By Jon T. Pitts

Mathematical No/ex, 27

Originally released in 1981.

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Y~) 45 z y x Figure 12. sin rz The surface Sr (r large) (sinh rx)(sinh ry) 46 Tompkins [MT] and Shiffrnan [SM] on the existence of unstable minimal surfaces which span curves bounding two locally area minimizing surfaces. 14 (or theorem Fin the introduction) with and (1) above, suppose circles Bt C k = 2 consists of the two parallel oppositely oriented. If t is sufficiently small, so that the circles are close, and if we choose T1 and to be the two parallel disks T2 Dt to be the area minimizing catenoid spanning and corresponding to Ut ).

LOa. The map Fig. lOb. ~ from The map p t 0 at t to 2/5 t 1/5 39 Fig. lOc. The map ~ at t 1/2 Fig. lOd. The map ~ at t 3/5 40 Fig. lOe. The map I/! at t 4/5 1 4/5 Fig. lOf. The map l/J from t 4/5 to t 1 41 1;3 STABLE MANIFOLDS. (1) Much of our regularity theory depends on a careful study of stable manifolds. 3, or theorem C in the introduction). Here is an example in ~3 to illustrate this theorem. For each positive number Bt For any t = f (x,y,z) t : x 2 , define + y 2 = B t lzl 1 : = t}. , there is always one minimal surface spanning Bt ; namely, the surface Dt consisting of two parallel disks, Dt = [ (x,y,z ) Moreover, for all t : x z+ Y2 < 1, Bt , both catenoids: [(x,y,z) Here t}.

To t 3/4 = Every curve l/>(t) from t oscillates enough to have length exactly, but most are not minimal. = 1/4 2" Thus it is generally not enough to select any critical surface, because it may not be minimal. (We have called this a technical difficulty because it can be eliminated; one can guarantee that every surface at the critical level is minimal (cf. ) The generic difficulty appears in the next example. Here M= over M ~2 • Let for which cp be any critical path of 1-cycles cp(l/2) is a great circle.