Boundary Regularity and Embedded Solutions for the Oriented Plateau Problem

Robert Hardt, Leon Simon
1979 Annals of Mathematics  
Any fixed C 2 Jordan curve F in R 3 is known to span an orientable minimal surface in several different senses. In the work of Douglas, Rado and Courant (see e.g. [3, IV, §4] ) the minimal surface occurs as an area-minimizing mapping from a fixed orientable surface of finite genus and may possibly have self-intersections. In the work of Fédérer and Fleming (see e.g. [4, §5]) the minimal surface, which occurs as the support of an area-minimizing rectifiable current, is necessarily embedded (away
more » ... from T) but was not previously known even to have finite genus. Our work in [7], which establishes complete boundary regularity for the latter surface, thus implies that there exists an orientable embedded minimal surface with boundary T. In fact: THEOREM 1. For any compact orientable n -1 dimensional C 2 embedded submanifold NofR n + l , there exists an orientable bounded stable minimal embedded C 1 i0i hypersurface with boundary for all 0 < a 2, C k,0L
doi:10.2307/1971233 fatcat:26bttxrx5zbnddpeai2zitj6ci