The Viscosity Method for Min–Max Free Boundary Minimal Surfaces

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 free boundary case. Namely, given a compact oriented surface $$\Sigma $$ , possibly with boundary, closed ambient Riemannian manifold $$({\mathcal {M}}^m,g)$$ and embedded submanifold $${\mathcal {N}}^n\subset {\mathcal {M}}$$ we study asymptotic behavior of (almost) critical maps $$\Phi for functional $$\begin{aligned} E_\sigma (\Phi ):={\text {area}}(\Phi )+\sigma {\text {length}}(\Phi |_{\partial \Sigma })+\sigma ^4\int _\Sigma |\mathrm {I\!I}^\Phi |^4\,{\text {vol}}_\Phi \end{aligned}$$ on immersions :\Sigma \rightarrow constraint (\partial )\subseteq {N}}$$ as $$\sigma 0$$ assuming an upper bound area suitable entropy condition. As consequence, any collection {F}}$$ subsets space smooth $$(\Sigma ,\partial )\rightarrow ({\mathcal {M}},{\mathcal {N}})$$ be stable under isotopies this space, show that min–max value \inf _{A\in {F}}}\max _{\Phi \in A}{\text is sum areas finitely many branched minimal _{(i)}:\Sigma _{(i)}\rightarrow _{(i)}(\partial _{(i)})\subseteq $$\partial _\nu \Phi _{(i)}\perp T{\mathcal along _{(i)}$$ whose (connected) domains can different from but cannot have more complicated topology. Contrary other frameworks, present one applies in arbitrary codimension. adopt point view which exploits extensively diffeomorphism invariance $$E_\sigma and, way, simplify streamline several arguments initial work (Rivière 2017). Some parts generalize higher-dimensional domains, get integral stationary varifold limit.