Bubbling analysis and geometric convergence results for free boundary minimal surfaces
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 621-664.

We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their limit, and of the bubbles or half-bubbles that occur as blow-up models. We present various geometric applications of these tools, including a description of the behaviour of index one free boundary minimal surfaces inside a 3-manifold of non-negative scalar curvature and strictly mean convex boundary. In particular, in the case of compact, simply connected, strictly mean convex domains in 3 unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified.

Nous étudions le comportement à la limite de suites de surfaces minimales à bord libre d’indice et de volume bornés, en présentant une analyse détaillée de la dégénérescence au voisinage des points de concentration de courbure. Nous en déduisons une identité générale de quantification pour la fonctionnelle de courbure totale, valable en dimension inférieure à 8 et applicable à des hypersurfaces limites qui peuvent être impropres. En dimension 3, cette identité peut être combinée au théorème de Gauss-Bonnet pour fournir une contrainte reliant la topologie des surfaces minimales à bord libre dans une suite convergente, celle de leur limite, et celle des bulles ou demi-bulles qui apparaissent comme modèles d’explosion. Nous présentons diverses applications de ces outils, notamment une description du comportement des surfaces minimales à bord libre d’indice 1 dans une variété de dimension 3 de courbure scalaire positive ou nulle et à bord strictement convexe en moyenne. En particulier, dans le cas de domaines de 3 compacts, simplement connexes et strictement convexes en moyenne, il y a convergence inconditionnelle pour tous les types topologiques exceptés le disque et l’anneau et, dans ces cas, nous classifions les dégénérescences possibles.

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DOI: 10.5802/jep.102
Classification: 53A10,  53C42,  49Q05
Keywords: Surfaces minimales à bord libre, analyse des bulles, quantification, compacité géométrique
Ambrozio, Lucas 1; Buzano, Reto 2; Carlotto, Alessandro 3; Sharp, Ben 4

1 Department of Mathematics, University of Warwick Coventry CV4 7AL, United Kingdom
2 School of Mathematical Sciences, Queen Mary University of London London E1 4NS, United Kingdom
3 Department of Mathematics, ETH 8092 Zürich, Switzerland
4 School of Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom
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Ambrozio, Lucas; Buzano, Reto; Carlotto, Alessandro; Sharp, Ben. Bubbling analysis and geometric convergence results for free boundary minimal surfaces. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 621-664. doi : 10.5802/jep.102. http://www.numdam.org/articles/10.5802/jep.102/

[1] Ambrozio, Lucas; Buzano, Reto; Carlotto, Alessandro; Sharp, Ben Geometric convergence results for closed minimal surfaces via bubbling analysis, 2018 | arXiv

[2] Ambrozio, Lucas; Carlotto, Alessandro; Sharp, Ben Compactness of the space of minimal hypersurfaces with bounded volume and p-th Jacobi eigenvalue, J. Geom. Anal., Volume 26 (2016) no. 4, pp. 2591-2601 | DOI | MR | Zbl

[3] Ambrozio, Lucas; Carlotto, Alessandro; Sharp, Ben Compactness analysis for free boundary minimal hypersurfaces, Calc. Var. Partial Differential Equations, Volume 57 (2018) no. 1, 22, 39 pages | DOI | MR | Zbl

[4] Ambrozio, Lucas; Carlotto, Alessandro; Sharp, Ben Index estimates for free boundary minimal hypersurfaces, Math. Ann., Volume 370 (2018) no. 3-4, pp. 1063-1078 | DOI | MR | Zbl

[5] Buzano, Reto; Sharp, Ben Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area, Trans. Amer. Math. Soc., Volume 370 (2018) no. 6, pp. 4373-4399 | DOI | MR | Zbl

[6] do Carmo, M.; Peng, C. K. Stable complete minimal surfaces in R 3 are planes, Bull. Amer. Math. Soc. (N.S.), Volume 1 (1979) no. 6, pp. 903-906 | DOI | MR | Zbl

[7] Chen, Jingyi; Fraser, Ailana; Pang, Chao Minimal immersions of compact bordered Riemann surfaces with free boundary, Trans. Amer. Math. Soc., Volume 367 (2015) no. 4, pp. 2487-2507 | DOI | MR | Zbl

[8] Cheng, Shiu Yuen; Tysk, Johan Schrödinger operators and index bounds for minimal submanifolds, Rocky Mountain J. Math., Volume 24 (1994) no. 3, pp. 977-996 | DOI | Zbl

[9] Chodosh, Otis; Maximo, Davi On the topology and index of minimal surfaces, J. Differential Geom., Volume 104 (2016) no. 3, pp. 399-418 http://projecteuclid.org/euclid.jdg/1478138547 | DOI | MR | Zbl

[10] Courant, R. The existence of minimal surfaces of given topological structure under prescribed boundary conditions, Acta Math., Volume 72 (1940), pp. 51-98 | DOI | MR | Zbl

[11] Courant, Richard Dirichlet’s principle, conformal mapping, and minimal surfaces, Springer-Verlag, New York-Heidelberg, 1977 | Zbl

[12] De Lellis, Camillo; Ramic, Jusuf Min-max theory for minimal hypersurfaces with boundary, Ann. Inst. Fourier (Grenoble), Volume 68 (2018) no. 5, pp. 1909-1986 http://aif.cedram.org/item?id=AIF_2018__68_5_1909_0 | DOI | MR | Zbl

[13] Ejiri, Norio; Micallef, Mario Comparison between second variation of area and second variation of energy of a minimal surface, Adv. Calc. Var., Volume 1 (2008) no. 3, pp. 223-239 | DOI | MR | Zbl

[14] Fischer-Colbrie, D. On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math., Volume 82 (1985) no. 1, pp. 121-132 | DOI | MR | Zbl

[15] Fischer-Colbrie, Doris; Schoen, Richard The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math., Volume 33 (1980) no. 2, pp. 199-211 | DOI | MR | Zbl

[16] Folha, Abigail; Pacard, Frank; Zolotareva, Tatiana Free boundary minimal surfaces in the unit 3-ball, Manuscripta Math., Volume 154 (2017) no. 3-4, pp. 359-409 | DOI | MR | Zbl

[17] Fraser, Ailana; Li, Martin Man-chun Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary, J. Differential Geom., Volume 96 (2014) no. 2, pp. 183-200 http://projecteuclid.org/euclid.jdg/1393424916 | DOI | MR | Zbl

[18] Fraser, Ailana; Schoen, Richard Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math., Volume 203 (2016) no. 3, pp. 823-890 | DOI | MR | Zbl

[19] Fraser, Ailana M. On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math., Volume 53 (2000) no. 8, pp. 931-971 | DOI | MR | Zbl

[20] Freidin, Brian; Gulian, Mamikon; McGrath, Peter Free boundary minimal surfaces in the unit ball with low cohomogeneity, Proc. Amer. Math. Soc., Volume 145 (2017) no. 4, pp. 1671-1683 | DOI | MR | Zbl

[21] Grüter, M.; Jost, J. On embedded minimal disks in convex bodies, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 3 (1986) no. 5, pp. 345-390 | DOI | MR | Zbl

[22] Guang, Q.; Zhou, X. Compactness and generic finiteness for free boundary minimal hypersurfaces, 2018 | arXiv

[23] Hoffman, D.; Meeks, W. H. III The strong halfspace theorem for minimal surfaces, Invent. Math., Volume 101 (1990) no. 2, pp. 373-377 | DOI | MR | Zbl

[24] Jorge, Luquésio P.; Meeks, William H. III The topology of complete minimal surfaces of finite total Gaussian curvature, Topology, Volume 22 (1983) no. 2, pp. 203-221 | DOI | MR | Zbl

[25] Jost, Jürgen Existence results for embedded minimal surfaces of controlled topological type. I, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 13 (1986) no. 1, pp. 15-50 | MR | Zbl

[26] Jost, Jürgen Existence results for embedded minimal surfaces of controlled topological type. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 13 (1986) no. 3, pp. 401-426 | MR | Zbl

[27] Jost, Jürgen Existence results for embedded minimal surfaces of controlled topological type. III, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 14 (1987) no. 1, pp. 165-167 | MR | Zbl

[28] Kapouleas, N.; Li, N. Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and equatorial disk, 2017 | arXiv

[29] Kapouleas, N.; Wygul, D. Free-boundary minimal surfaces with connected boundary in the 3-ball by tripling the equatorial disc, 2017 | arXiv

[30] Ketover, D. Free boundary minimal surfaces of unbounded genus, 2016 | arXiv

[31] Li, Martin Man-chun A general existence theorem for embedded minimal surfaces with free boundary, Comm. Pure Appl. Math., Volume 68 (2015) no. 2, pp. 286-331 | DOI | MR | Zbl

[32] Li, N.; Zhou, X. Min-max theory for free boundary minimal hypersurfaces I – regularity theory, 2016 | arXiv

[33] Lima, V. Bounds for the Morse index of free boundary minimal surfaces, 2017 | arXiv

[34] López, Francisco J.; Ros, Antonio Complete minimal surfaces with index one and stable constant mean curvature surfaces, Comment. Math. Helv., Volume 64 (1989) no. 1, pp. 34-43 | DOI | MR | Zbl

[35] López, Francisco J.; Ros, Antonio On embedded complete minimal surfaces of genus zero, J. Differential Geom., Volume 33 (1991) no. 1, pp. 293-300 http://projecteuclid.org/euclid.jdg/1214446040 | DOI | MR | Zbl

[36] Maximo, Davi; Nunes, Ivaldo; Smith, Graham Free boundary minimal annuli in convex three-manifolds, J. Differential Geom., Volume 106 (2017) no. 1, pp. 139-186 http://projecteuclid.org/euclid.jdg/1493172096 | DOI | MR | Zbl

[37] Osserman, Robert On complete minimal surfaces, Arch. Rational Mech. Anal., Volume 13 (1963), pp. 392-404 | DOI | MR | Zbl

[38] Osserman, Robert Global properties of minimal surfaces in E 3 and E n , Ann. of Math. (2), Volume 80 (1964), pp. 340-364 | DOI | MR | Zbl

[39] Pogorelov, A. V. On the stability of minimal surfaces in Lobachevskiĭ space, Dokl. Akad. Nauk, Volume 354 (1997) no. 6, pp. 742-744 | MR | Zbl

[40] Ros, Antonio; Vergasta, Enaldo Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata, Volume 56 (1995) no. 1, pp. 19-33 | DOI | MR | Zbl

[41] Schoen, Richard; Simon, Leon Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math., Volume 34 (1981) no. 6, pp. 741-797 | DOI | MR | Zbl

[42] Schoen, Richard M. Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom., Volume 18 (1983) no. 4, p. 791-809 (1984) http://projecteuclid.org/euclid.jdg/1214438183 | DOI | MR | Zbl

[43] Sharp, Ben Compactness of minimal hypersurfaces with bounded index, J. Differential Geom., Volume 106 (2017) no. 2, pp. 317-339 | DOI | MR | Zbl

[44] Struwe, M. On a free boundary problem for minimal surfaces, Invent. Math., Volume 75 (1984) no. 3, pp. 547-560 | DOI | MR | Zbl

[45] Tysk, Johan Finiteness of index and total scalar curvature for minimal hypersurfaces, Proc. Amer. Math. Soc., Volume 105 (1989) no. 2, pp. 429-435 | DOI | MR | Zbl

[46] Wang, Goufang Birkhoff minimax principle for minimal surfaces with a free boundary, Math. Ann., Volume 314 (1999) no. 1, pp. 89-107 | DOI | MR | Zbl

[47] White, Brian Which ambient spaces admit isoperimetric inequalities for submanifolds?, J. Differential Geom., Volume 83 (2009) no. 1, pp. 213-228 http://projecteuclid.org/euclid.jdg/1253804356 | DOI | MR | Zbl

[48] White, Brian On the compactness theorem for embedded minimal surfaces in 3-manifolds with locally bounded area and genus, Comm. Anal. Geom., Volume 26 (2018) no. 3, pp. 659-678 | DOI | MR | Zbl

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