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- Title
The Viscosity Method for Min–Max Free Boundary Minimal Surfaces.
- Authors
Pigati, Alessandro
- Abstract
We adapt the viscosity method introduced by Rivière (Publ Math Inst Hautes Études Sci 126:177–246, 2017. https://doi.org/10.1007/s10240-017-0094-z) to the free boundary case. Namely, given a compact oriented surface Σ , possibly with boundary, a closed ambient Riemannian manifold (M m , g) and a closed embedded submanifold N n ⊂ M , we study the asymptotic behavior of (almost) critical maps Φ for the functional E σ (Φ) : = area (Φ) + σ length (Φ | ∂ Σ) + σ 4 ∫ Σ | I I Φ | 4 vol Φ on immersions Φ : Σ → M with the constraint Φ (∂ Σ) ⊆ N , as σ → 0 , assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection F of compact subsets of the space of smooth immersions (Σ , ∂ Σ) → (M , N) , assuming F to be stable under isotopies of this space, we show that the min–max value inf A ∈ F max Φ ∈ A area (Φ) is the sum of the areas of finitely many branched minimal immersions Φ (i) : Σ (i) → M with Φ (i) (∂ Σ (i)) ⊆ N and ∂ ν Φ (i) ⊥ T N along ∂ Σ (i) , whose (connected) domains Σ (i) can be different from Σ but cannot have a more complicated topology. Contrary to other min–max frameworks, the present one applies in an arbitrary codimension. We adopt a point of view which exploits extensively the diffeomorphism invariance of E σ and, along the way, we simplify and streamline several arguments from the initial work (Rivière 2017). Some parts generalize to closed higher-dimensional domains, for which we get an integral stationary varifold in the limit.
- Subjects
VISCOSITY; IMMERSIONS (Mathematics); MINIMAL surfaces; RIEMANNIAN manifolds; COMPACT spaces (Topology); TOPOLOGY; MATHEMATICS
- Publication
Archive for Rational Mechanics & Analysis, 2022, Vol 244, Issue 2, p391
- ISSN
0003-9527
- Publication type
Article
- DOI
10.1007/s00205-022-01761-9