The aim of this work is to present a direct min-max approach for constructing minimal surfaces in a given closed submanifold \(N^n\) of \(\mathbb{R}^m\). Let \(\mathcal{E}_{\Sigma, p}\) be the space of \(W^{2,2p}\)-immersions \(\vec{\Phi}\) of a given closed surface \(\Sigma\) for \(p>1\) into \(N^n\subset \mathbb{R}^m\). For such immersions consider the relaxed energy \[ \mathrm A^\sigma(\vec{\Phi}):=\;\mathrm{Area}(\vec{\Phi})+\sigma^2\int_\Sigma[1+|\vec{\mathbf{I}}_{\vec{\Phi}}|^2]^p \mathrm{dvol}_{g_{\vec{\Phi}}}, \] where \(g_{\vec{\Phi}}\) and \(\vec{\mathbf{I}}_{\vec{\Phi}}\) are respectively the first and second fundamental forms of \(\vec{\Phi}\) in \(N^n\). The energy A\(^\sigma\) is intrinsic in the sense that it is invariant under re-parametrization of \(\vec{\Phi}\), and for a fixed \(\sigma\neq 0\) the Lagrangian A\(^\sigma\) satisfies the Palais-Smale condition. The main results of this paper are the following theorems.
Theorem I.1. Let \(N^n\) be a closed \(n\)-dimensional submanifold of \(\mathbb{R}^m\) with \(3\leq n\leq m-1\). Let \(\Sigma\) be an arbitrary closed Riemannian 2-dimensional manifold. Let \(\sigma_k\to 0\) and let \(\vec{\Phi}_k\) be a sequence of critical points of the energy \(\mathrm{A}^{\sigma_k}(\vec{\Phi})\) in the space of \(W^{2,2p}\)-immersions \(\vec{\Phi}\) of \(\Sigma\) and satisfying the entropy condition \[ \sigma^2_k\int_\Sigma[1+|\vec{\mathbf{I}}_{\vec{\Phi}_k}|^2]^p \mathrm{dvol}_{g_{\vec{\Phi}_k}}=o\left(\frac{1}{\log\sigma_k^{-1}}\right). \] Then, modulo extraction of a subsequence, there exists a closed Riemann surface \((S, h_0)\) with \(\mathrm{genus}(S)\leq \mathrm{genus}(\Sigma)\) and a conformal integer target harmonic map \((\vec{\Phi}_\infty, N)\) from \(S\) into \(N^n\) such that \[ \lim_{k\to+\infty} \mathrm{A}^{\sigma_k}(\vec{\Phi}_k)=\frac 12\int_S N|d \vec{\Phi}_\infty|^2_{h_0}\mathrm{dvol}_{h_0}. \] Moreover, the oriented varifold associated to \(\vec{\Phi}_k\) converges in the sense of Radon measures towards the stationary integer varifold associated to \((\vec{\Phi}_\infty, N)\).
According to the main results of the paper [the author, Calc. Var. Partial Differ. Equ. 56, No. 4, Paper No. 117, 15 p. (2017; Zbl 1380.58013)], the limit \((\vec{\Phi}_\infty, N)\) is a smooth minimal branched immersion equipped with a smooth integer valued multiplicity.
Theorem I.2. Let \(\mathcal{A}\) be an admissible family in the space of \(W^{2,2p}\)-immersions of \(\Sigma\) into a closed submanifold \(N^n\) of \(\mathbb{R}^m\). Assume \[ \inf_{A\in\mathcal{A}}\max_{\vec{\Phi}\in A}\mathrm{Area}(\vec{\Phi})=\beta^0>0, \] then there exists a closed Riemann surface \((S, h_0)\) with \(\mathrm{genus}(S)\leq \mathrm{genus}(\Sigma)\) and a conformal integer target harmonic map \((\vec{\Phi}_\infty, N)\) from \(S\) into \(N^n\) such that \[ \frac 12\int_S N|d \vec{\Phi}_\infty|^2_{h_0}\mathrm{dvol}_{h_0}=\beta^0. \]