Orthogonal stance of a minimal surface against its bounding surfaces (Q1824201)

Suppose a minimal surface is spanning a certain nice surface besides the usual boundary curves. Then there arises a question: Under some additional condition, e.g., that the minimal surface considered is stable, does it touch the boundary surfaces orthogonally along the intersecting traces? Many soap film experiments such as those Gergonne observed, seem to have convinced us of the affirmative answer. On the other hand, mathematical descriptions suggesting these circumstances can also be found somewhere; indeed a mean orthogonal intersection in the weak sense is shown by \textit{R. Courant} [Dirichlet's principle, conformal mapping and minimal surfaces (1950; Zbl 0040.346), pp. 207-208]. The present study is concerned with one of the simplest case in this circle of ideas, namely, with an oriented minimal surface S of disk type, which spans partly a given Jordan arc \(\gamma\), and whose remaining boundary arc complementary to \(\gamma\) lies on a sufficiently smooth surface T prescribed. Under these circumstances the minimal surface in question turns out to meet the base surface T orthogonally at almost all points of the intersection arc, which is our main assertion to be proved in the ultimate.