Bubbling analysis and geometric convergence results for free boundary minimal surfaces

Lucas Ambrozio; Reto Buzano; Alessandro Carlotto; Ben Sharp

doi:10.5802/jep.102

Bubbling analysis and geometric convergence results for free boundary minimal surfaces
[Analyse des bulles et résultats de convergence géométrique pour des surfaces minimales à bord libre]

Lucas Ambrozio ¹ ; Reto Buzano ² ; Alessandro Carlotto ³ ; Ben Sharp 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

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 621-664.

Résumé
Abstract

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 $ℝ^{3}$ 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.

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 $ℝ^{3}$ unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified.

Reçu le : 2019-02-24
Accepté le : 2019-08-04
Publié le : 2019-09-24

Zbl

DOI : 10.5802/jep.102

Classification : 53A10, 53C42, 49Q05
Mots clés : Surfaces minimales à bord libre, analyse des bulles, quantification, compacité géométrique

Affiliations des auteurs :

Lucas Ambrozio ¹ ; Reto Buzano ² ; Alessandro Carlotto ³ ; Ben Sharp ⁴

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Bubbling analysis and geometric convergence results for free boundary minimal surfaces},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {621--664},
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     zbl = {07114035},
     language = {en},
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Lucas Ambrozio; Reto Buzano; Alessandro Carlotto; Ben Sharp. Bubbling analysis and geometric convergence results for free boundary minimal surfaces. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 621-664. doi : 10.5802/jep.102. https://jep.centre-mersenne.org/articles/10.5802/jep.102/

Bibliographie
Cité par

[1] L. Ambrozio, R. Buzano, A. Carlotto & B. Sharp - “Geometric convergence results for closed minimal surfaces via bubbling analysis”, 2018 | arXiv | Zbl

[2] L. Ambrozio, A. Carlotto & B. Sharp - “Compactness of the space of minimal hypersurfaces with bounded volume and $p$ -th Jacobi eigenvalue”, J. Geom. Anal. 26 (2016) no. 4, p. 2591-2601 | DOI | MR | Zbl

[3] L. Ambrozio, A. Carlotto & B. Sharp - “Compactness analysis for free boundary minimal hypersurfaces”, Calc. Var. Partial Differential Equations 57 (2018) no. 1, article ID 22, 39 pages | DOI | MR | Zbl

[4] L. Ambrozio, A. Carlotto & B. Sharp - “Index estimates for free boundary minimal hypersurfaces”, Math. Ann. 370 (2018) no. 3-4, p. 1063-1078 | DOI | MR | Zbl

[5] R. Buzano & B. Sharp - “Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area”, Trans. Amer. Math. Soc. 370 (2018) no. 6, p. 4373-4399 | DOI | MR | Zbl

[6] M. do Carmo & C. K. Peng - “Stable complete minimal surfaces in $R^{3}$ are planes”, Bull. Amer. Math. Soc. (N.S.) 1 (1979) no. 6, p. 903-906 | DOI | MR | Zbl

[7] J. Chen, A. Fraser & C. Pang - “Minimal immersions of compact bordered Riemann surfaces with free boundary”, Trans. Amer. Math. Soc. 367 (2015) no. 4, p. 2487-2507 | DOI | MR | Zbl

[8] S. Y. Cheng & J. Tysk - “Schrödinger operators and index bounds for minimal submanifolds”, Rocky Mountain J. Math. 24 (1994) no. 3, p. 977-996 | DOI | Zbl

[9] O. Chodosh & D. Maximo - “On the topology and index of minimal surfaces”, J. Differential Geom. 104 (2016) no. 3, p. 399-418 | DOI | MR | Zbl

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

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

[12] C. De Lellis & J. Ramic - “Min-max theory for minimal hypersurfaces with boundary”, Ann. Inst. Fourier (Grenoble) 68 (2018) no. 5, p. 1909-1986 | DOI | MR | Zbl

[13] N. Ejiri & M. Micallef - “Comparison between second variation of area and second variation of energy of a minimal surface”, Adv. Calc. Var. 1 (2008) no. 3, p. 223-239 | DOI | MR | Zbl

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

[15] D. Fischer-Colbrie & R. Schoen - “The structure of complete stable minimal surfaces in $3$ -manifolds of nonnegative scalar curvature”, Comm. Pure Appl. Math. 33 (1980) no. 2, p. 199-211 | DOI | MR | Zbl

[16] A. Folha, F. Pacard & T. Zolotareva - “Free boundary minimal surfaces in the unit 3-ball”, Manuscripta Math. 154 (2017) no. 3-4, p. 359-409 | DOI | MR | Zbl

[17] A. Fraser & 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, p. 183-200 | DOI | MR | Zbl

[18] A. Fraser & R. Schoen - “Sharp eigenvalue bounds and minimal surfaces in the ball”, Invent. Math. 203 (2016) no. 3, p. 823-890 | DOI | MR | Zbl

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

[20] B. Freidin, M. Gulian & P. McGrath - “Free boundary minimal surfaces in the unit ball with low cohomogeneity”, Proc. Amer. Math. Soc. 145 (2017) no. 4, p. 1671-1683 | DOI | MR | Zbl

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

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

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

[24] L. P. Jorge & W. H. Meeks - “The topology of complete minimal surfaces of finite total Gaussian curvature”, Topology 22 (1983) no. 2, p. 203-221 | DOI | MR | Zbl

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

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

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

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

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

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

[31] M. M.-c. Li - “A general existence theorem for embedded minimal surfaces with free boundary”, Comm. Pure Appl. Math. 68 (2015) no. 2, p. 286-331 | DOI | MR | Zbl

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

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

[34] F. J. López & A. Ros - “Complete minimal surfaces with index one and stable constant mean curvature surfaces”, Comment. Math. Helv. 64 (1989) no. 1, p. 34-43 | DOI | MR | Zbl

[35] F. J. López & A. Ros - “On embedded complete minimal surfaces of genus zero”, J. Differential Geom. 33 (1991) no. 1, p. 293-300 | DOI | MR | Zbl

[36] D. Maximo, I. Nunes & G. Smith - “Free boundary minimal annuli in convex three-manifolds”, J. Differential Geom. 106 (2017) no. 1, p. 139-186 | DOI | MR | Zbl

[37] R. Osserman - “On complete minimal surfaces”, Arch. Rational Mech. Anal. 13 (1963), p. 392-404 | DOI | MR | Zbl

[38] R. Osserman - “Global properties of minimal surfaces in $E^{3}$ and $E^{n}$ ”, Ann. of Math. (2) 80 (1964), p. 340-364 | DOI | MR | Zbl

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

[40] A. Ros & E. Vergasta - “Stability for hypersurfaces of constant mean curvature with free boundary”, Geom. Dedicata 56 (1995) no. 1, p. 19-33 | DOI | MR | Zbl

[41] R. Schoen & L. Simon - “Regularity of stable minimal hypersurfaces”, Comm. Pure Appl. Math. 34 (1981) no. 6, p. 741-797 | DOI | MR | Zbl

[42] R. M. Schoen - “Uniqueness, symmetry, and embeddedness of minimal surfaces”, J. Differential Geom. 18 (1983) no. 4, p. 791-809 (1984) | DOI | MR | Zbl

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

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

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

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

[47] B. White - “Which ambient spaces admit isoperimetric inequalities for submanifolds?”, J. Differential Geom. 83 (2009) no. 1, p. 213-228 | DOI | MR | Zbl

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

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