An area-minimizing surface (rectifiable current) bounded by a smooth curve in 
is a smooth submanifold with boundary.
Regularity Theorem
See also
Minimal Surface, Rectifiable CurrentExplore with Wolfram|Alpha
References
Morgan, F. "What is a Surface?" Amer. Math. Monthly 103, 369-376, 1996.Referenced on Wolfram|Alpha
Regularity TheoremCite this as:
Weisstein, Eric W. "Regularity Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularityTheorem.html