Conservation law of harmonic mappings in supercritical dimensions
\textit{T. Rivière} [Invent. Math. 168, No. 1, 1--22 (2007; Zbl 1128.58010)] showed that harmonic maps into closed manifolds, \(H\)-systems, and more generally the Euler-Lagrange equations of a large class of conformally invariant variational functionals for maps into manifolds can be written in the form\N\[\N\Delta u^i = \sum_{j=1}^N \Omega_{ij} \cdot \nabla u^j, \quad i=1,\ldots,N\N\]\Nwhere \(u \in W^{1,2}(\mathbb{B}^n,\mathbb{R}^N)\) and \(\Omega_{ij} = - \Omega_{ji} \in L^2(\mathbb{B}^n,\mathbb{R}^n)\) for \(1 \leq i,j \leq N\).\N\NFurthermore, in two dimensions \(n=2\) and if\N\[\N\|\Omega\|_{L^2(\mathbb{B}^n)} \ll 1\N\]\Nhe showed that one can distort Uhlenbeck's Coulomb Gauge into a invertible map \(A,A^{-1} \in W^{1,2} \cap L^\infty(\mathbb{B}^2,GL(N))\) and \(B \in W^{1,2}(\mathbb{B}^2,\mathbb{R}^{N \times N})\) such that\N\[\N\mathrm{div}(A \nabla u+B \nabla^\perp u) = 0,\N\]\Ni.e. one can write all those equations from above in divergence form which has implications on the regularity and bubbling theory.\N\NThere have been attempts on obtaining similar results in higher dimensions, however they all need ``stronger than natural'' growth conditions on \(\Omega\). For example, Keller proved analogous statements in arbitrary dimensions under the assumption that \(\Omega\) is small in a Besov-Morrey space. The issue is that in higher dimensions there is no good (in the sense of natural) replacement for Wente's inequality which is used crucially in the construction of \(A\) and \(B\).\N\NIn the work at hand, the authors instead assume that that \(\Omega\) is small in the \(L^{n,2}\)-norm, where \(L^{n,2}\) denotes the Lorentz space. The main point is that under these assumptions an analogon of Wente's inequality becomes available.
scientific article; zbMATH DE number 7915251