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 $$\Sigma $$ , possibly with boundary, a closed ambient Riemannian manifold $$({\mathcal {M}}^m,g)$$ and a closed embedded submanifold $${\mathcal {N}}^n\subset {\mathcal {M}}$$ , we study the asymptotic behavior of (almost) critical maps $$\Phi $$ for the 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 $$\Phi :\Sigma \rightarrow {\mathcal {M}}$$ with the constraint $$\Phi (\partial \Sigma )\subseteq {\mathcal {N}}$$ , as $$\sigma \rightarrow 0$$ , assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection $${\mathcal {F}}$$ of compact subsets of the space of smooth immersions $$(\Sigma ,\partial \Sigma )\rightarrow ({\mathcal {M}},{\mathcal {N}})$$ , assuming $${\mathcal {F}}$$ to be stable under isotopies of this space, we show that the min–max value $$\begin{aligned} \inf _{A\in {\mathcal {F}}}\max _{\Phi \in A}{\text {area}}(\Phi ) \end{aligned}$$ is the sum of the areas of finitely many branched minimal immersions $$\Phi _{(i)}:\Sigma _{(i)}\rightarrow {\mathcal {M}}$$ with $$\Phi _{(i)}(\partial \Sigma _{(i)})\subseteq {\mathcal {N}}$$ and $$\partial _\nu \Phi _{(i)}\perp T{\mathcal {N}}$$ along $$\partial \Sigma _{(i)}$$ , whose (connected) domains $$\Sigma _{(i)}$$ can be different from $$\Sigma $$ 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 $$E_\sigma $$ 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.