On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary

Abstract

Geometrically speaking these conditions express the fact that x=r(w) represents a surface in R 3 of constant mean curvature H, having finite area A(t) and an assigned boundary F*. Under various conditions on F* and H existence proofs for such surfaces have been given in the literature ([1], [5], [6], [8], [9], and [10]; see also [7] for a complete discussion of the non-parametric case). If one considers the totality of all boundary curves F* lying in the unit ball I~[ < 1, the sharpest result hitherto obtained is due to HILDEBRANDT [6], who proved that in this case the class ~ (F* , H) is nonempty, provided that [HI < 1. On the other hand, from geometric considerations it is very plausible that for a given curve F* the surfaces x =z(w) should cease to exist, if [HI exceeds a certain critical number c(F*). The purpose of this note is to substantiate this fact by estimating the quantity c(F*) for a class of boundary curves F*. Introduce the numbers