Bubbling analysis and geometric convergence results for free boundary minimal surfaces

Ambrozio, Lucas; Buzano, Reto; Carlotto, Alessandro; Sharp, Ben

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

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.

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

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