The regularity of conformal target harmonic maps (Q2406081)

Let \(\Sigma\) be a smooth closed surface, and let \(N^n\) be a closed oriented submanifold of an Euclidean space \({\mathbb R}^m\). For an immersion \(\vec{\Phi} : \Sigma \to N\) and a small parameter \(\sigma\), consider \[ A^\sigma(\vec{\Phi}) = \text{Area}(\vec{\Phi}) + \sigma^2 \int_\Sigma \left[1+ |\text{II}_{\vec{\Phi}}|^2\right]^p\, dv_{g_{\vec{\Phi}}}, \] where \(\text{II}_{\vec{\Phi}}\) is the second fundamental form of the immersion \(\vec{\Phi}\) and \(dv_{g_{\vec{\Phi}}}\) is the volume form associated to the induced metric. For \(p>1\) and \(\sigma>0\), by applying the Palais-Smale deformation theory, one can produce critical points \(\vec{\Phi}_\sigma\) to \(A^\sigma\). It is proved in [\textit{A. Michelat} and \textit{T. Rivière}, ESAIM, Control Optim. Calc. Var. 22, No. 4, 1282--1324 (2016; Zbl 1353.49006)] that, for a sequence of parameters \(\sigma_k \to 0\), the sequence of integer rectifiable varifold associated to the immersion of \(\Sigma\) by \(\vec{\Phi}_{\sigma_k}\) does not necessarily converge to a stationary integer rectifiable varifold. However, assuming an additional entropy estimate, the author proved in [Publ. Math., Inst. Hautes Étud. Sci. 126, 177--246 (2017; Zbl 1387.53084)] that the immersion of \(\Sigma\) by \({\vec{\Phi}_{\sigma_k}}\) converges to a stationary integer rectifiable varifold given by the image of a smooth Riemann surface \(S\) by a weakly conformal \(W^{1,2}\) map \(\vec{\Phi}\) into \(N\) equipped by an integer multiplicity. In this paper, the author proves that if this multiplicity is constant, such a map is smooth and satisfies the harmonic map equation. In other words, any weakly conformal \(W^{1,2}\) map from a Riemann surface \(S\) into a closed oriented submanifold \(N\) of an Euclidean space \({\mathbb R}^m\) realizes, for almost every domain, a stationary varifold if and only if it is a smooth conformal harmonic map from \(S\) into \(N\).