THE ENVELOPE OF HOLOMORPHY OF A TWO-MANIFOLD IN C2 61
of a fc-manifold in C", k>ιι, with a one-higher dimensional envelope
of holomorphy. For real codimension 2, all such embeddings have this
property. One could conjecture that:
a) All compact submanifolds of C", of real dimension > n, have an
envelope of holomorphy of at least one higher dimension.
b) All compactsubmanifoldsofC",ofrealdimension «,have an envelope
of holomorphy of at least one higher dimension, provided that the mani-
folds are not totally real.
It is possible that the results mentioned in Remark No. 3 will be applicable
in proving a) for five-dimensional submanifolds of C4.
Added in Proof: S. Greenfield has recently given an affirmative an-
swer to the question in Remark 2.
REFERENCES
[7] Bishop, E., Differentiable manifolds in complex Euclidean space,
Duke Math. J. 32 (1965), 1-22.
[2] Bochner, S. and W. T. Martin, Functions of Several Complex
Variables, Princeton University Press, Princeton, N. J. (1948).
[3] Chern, S. S. and E. Spanier, A theorem on orientable surfaces in
four dimensional space, Comm. Math. HeIv. 25 (1951), 205-209.
[4] Freeman, M., Local holomorphic convexity of a two-manifold in C2
(Complex Analysis 1969), Rice Univ. Studies 56, No. 2 (1970),
pp. 171-180.
[5] Greenfield, S. J., Cauchy-Riemann equations in several variables,
Ann. Scuola Norm. Supp. Pisa 22 (1968), 275-314.
[6‘] Harvey, R. and R. O. Wells, Jr., Compact holomorphically convex
subsets of a Stein manifold, Trans. Amer. Math. Soc. 136 (1969),
509-516.
[7] Hunt, L. R., The local envelope of holomorphy of an «-manifold in
C (to appear).
[<S] Levine, H., Singularities of differentiable mappings I, Math. Inst, der
Univ., Bonn (1960), 1-33.
[9] Milnor, J., Morse Theory, Ann. Math. Studies 51, Princeton, N. J.
(1963).
[70] Nirenberg, R. and R. O. Wells, Jr., Approximation theorems on
differentiable submanifolds of a complex manifold, Trans. Amer.
Math. Soc. 42 (1969), 15-36.
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