A Lipschitz decomposition of minimal surfaces
A Jordan curve \(\Gamma\) in \(\mathbb{R}^ n\) satisfies an \(M\)-cord-arc condition if the length \(l(z,w)\) of the shorter subarc of \(\Gamma\) joining the points \(z\), \(w\in\Gamma\) is bounded by \(M| z-w|\). The authors prove the following decomposition theorem for classical minimal surfaces (i.e. conformal, branched, minimal immersions) \(F:D\to\mathbb{R}^ n\), \(D\) the unit disc: there is a universal constant \(M\) such that whenever \(F\) is a minimal surface with boundary a rectifiable Jordan curve \(\Gamma\), then there exists a locally finite partition of \(D\) into non-overlapping simply connected subdomains \(D_ j\) such that (i) \(F\) is an embedding on \(\overline D_ j\), (ii) \(F(\partial D_ j)\) is an \(M\)- chord-arc curve, (iii) \(\sum l(F(\partial D_ j))\leq Ml(\Gamma)\).
scientific article; zbMATH DE number 120217