Abstract
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.
Highlights
Abstract. — 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
Let us denote by Π1 the closed half-space bounded by Π that contains the interior of Σ and by Π2 the other closed half-space bounded by Π
We argue by contradiction as follows
Summary
Let Σn ⊂ Rn+1 be a half-bubble in the sense above. Let us denote by Π1 the closed half-space bounded by Π that contains the interior of Σ and by Π2 the other closed half-space bounded by Π. — In the setting above, we define index(Σ) to be the largest dimension of a linear subspace of Cc∞(Σ) where Q|A|2 is negative definite. — A two-dimensional half-bubble has finite Morse index and this equals IndE (Q|A|2 ) where Σis the double of Σ. — A two-dimensional half-bubble has finite Morse index with Dirichlet boundary conditions and this equals IndO(Q|A|2 ), where Σis the double of Σ. A negative result would follow by proving that a half-bubble of Morse index 2 has vanishing Morse index with Dirichlet boundary conditions. We can extend to half-bubbles some further classification results that are wellknown for complete minimal surfaces in R3 This discussion parallels the one presented in the previous subsection, which was based on index-theoretic criteria instead. The conclusion comes at once by considering all possible planes of symmetry of a catenoid
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