Chern - A Great Geometer of the Twentieth Century by Shing-Tung Yau (Chief Editor)

By Shing-Tung Yau (Chief Editor)

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The conclusion of the decomposition theorem is that the behavior of Dt is typical of stable hypersurfaces near a cone. (2) I am grateful to K. Steffen for the following pretty example which shows the subtlety of the stability 43 hypothesis in the decomposition theorem. We exhibit a sequence of regular imbedded two dimensional minimal submanifolds of ~3 which converges (in the weak topology on varifolds) to a surface which is the union of two orthogonal planes. In figure 11, we have drawn a portion of the graph of Scherk's second minimal surface of infinite genus, denoted sl .

J = 1,2, •... is the sum of two oppositely oriented homologically area minimizing circles on the torus V Since the varifold = lim T. (as varifolds), the j-"' surface M . • also converges to V w(3-l - 3-l-j) (as varifolds), but no current in this sequence is locally area minimizing in M. 1(4). The critical surface, we recall, is a figure eight, which we shall denote by V . Let T1 , T2 , •.. be a sequence of cycles in the critical path approaching V from the left, as illustrated in 30 figure 7a .

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}.

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