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Hulls of deformations in ${\bf C}\sp{n}$

H. Alexander

1981
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Transactions of the American Mathematical Society
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A problem of E. Bishop on the polynomially convex hulls of deformations of the torus is considered. Let the torus T2 be the distinguished boundary of the unit polydisc in C2. If t1-» T2 is a smooth deformation of T2 in C2 and g0 is an analytic disc in C2 with boundary in T2, a smooth family of analytic discs t h» g, is constructed with the property that the boundary of g, lies in T2. This construction has implications for the polynomially convex hulls of the tori T2. An analogous problem for a
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... -sphere in C2 is also considered. Introduction. By a result of A. Browder [3], a compact real orientable manifold X of dimension > n in C" is never polynomially convex. Beginning with the work of E. Bishop [1], a great deal of effort has gone into giving a constructive explanation of this phenomenon. Bishop showed that there exist analytic discs in C with boundaries in X near certain "exceptional" points of X. His method was amplified by a number of others; a report on these developments is given in [6] . We shall consider a related problem which has been attributed to E. Bishop in [2, p. 234, Problem 17]. Let {F,} be a deformation of the torus T2 in C2; i.e., F0 = identity and {F,} is a family of diffeomorphisms of T2 into C2 which vary smoothly with t. Put T2 = Ft(T2); T2 is a torus in C2 which is "close" to T2 for small /. By Browder's result T2 is not polynomially convex. Also, every point of the closed unit polydisc U2, the polynomially convex hull of F" = T2, hes on some analytic disc g0 with boundary in T2. The problem is to show that associated to the smooth deformation {F,} there exists a smooth family of analytic discs {g,} such that the boundary of g, lies in T2. We shall construct such a family {g,}, at least for t sufficiently small. This will account for the nonpolynomial convexity of T2 in the most direct manner. As T2 is deformed to T2, its hull U2 is deformed to a set D, which is close to U2 ( = D0) and which is related to T2 in a way that parallels the relationship of U2 to T2; namely, D, has a topological boundary composed of analytic discs whose own boundaries he in T2 and thus D, has T2 as a distinguished boundary. Eric Bedford [7] has proved that D, is in fact the polynomially convex hull of T2. We shall also obtain an analogous result for a particular 2-sphere in C2: S2 = {(zx,z2) E C2: lmz2 = 0, \zx\2 + (Rez2)2= l}.

doi:10.1090/s0002-9947-1981-0613794-x
fatcat:rzvnxw4ng5ailmwdiujiueno5m