A BIFURCATIOI \ PROCESS

The discussion to follow will deal with aspects of Plateau's problem about which no exhaustive information is available today, and particular attention will be paid to the phenomenon of non-uniqueness. It is well known that the area of a minimal surface, its suggestive name notwith-standing, need not furnish a minimum (absolute or relative) among the areas of all surfaces having the same boundary. Let us consider a minimal surface S: {t:t(u,a)iu2*a2<R2} which lies imbedded in Euclidean 3-space. We shall assume that u and a ate isothermal parameters on S so that t?*(u,u) : r?(u,a) E(u,a)> O and q(u,o)t(u,a) : I(u,a) : 0. Let us further denote by ff(u,a) the unit normal vector of our surface and by K(u, a) its Gaussian curvature. On B we consider the one-parameter family of Jordan currreg T, : {r, : r(rcos0, rsind); 0 <8{2n}, 0<r<R, which bound expanding portion