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The viscosity method for min-max free boundary minimal surfaces

We adapt the viscosity method introduced by Rivière to the free boundary case. Namely, given a compact oriented surface $Σ$, 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 $Φ$ for the functional \begin{align*} &E_σ(Φ):=\operatorname{area}(Φ)+σ\operatorname{length}(Φ|_{\partialΣ})+σ^4\int_Σ|{\mathrm {I\!I}}^Φ|^4\,\operatorname{vol}_Φ\end{align*} on immersions $Φ:Σ\to\mathcal{M}$ with the constraint $Φ(\partialΣ)\subseteq\mathcal{N}$, as $σ\to 0$, 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 $(Σ,\partialΣ)\to(\mathcal{M},\mathcal{N})$, assuming $\mathcal{F}$ to be stable under isotopies of this space we show that the min-max value \begin{align*} &β:=\inf_{A\in\mathcal{F}}\max_{Φ\in A}\operatorname{area}(Φ) \end{align*} is the sum of the areas of finitely many branched minimal immersions $Φ_{(i)}:Σ_{(i)}\to\mathcal{M}$ with $\partial_νΦ_{(i)}\perp T\mathcal{N}$ along $\partialΣ_{(i)}$, whose (connected) domains $Σ_{(i)}$ can be different from $Σ$ but cannot have a more complicated topology. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_σ$ and, along the way, we simplify several arguments from the original work. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.