Embeddedness of Minimal Surfaces with Total Boundary Curvature at Most

This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4 must be smoothly embedded. Related results are proved for varifolds and for soap lm surfaces. In a celebrated paper N3] of 1973, Nitsche proved that if ? is an analytic simple closed curve in R 3 with total curvature at most 4, then ? bounds exactly one minimal disk M. Furthermore, that disk is smoothly immersed: it has no branch points, either in the interior or at the boundary. His analysis left open the following questions: (i) Must M in fact be embedded? (ii) If ? bounds other minimal surfaces, must they also be free of branch points, or even be smoothly embedded? In this paper, we show the answer to both questions is \yes", even for curves in R N. Regarding (ii), we give an example of such a ? in R 3 that does indeed bound at least two other minimal surfaces, namely MM obius strips. We conjecture that any such ? can bound at most two MM obius strips, and no surfaces of other topological types. See section 5. Before stating our main result, we review some terminology. The total curvature of a polygonal curve is the sum of the exterior angles at the vertices. For an arbitrary continuous curve, the total curvature is the supremum of the total curvatures of inscribed polygonal curves. This deenition, suggested by Fox, was introduced and analyzed in a paper by Milnor Mi]. In case the curve is piecewise smooth, this deenition agrees with the classical one: the integral of the norm of the curvature vector with respect to arclength plus the sum of the exterior angles at the vertices. Any bounded curve of nite total curvature is rectiiable (i.e., has nite arclength; see x10.1). If is an arclength parametrization of such a curve, then the right and left derivatives T + and T ? of exist and are unit vectors at each interior point 1991 Mathematics Subject Classiication. Primary 53A10, secondary 49F10.