极小奇异曲面的局部单值化与自由边界正则性
Local uniformization and free boundary regularity of minimal singular surfaces
摘要：
为了构造具有奇异性的极小曲面，我们继续研究了二维黎曼单纯复形的调和映射，得到了关于自由边界曲线实解析性的先验正则性结果.自由边界是三个圆盘型极小曲面相交的奇异集。这里的奇异极小曲面的配置是通过一个加权能量泛函的最小化，在精神上的道格拉斯的方法高原问题。利用调和映射的自由边界正则性，构造了奇异曲面的一个局部单值化，作为自由边界上一点邻域的奇异切锥参数化.此外，讨论了局部单值化在极小曲面实解析延拓中的应用。
英文摘要：
In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free boundary curve. The free boundary is the singular set along which three disk-type minimal surfaces meet. Here the configuration of the singular minimal surface is obtained by a minimization of a weighted energy functional, in the spirit of J.Douglas' approach to the Plateau Problem. Using the free boundary regularity of the harmonic map, we construct a local uniformization of the singular surface as a parameterization of a neighborhood of a point on the free boundary by the singular tangent cone. In addition, applications of the local uniformization are discussed in relation to H.Lewy's real analytic extension of minimal surfaces.
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