Sharp, B., Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area, Trans. Amer. Math. Soc., 370 (2018), 4373-4399. Zbl1390.53008MR3811532DOI10.1090/tran/7168
12. Caffarelli, L., Jerison, D., Kenig, C., Global energy minimizers for free boundary problems and full regularity in three dimensions. Noncompact problems at the intersection of geometry, analysis, and topology, 83-97, Contemp. Math., 350, Amer. Math. Soc., Providence, RI, 2004. Zbl1330.35545MR2082392DOI10.1090/conm/350/06339
13. Calabi, E., Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry1 (1967), 111-125. Zbl0171.20504MR233294
14. Cavalcante, M., de Oliveira, D., Index estimates for free boundary constant mean curvature surfaces, preprint (arXiv: 1803.05995). MR4077688DOI10.2140/pjm.2020.305.153
15. Cheng, S. Y. and Tysk, J., Schrödinger operators and index bounds for minimal submanifolds, Rocky Mountain J. Math.24 (1994), no. 3, 977-996. Zbl0818.53075MR1307586DOI10.1216/rmjm/1181072383
16. Chen, J., Fraser, A., Pang, C., Minimal immersions of compact bordered Riemann surfaces with free boundary, Trans. Amer. Math. Soc., 367 (2014), no. 4, 2487-2507. Zbl1308.58007MR3301871DOI10.1090/S0002-9947-2014-05990-4
17. Chodosh, O., Ketover, D., Maximo, D., Minimal surfaces with bounded index, Invent. Math.209 (2017), no. 3, 617-664. Zbl1378.53072MR3681392DOI10.1007/s00222-017-0717-5
18. Chodosh, O., Maximo, D., On the topology and index of minimal surfaces, J. Differential Geom.104 (2016), no. 3, 399-418. Zbl1357.53016MR3568626
19. Choi, H., Schoen, R., The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math.81 (1985), no. 3, 387-394. Zbl0577.53044MR807063DOI10.1007/BF01388577
20. Choi, H., Wang, A., A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geom.18 (1983), no. 3, 559-562. Zbl0523.53055MR723817
21. Colding, T., De Lellis, C., Singular limit laminations, Morse index, and positive scalar curvature, Topology44 (2005), no. 1, 25-45. Zbl1096.53037MR2103999DOI10.1016/j.top.2004.01.007
22. Colding, T., Minicozzi, W., A course in minimal surfaces, Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp. Zbl1242.53007MR2780140DOI10.1090/gsm/121
23. Courant, R., The existence of minimal surfaces of given topological structure under prescribed boundary conditions, Acta Math.72 (1940), 51-98. Zbl0023.39901MR2478DOI10.1007/BF02546328
24. Courant, R., Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces. Appendix by M. Schiffer, Interscience Publishers, Inc., New York, N.Y., 1950. xiii+330 pp. MR36317
25. De Lellis, C., Ramic, J., Min-max theory for minimal hypersurfaces with boundary, preprint (arXiv:1611.00926). Zbl1408.53079MR3893761
26. Devyver, B., Index of the critical catenoid, preprint (arXiv: 1609.02315). Zbl1412.53015MR3928806DOI10.1007/s10711-018-0353-2
27. M. do Carmo, Peng, C. K., Stable complete minimal surfaces in ${ℝ}^3$ are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 903-906. Zbl0442.53013MR546314DOI10.1090/S0273-0979-1979-14689-5
28. Ejiri, N. and Micallef, M., Comparison between second variation of area and second variation of energy of a minimal surface, Adv. Calc. Var.1 (2008), no. 3, 223-239. Zbl1163.58006MR2458236DOI10.1515/ACV.2008.009
29. Folha, A., Pacard, F., Zolotareva, T., Free boundary minimal surfaces in the unit 3-ball, Manuscripta Math.154 (2017), no. 3-4, 359-409. Zbl1381.35040MR3713919DOI10.1007/s00229-017-0924-9
30. Fraser, A., On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math.53 (2000), no. 8, 931-971. Zbl1039.58013MR1755947DOI10.1002/1097-0312(200008)53:8<931::AID-CPA1>3.3.CO;2-0
31. Fraser, A., Index estimates for minimal surfaces and k-convexity, Proc. Amer. Math. Soc.135 (2007), no. 11, 3733-3744. Zbl1127.58007MR2336590DOI10.1090/S0002-9939-07-08894-6
32. Fraser, A., Li, M., 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. Zbl1295.53062MR3178438
33. Fraser, A., Schoen, R., The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math.226 (2011), no. 5, 4011-4030. Zbl1215.53052MR2770439DOI10.1016/j.aim.2010.11.007
34. Fraser, A., Schoen, R., Minimal surfaces and eigenvalue problems. Geometric analysis, mathematical relativity, and nonlinear partial differential equations, 105-121, Contemp. Math.599, Amer. Math. Soc., Providence, RI, 2013. Zbl1321.35118MR3202476DOI10.1090/conm/599/11927
35. Fraser, A., Schoen, R., Uniqueness theorems for free boundary minimal disks in space forms, Int. Math. Res. Not. IMRN2015, no. 17, 8268-8274. Zbl1325.53076MR3404014DOI10.1093/imrn/rnu192
36. Fraser, A., Schoen, R., Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math.203 (2016), no. 3, 823-890. Zbl1337.35099MR3461367DOI10.1007/s00222-015-0604-x
37. Freidin, B., Gulian, M., McGrath, P., Free boundary minimal surfaces in the unit ball with low cohomgeneity, Proc. Amer. Math. Soc.145 (2017), no. 4, 1671-1683. Zbl1360.49034MR3601558DOI10.1090/proc/13424
38. Grüter, M., Jost, J., On embedded minimal disks in convex bodies, Ann. Inst. H. Poincaré Anal. Non Linéaire3 (1986), no. 5, 345-390. Zbl0617.49017MR868522
39. Guang, Q., Zhou, X., Compactness and generic finiteness for free boundary minimal hypersurfaces, preprint (arXiv:1803.01509). Zbl07326754MR231707DOI10.1002/malq.19680141304
40. Hoffman, D., Meeks, W. H., The strong halfspace theorem for minimal surfaces, Invent. Math.101 (1990), no. 2, 373-377. Zbl0722.53054MR1062966DOI10.1007/BF01231506
41. Hsiang, W.-Y., Minimal cones and the spherical Bernstein problem. I, Ann. of Math. (2) 118 (1983), no. 1, 61-73. Zbl0522.53051MR707161DOI10.2307/2006954
42. Hsiang, W.-Y., Minimal cones and the spherical Bernstein problem. II, Invent. Math.74 (1983), no. 3, 351-369. Zbl0532.53045MR724010DOI10.1007/BF01394241
43. Kapouleas, N., Li, M.Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and equatorial disk, preprint (arXiv: 1709.08556). 
44. Kapouleas, N., Wygul, D., Free-boundary minimal surfaces with connected boundary in the 3-ball by tripling the equatorial disc, preprint (arXiv: 1711.00818). 
45. Irie, K., Marques, F. C., Neves, A., Density of minimal hypersurfaces for generic metrics, Ann. Math.187 (2018), no. 3, 963-972. Zbl1387.53083MR3779962DOI10.4007/annals.2018.187.3.8
46. Jorge, L., Meeks, W. H., The topology of complete minimal surfaces of finite total Gaussian curvature, Topology22 (1983), no. 2, 203-221. Zbl0517.53008MR683761DOI10.1016/0040-9383(83)90032-0
47. Jost, J., Existence results for embedded minimal surfaces of controlled topological type. I, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 1, 15-50. Zbl0619.49019MR863634
48. Jost, J., Existence results for embedded minimal surfaces of controlled topological type. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 3, 401-426. Zbl0669.49024MR881099
49. Jost, J., Existence results for embedded minimal surfaces of controlled topological type. III, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 1, 165-167. Zbl0681.49039MR937541
50. Ketover, D., Free boundary minimal surfaces of unbounded genus, preprint (arXiv:1612.08691). Zbl07088294
51. Li, M., A general existence theorem for embedded minimal surfaces with free boundary, Comm. Pure Appl. Math.68 (2015), no. 2, 286-331. Zbl1321.53074MR3298664DOI10.1002/cpa.21513
52. Li, M., Zhou, X., Min-max theory for free boundary minimal hypersurfaces I - regularity theory, preprint (arXiv:1611.02612). 
53. Lima, V., Bounds for the Morse index of free boundary minimal surfaces, preprint (arXiv: 1710.10971). 
54. Liokumovich, Y., Marques, F. C., Neves, A., Weyl law for the volume spectrum, Ann. Math.187 (2018), no. 3, 933-962. Zbl1390.53034MR3779961DOI10.4007/annals.2018.187.3.7
55. Marques, F., Minimal surfaces - variational theory and applications, Proceedings of the International Congress of Mathematicians, Seoul2014. Zbl1373.53004MR3728473
56. Marques, F., Neves, A., Existence of infinitely many minimal hypersurfaces in positive Ricci curvature, Invent. Math.209 (2017), no. 2, 577-616. Zbl1390.53064MR3674223DOI10.1007/s00222-017-0716-6
57. Marques, F., Neves, A., Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683-782. Zbl1297.49079MR3152944DOI10.4007/annals.2014.179.2.6
58. Marques, F., Neves, A., Morse index and multiplicity of min-max minimal hypersurfaces, Camb. J. Math.4 (2016), no. 4, 463-511. Zbl1367.49036MR3572636DOI10.4310/CJM.2016.v4.n4.a2
59. Marques, F., Neves, A., Rigidity of min-max minimal spheres in three-manifolds, Duke Math. J.161 (2012), no. 14, 2725-2752. Zbl1260.53079MR2993139DOI10.1215/00127094-1813410
60. Marques, F. C., Neves, A., Song, A., Equidistribution of minimal hypersurfaces for generic metrics, preprint (arXiv: 1712.06238). Zbl1419.53061MR3953507DOI10.1007/s00222-018-00850-5
61. Máximo, D., Nunes, I., Smith, G., Free boundary minimal annuli in convex three-manifolds, J. Differential Geom.106 (2017), no. 1, 139-186. Zbl1386.53071MR3640009
62. McGrath, P., A characterization of the critical catenoid, preprint (arXiv:1603.04114v2). Zbl06971405MR3798860DOI10.1512/iumj.2018.67.7251
63. Nadirashvili, N., Penskoi, A., Free boundary minimal surfaces and overdetermined boundary value problems, preprint (arXiv: 1812.08943). Zbl07317681MR4174046DOI10.1007/s11854-020-0129-0
64. Nitsche, J., Stationary partitioning of convex bodies, Arch. Rational Mech. Anal.89 (1985), no. 1, 1-19. Zbl0572.52005MR784101DOI10.1007/BF00281743
65. Osserman, R., On complete minimal surfaces, Arch. Rational Mech. Anal.13 (1963), 392-404. Zbl0127.38003MR151907DOI10.1007/BF01262706
66. Osserman, R., Global properties of minimal surfaces in $E^3$ and $E^n$, Ann. of Math. (2) 80 (1964), 340-364. Zbl0134.38502MR179701DOI10.2307/1970396
67. Pitts, J., Existence and regularity of minimal surfaces on Riemannian manifolds, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo. Mathematical Notes27 (1981). Zbl0462.58003MR626027
68. Ros, A., One-sided complete stable minimal surfaces, J. Differential Geom.74 (2006), no. 1, 69-92. Zbl1110.53009MR2260928
69. Ros, A., Stability of minimal and constant mean curvature surfaces with free boundary, Mat. Contemp.35 (2008), 221-240. Zbl1206.53011MR2584186
70. Ros, A., Souam, R., On stability of capillary surfaces in a ball, Pacific J. Math.178 (1997), no. 2, 345-361. Zbl0930.53007MR1447419DOI10.2140/pjm.1997.178.345
71. Ros, A., Vergasta, E., Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata56 (1995), no. 1, 19-33. Zbl0912.53009MR1338315DOI10.1007/BF01263611
72. Savo, A., Index bounds for minimal hypersurfaces of the sphere, Indiana Univ. Math. J.59 (2010), no. 3, 823-837. Zbl1209.53052MR2779062DOI10.1512/iumj.2010.59.4013
73. Schoen, R., Simon, L., Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math.34 (1981), no. 6, 741-797. Zbl0497.49034MR634285DOI10.1002/cpa.3160340603
74. Sharp, B., Compactness of minimal hypersurfaces with bounded index, J. Differential Geom.106 (2017), no. 2, 317-339. Zbl1390.53065MR3662994DOI10.4310/jdg/1497405628
75. Smith, G., Zhou, D., The Morse index of the critical catenoid, preprint (arXiv:1609.01485). Zbl1418.53010MR3978532DOI10.1007/s10711-018-0412-8
76. Struwe, M., On a free boundary problem for minimal surfaces, Invent. Math.75 (1984), no. 3, 547-560. Zbl0537.35037MR735340DOI10.1007/BF01388643
77. Tysk, J., Finiteness of index and total scalar curvature for minimal hypersurfaces, Proc. Amer. Math. Soc.105 (1989), no. 2, 428-435. Zbl0661.53039MR946639DOI10.2307/2046961
78. Wang, G., Birkhoff minimax principle for minimal surfaces with a free boundary, Math. Ann.314 (1999), no. 1, 89-107. Zbl0938.58015MR1689264DOI10.1007/s002080050287
79. White, B., Which ambient spaces admit isoperimetric inequalities for submanifolds?, J. Diff. Geom.83 (2009), no. 1 213-228. Zbl1179.53061MR2545035