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In continuing the study of harmonic mapping from 2-dimensional Riemanniansimplicial complexes in order to construct minimal surfaces with singularity,we obtain an a-priori regularity result concerning the real analyticity of thefree boundary curve. The free boundary is the singular set along which threedisk-type minimal surfaces meet. Here the configuration of the singular minimalsurface is obtained by a minimization of a weighted energy functional, in thespirit of J.Douglas' approach to the Plateau Problem. Using the free boundaryregularity of the harmonic map, we construct a local uniformization of thesingular surface as a parameterization of a neighborhood of a point on the freeboundary by the singular tangent cone. In addition, applications of the localuniformization are discussed in relation to H.Lewy's real analytic extension ofminimal surfaces.

Local uniformization and free boundary regularity of minimal singular surfaces