On the nonexistence of boundary branch points for minimal surfaces spanning smooth contours II

Abstract

This is the second in a series of two papers discussing the elementary but beautiful and fundamental question (open for some eighty years) of whether or not a minimal surface spanning a sufficiently smooth curve, which is a local minimizer, is immersed up to and including the boundary. We show that C k minimizers of energy or area cannot have nonexceptional boundary branch points.Mathematics Subject Classification (2010). 49Q05, 53A10, 58E12. Keywords. Plateau's problem, branch points.We investigate the question of whether a relative minimum (in any C r topology, r ≥ 0) of Dirichlet's energy on area in the Classical Plateau problem involving sufficiently smooth contours Γ can have a boundary branch point. In [2] the authors defined the integer index m > 0 of a branch point. It is shown that if the curvature and torsion of Γ are both nonzero, then for smooth curves there is an a priori estimate 1 + m ≤ n(n + 1), where n is the order of the branch point. In the case Γ is analytic, m is always finite, independent of the assumptions on the curvature and torsion. The cases 1 + m = 2(n + 1) and 1 + m = 3(n + 1) are called exceptional. The notion of exceptional is the infinitesimal notion of analytically false [2].If a minimal surfaceX has a boundary branch point of order n and index m satisfying 2m − 2 < 3n, then Wienholtz [18] showed thatX cannot be a minimizer in any C r topology. Wienholtz's condition implies that the branch point is nonexceptional. In [14] we handled the case 3n ≤ 2m−2 ≤ 5n excluding m + 1 = 2(n + 1) and n = 2, m = 4. Here we consider all the remaining cases including the exceptional ones. We provide an argument that the exceptional cases are, in fact, undecidable. In the nonexceptional cases we show (with one exception) that some first nonvanishing derivative of Dirichlet's energy can be made negative. This paper, [15] and [18] are the first time Journal of Fixed Point Theory and Applications 254A. J. Tromba JFPTA that higher-order derivatives of global energy have been considered. Reducing energy implies that area can also be reduced. Since derivatives higher than the third are nonintrinsic, the method is to develop an algorithm for finding curves along which the calculation of higher-order derivatives reduces to Cauchy's integral theorem. This method works to rule out exceptional interior branch points for minima, but it completely breaks down in the nonanalytic case for exceptional boundary branch points. In the latter case, the algorithm leads to a curve along which all derivatives of energy are zero.