On the regularity of minimal surfaces with free boundaries in Riemannian manifolds
The author proves regularity for stationary minimal surfaces with a free boundary in a Riemannian manifold by the way extending earlier results obtained for example by \textit{M. Grüter}, \textit{S. Hildebrandt} and \textit{J. C. Nitsche} [Manuscr. Math. 35, 387-410 (1981; Zbl 0483.49037)] for supporting surfaces \(\Sigma\) located in Euclidean space \({\mathbb{R}}^ 3\). Now \({\mathbb{R}}^ 3\) is replaced by an n-dimensional Riemannian manifold X with bounded sectional curvature and with injectivity radius bounded away from zero, and \(\Sigma\) denotes a \(C^ 2\)-hypersurface in X with bounded second fundamental form. Moreover, \(\Sigma\) has a uniform tubular neighborhood on which a smooth nearest point retraction is defined. In this case a stationary solution of the minimal surface problem with free boundary \(\Sigma\) is of class \(C^{1,\alpha}\) up to the boundary. The proof of this theorem uses methods developed by \textit{M. Grüter} and \textit{J. Jost} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, No.1, 129-169 (1986)] and exploits geometric constructions from \textit{J. Jost} and \textit{H. Karcher} [Manuscr. Math. 40, 27-77 (1982; Zbl 0502.53036)].
scientific article; zbMATH DE number 3969498