The Viscosity Method for Min–Max Free Boundary Minimal Surfaces

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 (Mm, g) and a closed embedded submanifold Nn⊂ M, we study the asymptotic behavior of (almost) critical maps Φ for the functional Eσ(Φ):=area(Φ)+σlength(Φ|∂Σ)+σ4∫Σ|IIΦ|4volΦon 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 infA∈FmaxΦ∈Aarea(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 ∂νΦ (i)⊥ TN 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 Eσ 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.