Boundary Values and Mapping Degree - Project Euclid

Michigan Math. J. 47 (2000)

Boundary Values and Mapping Degree E d ga r L e e St o u t

Introduction This note is an addendum to the paper of Alexander and Wermer [2], in which the authors relate the theory of linking numbers to the question of finding an analytic variety bounded by a given real, odd-dimensional submanifold of C N. We give a characterization of the boundary values of holomorphic functions on certain domains in C N in similar terms. In fact, the work of Alexander and Wermer contains such a characterization in the case of functions of class C 1. It seems that the methods used in [2] require this degree of smoothness, but we have found that it is possible to obtain a result that characterizes the continuous functions that are boundary values of holomorphic functions that is entirely in the spirit of [2]. Specifically, we shall prove the following result. Main Theorem. Let be a bounded domain in C N with boundary of class C 2 , ¯ has a Stein neighborhood basis. A continuous function f on and assume that ¯ that is holomorphic on if and b is of the form F b for a function F ∈ C () only if the following condition is met. (∗) With 0f the graph {(z, f (z)) : z ∈ b}, a compact subset of C N +1, if Q is ¯ × C with a C N -valued holomorphic map defined on a neighborhood of −1 N Q (0) ∩ 0f = ∅, then the degree of the map b → C \ {0} given by z 7 → Q(z, f (z)) is nonnegative. Recall that a closed set E in C N is said to have a Stein neighborhood basis if it is the intersection of a sequence of domains of holomorphy in C N. If E is the closure of a strictly pseudoconvex domain or a polydisc in C N, then it has a Stein neighborhood basis. The case of the main theorem in which f is of class C 1 is contained in [2] as a very special case of the main results of that paper. The main theorem seems to be new, even in the setting of classical function theory, where a version of the result is the following. Let U denote the open unit disc in the complex plane. Corollary. A continuous function f on bU extends holomorphically through U if and only if, for each polynomial p(z) = p(z1, z 2 ) in two complex variables Received November 12, 1999.

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such that, with ψf,p (ζ) = p(ζ, f (ζ)), if ψf,p is zero-free on bU then the change in argument around bU of the function ψf,p is nonnegative. In particular, if the function f is smooth Rthen the condition is that (with p and ψf,p as in the corollary) the integral (1/2πi) bU (dψf,p /ψf,p ) be nonnegative. The proof of the Main Theorem proceeds by induction on dimension. The case of planar domains is based on some results on polynomial convexity; the higherdimensional case depends on a suitable slicing argument. The arguments make systematic use of the Bochner–Martinelli kernels. We preface the proof with a section that assembles some information on degree theory.

1. Degree Theory Our discussion draws essentially on the theory of the degree of mappings. We will restrict attention to the theory of the degree of mappings from sets in manifolds to Euclidean spaces. Fix an oriented smooth manifold N of dimension N. To each triple (f, , y) ¯ consisting of a relatively compact open set in N , a continuous map f from into RN, and a point y ∈ RN \ f (b), there is assigned an integer d(f, , y), the degree of f. The function d has the following properties: (d1) d(id, , y) = 1 if y ∈ ; (d2) d(f, , y) = d(f, 1, y) + d(f, 2 , y) whenever 1 and 2 are disjoint ¯ \ (1 ∪ 2 )); open subsets of such that y ∈ / f ( ¯ → RN (d3) d(h(t, ·), , y(t)) is independent of t ∈ [0, 1] whenever h : [0, 1] × N is continuous, and y : [0, 1] → R is continuous and satisfies y(t) ∈ / h(t, b) for all t ∈ [0, 1]; (d4) given (f, , y) and (g, , y) as before, if f b = g b then d(f, , y) = d(g, , y); ¯ \ 1 ). / f ( (d5) if 1 ⊂ (1 open), then d(f, , y) = d(f, 1, y) if y ∈ Property (d4) implies that one can assign a degree to (f, , y) when the continu/ f (b). ous map f to RN is defined only on b and satisfies the condition that y ∈ Degree theory in the form that we shall need is developed in [3] and [17]. An axiomatic development (for maps from RN to itself ) is given in [4]. If is a domain in the plane with boundary a finite collection of mutually dis¯ then joint simple closed curves and if f ∈ C (), 1 1 1 b log f = 1 b Arg f. d(f, , 0) = 2πi 2π Given a bounded domain in N with smooth boundary, if f : b → RN \ {0} is a smooth map then the degree is given by an integral formula, Z 1 f ∗ τ, d(f, , 0) = SN −1 b with SN −1 = 2π N/2/0(N/2) (the area of the unit sphere S N −1 in RN ) and with τ the form given by

Boundary Values and Mapping Degree

τ=

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N 1 X (−1)N −1xj dx1 ∧ · · · ∧ [j ] ∧ · · · ∧ dxN , |x|N j =1

where the expression [j ] indicates the omission of the j th term; see [17]. The (N − 1)th de Rham cohomology group with complex coefficients of RN \ {0} is isomorphic to C and so, if ϑ is any closed smooth (N − 1)-form on RN \ {0} that is not exact, then Z f ∗ϑ d(f, , 0) = c(ϑ) b

for a constant c(ϑ) that depends only on the form ϑ, not on the domain or on the map f. The constant c(ϑ) is determined by the condition that the form τ − c(ϑ)ϑ be exact. Given ϑ, we can determine c(ϑ) by taking to be the unit ball in RN and −1 f R the identity map. It follows that c(ϑ) is determined by the equation c(ϑ) = S N −1 ϑ. It will be convenient to use the following notation: If g = (g1, . . . , gr ) is a vector of complex-valued functions on C N, then ω(g) = ω(g1, . . . , gr ) = dg1 ∧· · ·∧ dgr and r X (−1) j −1gj dg1 ∧ · · · ∧ [j ] ∧ · · · ∧ dgr . ω 0(g) = ω 0(g1, . . . , gr ) = j =1

In particular, with gj = zj this yields ω(z) = dz1 ∧ · · · ∧ dz r . Similarly, ω 0( z¯ ) =

r X

(−1) j −1z¯j d z¯ 1 ∧ · · · ∧ [j ] ∧ · · · ∧ d z¯ r .

j =1

Recall that the Bochner–Martinelli kernel βN is the (N, N − 1)-form on C N defined by 1 N −2N ¯ 2 2 N −1 ¯ |z| ) ∂|z| ∧ (∂∂|z| βN = 2πi = cN |z|−2Nω 0( z¯ 1, . . . , z¯ N ) ∧ ω(z1, . . . , zN ), where cN denotes the constant (−1)(1/2)N(N−1)(N −1)!/(2πi)N. This form is closed ¯ and ∂-closed on C N \ {0}. It is not exact. The Bochner–Martinelli kernel gives an integral formula: If is a bounded domain in C N with smooth boundary, and if ¯ then g is a function holomorphic on a neighborhood of , Z g(0) if 0 ∈ , gβN = ¯ 0 if 0 ∈ / . b As follows from the preceding remarks, the Bochner–Martinelli kernel can be used to compute the degree of certain maps in C N. Given a bounded domain in C N with smooth boundary and given a smooth map f : b → C N \ {0}, the degree of f is given by Z d(f, , 0) =

b

f ∗βN .

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2. A Theorem in the Plane For X a compact subset of C, we shall use R (X) to denote the uniform closure in the space C (X) of complex-valued, continuous functions on X of the space of rational functions on C that have no poles on X. For a given set X, this space may or may not coincide with the space A(X) of functions that are continuous on X and holomorphic on its interior. (For a systematic discussion of the question of the equality of R (X) and A(X) for compacta X in the plane, see [7] and [18].) We shall also use the notation that, for a closed set E ⊂ C N, QE is the space of functions defined and holomorphic on a neighborhood of E × C in C N +1. The neighborhood depends on the particular function; it is not assumed to be a product set. The result to be established in this section is the following. Theorem. Let be a bounded connected open set in C. Assume that each point ¯ If f ∈ C (b), then there is a funcof b is a peak point for the algebra R (). ¯ and holomorphic on with F = f if and only tion F that is continuous on b if the following condition holds. (†) For every p ∈ Q ¯ such that for no ζ ∈ b does the quantity ψf,p (ζ) = p(ζ, f (ζ)) vanish, the degree of the map ζ 7→ ψf,p (ζ) from b to C \ {0} is nonnegative. The theorem applies in particular to all domains for which b consists of a finite number of mutually disjoint simple closed curves. (No regularity is required in this case beyond that imposed by the condition of being a simple closed curve; in particular, each of the curves might have locally finite area.) The condition is satisfied also by certain infinitely connected domains—for example, one obtained by excising from the open unit disc the union of countably many mutually disjoint closed subdiscs whose centers cluster only on the unit circle. Other (more exotic) examples can be found in [7] and [18]. Proof. We assume, as we can without loss of generality, that the origin lies in the domain . ¯ is holoIt is evident that, in the geometric situation of the theorem, if f ∈ C () morphic in and if p ∈ Q ¯ has the property that ψf,p does not vanish on b, then the degree of ζ 7 → ψf,p (ζ) from b to C \ {0} is nonnegative. Let o b be a domain with smooth boundary and with ψf,p zero-free on \ o . Then d(ψf,p , , 0) = d(ψf,p , o , 0) by property (d5) of the mapping degree. However, since ψf,p is holomorphic in , it follows that d(ψf,p , o , 0) is the nonnegative integer (1/2πi)1 b o Arg ψf,p . Denote by 0f the graph 0f = {(z, f (z)) : z ∈ b}. By Q ¯ (0f ) we denote the uniform closure in C (0f ) of the (restrictions to 0f of the elements of the) algebra Q ¯ . This is a uniform algebra on the set 0f . We denote the hull of the set 0f with respect to the algebra Q ¯ by Q ¯ − hull 0f . ¯ × C is not in Q ¯ − hull 0f if and only if there exists By definition, a point z ∈


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a q ∈ Q ¯ with q(z) = 1 > supp 0f |q|. The set Q ¯ − hull 0f is a compact set; it is the spectrum (or maximal ideal space) of the algebra Q ¯ (0f ). Define π1, π2 : C 2 → C to be the projections given by π1(z1, z 2 ) = z1 and ¯ π2 (z1, z 2 ) = z 2 , respectively. Then b ⊂ π1(Q ¯ − hull 0f ) ⊂ . We begin by showing that the set Q ¯ − hull 0f is not contained in π −1(b). Suppose it is. Let δo = inf{1/|z| : z ∈ b}, and let h ∈ C (b) be the function given by h(z1, z 2 ) = 1/z1 . Because Q ¯ − hull 0f is the spectrum of the algebra Q ¯ (0f ), we have that π1(Q ¯ − hull 0f ) is the spectrum of the element π1 0f of the algebra Q ¯ (0f ). Consequently, if g ∈ O (π1(Q ¯ −hull 0f )), then g B π1 ∈ Q ¯ (0f ). As each point of b is a peak point for R (b), it follows that R (b) = C (b) (see [7, p. 543]), whence there is a function q ∈ Q ¯ with |q(z) − h(z)| < 21 δo for all z ∈ 0f . As maps from 0f to C \ {0}, the functions q and h are homotopic. Consequently, the degrees of the maps ζ 7 → q(ζ, f (ζ)) and ζ 7 → h(ζ, f (ζ)) from b to C \ {0} are the same. However, the former degree is nonnegative (by hypothesis), and the latter is −1, a contradiction. Thus, π1(Q ¯ − hull 0f ) must meet the connected open set . The maximum principle then implies that π1(Q − hull 0f ) contains . Next, the set (Q ¯ − hull 0f ) ∩ π1−1(b) coincides with the set 0f . To see this, −1 suppose z o to be a point in ((Q ¯ − hull 0f ) ∩ π R 1 (b)) \ 0f .o There is then a regular Borel probability measure µ on 0f with 0f q dµ = q(z ) for all q ∈ Q . (The measure is a representing measure for the point z o with respect to the alge¯ the measure µ bra Q ¯ (0f ).) Because each point of b is a peak point for R (), is supported on the set ({z1o } × C ) ∩ 0f . This set is a singleton, for 0f is a graph. It follows that µ is a point mass, whence z o must lie in 0f —a contradiction. ¯ In order to do We shall show that π1 carries Q ¯ − hull 0f injectively onto . this, we use the notion of maximum-modulus algebra. (See [1] and [13].) Set A = {f | Q ¯ − hull 0f \ 0f : f ∈ Q ¯ (0f )}, an algebra of continuous, complex-valued functions on the locally compact space X = Q − hull 0f \ 0f . Let p = π1 X . Then (A , X, , p) is a maximum-modulus algebra on X over with projection p in the sense of [1]. Introduce the function δ : → [0, ∞) by δ(ζ) = diameter π2 (π1−1(ζ) ∩ Q ¯ − hull 0f ). The compactness of Q ¯ − hull 0f implies the boundedness of the function δ. The function δ tends to 0 at b. Otherwise, there is a sequence {ζk }∞ k=1 of points in with ζ → b as k → ∞ and with diameter π2 π1−1(ζ) > η for some positive η and all k. If ζk → ζ o ∈ b, then π1 carries two distinct points of 0f onto the point ζ o . However, the map π1 is injective over 0f . According to [1, Thm. 11.7], the function log δ is subharmonic on . As it tends to −∞ at b, it must be identically −∞ on . That is, π1 is injective on ¯ → C whose graph is Q ¯ − hull 0f . Thus, there is a continuous function F : Q ¯ − hull 0f ; F agrees with f on b. The theory of maximum-modulus algebras implies that F is holomorphic on . To see this, we can invoke the general theory of maximum-modulus algebras

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as given in [1]. Alternatively, we can appeal to Rudin’s treatment [13] of local maximum-modulus algebras. If we set B = {ϕ ∈ C () : ϕ(z) = f (z, F(z)) for some f ∈ A and all z ∈ }, then B is a local maximum-modulus algebra in the sense of [13]. It contains the identity map—namely, the map z 7 → π1(z, F(z))—so each element of B is holomorphic on . In particular, since F(z) = π2 (z, F(z)), we find that F is holomorphic, as we wished to show. The theorem is proved. Remark. By construction, the function F is holomorphic on and continuous ¯ ¯ Nothing we have done implies that F ∈ R (). on . Remark. The hypothesis in the preceding theorem that each point of b be a ¯ cannot be completely abandoned. Let I be the R () peak point for the algebra closed interval − 21 , 21 and let = U \ I ; thus, b = bU ∪ I. Let f : bU → C be given by z if z ∈ bU, f (z) = 0 if z ∈ I. The function f is not the boundary value of any function holomorphic on , con¯ However, it does satisfy the condition that, if p(z1, z 2 ) is a polynotinuous on . mial in two complex variables such that the function ψf,p (z) = p(z, f (z)) has no zero on b, then the degree of the map ψf,p on b is nonnegative. The points of ¯ the interval I are not peak points for the algebra R ().

3. The Induction Step We have proved the Main Theorem, and somewhat more, in the 1-dimensional case; we now show that the N -dimensional case is a consequence of the (N − 1)dimensional case. Thus, we assume that the Main Theorem has been established in the case of domains in C N −1 and continuous functions on their boundary. Let be a bounded domain in C N with boundary of class C 2 , and let f be a continuous function on b that satisfies the hypotheses of the Main Theorem. As in the statement of that theorem, let 0f denote the graph of the function f, a subset of C N +1. Let 5 be a complex affine hyperplane in C N that meets b transversely. We will show that the restriction f (b∩5) satisfies the hypotheses of the theorem (in the (N − 1)-dimensional case) and so extends holomorphically into the slice ∩ 5. For notational convenience, we assume that 5 is C N −1 = C N −1 × {0} = {zN = 0} ⊂ C N. Let Q1, . . . , QN −1 be functions defined and holomorphic on a neighborhood W 0 ¯ × C such that their set of common zeros is disjoint from the graph of (C N −1 ∩ ) 0f0 = {(z1, . . . , zN −1, 0, f (z1, . . . , zN −1, 0)) : (z1, . . . , zN −1, 0) ∈ b}. ¯ has a Stein neighborhood basis, there is a Stein neighborhood W of Since ¯ × C such that W ∩ (C N −1 × C ) ⊂ W 0. Each of the functions Q1, . . . , QN −1 extends to be holomorphic on W ; we denote these extensions also by Q1, . . . , QN −1. Denote by Q 0 the map Q 0 = (Q1, . . . , QN −1) from W 0 to C N −1. We must show


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that, if 9 0 : C N −1 → C N −1 is given by 9 0(z1, . . . , zN −1) = Q 0(z1, . . . , zN −1, f (z1, . . . , zN −1, 0)), then the degree of 9 0 as a map from b ∩ 5 to C N −1 is nonnegative. ¯ × C by Let Q be the C N -valued map defined for (z, ζ) in a neighborhood of Q(z, ζ) = (Q 0(z1, . . . , zN −1, ζ), zN ). The set Q−1(0) is disjoint from 0f . By hypothesis, the map 9 : b → C N \ {0} given by 9(z1, . . . , zN ) = Q(z, f (z)) has nonnegative degree. The point is that the degree of the map 9 0 is the same as the degree of the map 9. This fact is contained in [2, Lemma 1.1]. There is a derivation of the fact, based on the Bochner–Martinelli integral, as follows. Because the function f is only continuous, a preliminary step is required. Let % be a defining function of class C 2 for the domain so that % is defined on a ¯ d% 6 = 0 on b, and = {% < 0}. We suppose that the neighborhood of , ¯ and smooth function f has been extended to a continuous function defined on in . For example, f might be harmonic in with the given values on b. If δ > 0 is small then, with δ = {% < −δ}, the set b δ is again a manifold of class C 2 , as is b δ ∩ C N −1. Moreover, given that δ is small enough, the ¯ δ . Then d(9, , 0) = function f will not assume the value 0 in the set \ d(9, δ , 0) and d(9 0 , ∩ C N −1, 0) = d(9 0 , δ ∩ C N −1, 0). The upshot of this is that, for the purpose of verifying that the degree of 9 is the same as the degree Rof 9 0 , we canR assume the function f to be of class C 2 . Thus, weRmust prove that b 9 ∗βN = b∩5 9 0∗βN −1. This is equivalent to proving that 0f Q∗βN = R 0∗ 0 0 Q βN −1. Note that the last two formulas cannot be written without the asf

sumption that f be smooth. We need a fact about the Bochner–Martinelli kernel βN on C N : It is exact in C N \ C N −1. This is well known and due originally to Martinelli [12]. It is a consequence of a simple, direct calculation. If 4 is the (N, N − 2)-form on C N \ {0} given by 4 = |z|−2(N −1) ω 0( z¯ 1, . . . , z¯ N −1) ∧ ω(z1, . . . , zN )

¯ = (−1)N −1(N − 1)zN |z|−2Nω 0( z¯ 1, . . . , z¯ N ) ∧ ω(z1, . . . , zN ). From then d4 = ∂4 this it follows that, on the set in C N where zN 6= 0, if γN denotes the constant (−1)N −1(−1)(1/2)N(N−1)(N − 2)!/(2πi)N, then 1 βN = d γN 4 . zN If we pull this formula back to C N +1 by way of the map Q, we obtain 1 1 ∗ Q 4 . Q∗βN = d γN −1 (2πi)N −1 zN Since ¯ 1, . . . , Q ¯ N −1) ∧ ω(Q1, . . . , QN −1) ∧ dzN ω 0(Q , Q∗ 4 = 2 {|Q1| + · · · + |QN −1|2 + |zN |2 }N −1 it follows that


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¯1 ∧ ··· ∧ Q ¯ N −1) Q∗βN = d γN −1ω 0(Q ∧ ω(Q1, . . . , QN −1){|Q1|2 + · · · + |QN −1|2 + |zN |2 }−(N −1) dzN ∧ . 2πizN Now apply Stokes’s theorem and Fubini’s theorem (in the form given, e.g., in [17]): Z Z ∗ Q βN = lim+ Q∗βN = lim+ −γN −1 0˜

Z

ε→0

Z

{|zN |=ε}

ε→0

˜ 0∩{|z N |>ε}

¯ 1, . . . , Q ¯ N −1) ∧ ω(Q1, . . . , QN −1) dzN ω (Q . {|Q1|2 + · · · + |QN −2 |2 + |zN |2 }N −1 2πizN 0

˜ C N :ζ =zN } 0∩{z∈

The negative sign in the last equation arises from considerations of orientation. R As ε → 0 + , the last quantity tends to 0˜ 0 Q 0∗βN −2 . Thus, we have the desired equality of the two degrees in question. The inductive hypothesis now implies that the function f continues holomorphically into the domain 5 ∩ . That is, we have shown that if 5 is a complex-affine hyperplane in C N that meets b transversely, then f (5∩b) extends holomorphically into the slice 5 ∩ . If N = 2, we are in the situation of the 1-dimensional extension property considered in [15] (cf. [8]). The result of [15] implies that, as desired, f extends holomorphically through . If N > 2, a simpler argument is possible. Given a point w ∈ , let 5 be a complex-affine hyperplane in C N through w that meets b transversely. Define F5 (w) to be the value of the holomorphic extension F5 of f (5∩) through 5 ∩ . Since N ≥ 3, the value of F5 (w) does not depend on the choice of 5; denote this value by F(w). This gives a well-defined function F on that is holomorphic and assumes the values f on b. The Main Theorem is proved.

4. Extensions We shall now consider certain extensions of the work just given. A. The Convex Case It is more or less evident that an analog of the Main Theorem can be established in the setting where D is a bounded convex domain in C N and f ∈ C (bD) is a continuous function that satisfies condition (∗) of the Main Theorem. (We consider arbitrary convex domains, not only those with smooth boundaries; thus, we admit polydiscs or convex polyhedra among other examples.) The basis for this assertion is as follows. The compact set D¯ does have a neighborhood basis that consists of Stein domains. The slices of D by complex lines are convex 1-dimensional domains, so the result of Section 2 applies to them. The necessary 1-dimensional extension result, tailored to the setting of convex domains,


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is contained in [8, Thm. 3.2.1]. Thus, the only point that needs to be verified in this case is a fact about degrees. Precisely put, what has to be established is the following. Let 1 be a bounded convex domain in C N. If f ∈ C (b1) satisfies condition (∗) and if 5 is a complex hyperplane that meets 1, then f 5 , the restriction of f to b1 ∩ 5, has the corresponding property. This implication can be established as follows. Without loss of generality, we suppose 5 to be the coordinate axis {z ∈ C N : zN = 0}. Let 1N denote the intersection 1 ∩ 5. Fix a holomorphic mapping Q 0 from a neighborhood ¯ N × C into C N −1 such that the zero locus of the map 9 0 given by 9 0 = of 1 0 (Q (z1, . . . , zN −1, 0, f (z1, . . . , zN −1, 0)) is disjoint from the graph of f 5 . We are to prove that the degree of the map 9 0 from b1 ∩ 5 to C N −1 is nonnegative. The map Q 0 can be extended to a holomorphic C N −1-valued map (still denoted ¯ × C. Let Q be the C N -valued map defined by Q 0 ) defined on a neighborhood of 1 ¯ × C by (z, ζ) 7 → ((Q 0(z1, . . . , zN −1, 0, ζ), zN ). Then the on a neighborhood of 1 N C -valued map 9 defined on b1 by 9(z) = Q(z, f (z)) is zero-free; its degree as a map to C N \ {0} is nonnegative. Note that we can extend the map f through 1 as a smooth function. Consider a smoothly bounded convex domain D that is a relatively compact subset of 1 and that is large enough for the quantity Q(z, f (z)) to be zero-free ¯ \ D. Then the degree of the map 9 from b1 to C N \ {0} is on the compact set 1 the same as the degree of the map 9D : bD → C N \ {0}. Also, the degree of the map 9 0 from b1 ∩ 5 to C N −1 \ {0} is the same as the degree of the map 9D0 from bD ∩ 5 to C N −1 \ {0}. The analysis using the Bochner–Martinelli integral invoked in Section 3 shows the degree of 9D to be the same as the degree of 9D0 . Thus, a result analogous to the Main Theorem is established in the case of arbitrary bounded convex domains. B. The Manifold Case The work carried out in Sections 2 and 3 is set in the context of domains in C N. But little effort is required to extend it to certain manifold situations. Toward this end, if R is an open Riemann surface and if X is a compact subset of R, then we define R (X) to be the closure in C (X) of the (restrictions to X of ) functions defined and holomorphic on various neighborhoods of X in R. Runge’s theorem implies that, when R is the complex plane, this notion of R (X) coincides with that used in Section 2. If D is a relatively compact domain in R for which each point of bD is a peak ¯ then the continuous functions f on bD that extend point for the algebra R (D), holomorphically through D admit a characterization precisely parallel to that given in the Theorem of Section 2: They are those functions satisfying the condition (†R ) that, if p is a function holomorphic on a neighborhood in R × C of the set D¯ × C such that p−1(0) is disjoint from the graph 0f = {(z, f (z)) : z ∈ bD}, then the degree of the map z 7 → ϕp (z) = p(z, f (z)) from bD into C is nonnegative. The argument of Section 2 applies, mutatis mutandis, to deal with the present situation.

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Once we have this, we can formulate and prove a result on domains in manifolds as follows. Fix a complex manifold M of dimension M and in it a relatively compact domain D with bD of class C 2 . We assume that D¯ has a Stein neighborhood basis. It then entails no loss of generality to suppose that M is itself a Stein manifold and so, by the embedding theorem, that M is a closed submanifold of C N for some suitably large N. In this setting, we have the following result. The continuous function f on bD extends holomorphically through D if and only if it satisfies the condition (∗M ): For every holomorphic C M -valued map Q defined on a neighborhood of D¯ × C in M × C such that Q−1(0) is disjoint from the graph 0f = {(z, f (z)) : z ∈ bD}, the map z 7 → Q(z, f (z)) from bD into C M \ {0} has nonnegative degree. Unlike our proof of the Main Theorem, the proof we give for this is not based on induction on dimension. Rather, we shall apply some information about the Bochner–Martinelli kernel to show that it follows from the hypotheses that, for every nonsingular 1-dimensional complex submanifold 6 of a neighborhood of D¯ in M that meets bD transversely, the restriction f (6∩bD) extends holomorphically into the slice 6 ∩ D. Having established this point, by [16] we can conclude that f extends holomorphically through D. We establish the stated 1-dimensional extension property first in the particular case that the curve 6 is a complete intersection, so that there exists a map ϕ = (ϕ 1, . . . , ϕM−1) from a neighborhood of D¯ in M to C M−1 whose zero locus is 6 and whose differential has maximal rank at each point of 6. Consider a function G defined and holomorphic on a neighborhood of D¯ × C that satisfies the condition that the zero locus of G be disjoint from the partial graph {(z, f (z)) : z ∈ (6 ∩ bD)}. We want to show that the map z 7 → G(z, f (z)) from 6 ∩ bD to C \{0} has nonnegative degree. Having fixed G, there is no loss in assuming that f is smooth on bD. (This follows from argumentsRwe have already used.) Hence, we must prove that the winding number (1/2πi) 6∩bD (dG/G) is nonnegative. By construction, the holomorphic map 8 : D¯ × C → C M given by 8(z, ζ) = (ϕ(z), G(z, ζ)) = (ϕ 1(z), . . . , ϕM−1(z), G(z, ζ)) does not vanish R at any point of the graph 0f of f (on the whole of bD). Accordingly, the integral 0f 8∗βM is nonnegative. The map 8 is constructed so that it carries D¯ \ 6 into C M \ λ, where λ is the complex line {z ∈ C M : z1 = · · · = zM−1 = 0}. In Section 6, a primitive for the Bochner–Martinelli kernel is constructed in the domain C M \ λ; if 2 ∈ E 2M−2 (C M \ λ) is the form given in Section 6 (with M here replacing the N of Section 6), then 8∗βM = d8∗(cM 2), where (as in Section 1) cM denotes p the constant (−1)(1/2)M(M−1)(M − 1)!/(2πi)M. We shall write |ϕ| for the quantity |ϕ 1|2 + · · · + |ϕM−1|2 . Let ε > 0 be a small positive number. By Stokes’s theorem,


Z 0f

8∗βM = lim+ ε→0

= lim+ ε→0

Z Z

Z

∗

0f ∩{|ϕ(z)|>ε}

0f ∩{|ϕ|=ε}

363

8 βM = lim+ ε→0

b({0f ∩{|ϕ|>ε})

cM 2

cM 2(M − 1)(|ϕ|2 + |G|2 )M−1 M−2 X M − 1 × |G|2M−2r−4 |ϕ|2r−2M+2 r r=0

¯ − GdG) ¯ ∧ ω 0(ϕ) ¯ ∧ ω(ϕ) + (−1)M−1|G|2 ω(ϕ) ¯ ∧ ω(ϕ)]. × [(−1)M (GdG Since |ϕ| = ε on the path of integration, it follows from Stokes’s theorem and Fubini’s theorem for forms that Z Z Z cM 8∗βM = lim+ 2 2 M−1 ε→0 0f {ζ ∈ C M−1 :|ζ|=ε} 0f ∩ϕ −1(ζ) 2(M − 1)(ε + |G| ) M−2 X M − 1 ¯ − GdG) ¯ ¯ ∧ ω(ζ). × ω 0(ζ) G 2M−2r−4 |ε|2r−2M+2 (−1)M (GdG r r=0 ¯ ∧ ω(ϕ) Observe that the term in the primitive 8∗ 2 for 8∗βM that contains ω(ϕ) contributes nothing to the integral: It is a (2M − 2)-form in dϕ and so induces the 0-form on the ζ -path of integration (on which |ϕ| = ε). The ζ -path of integration is the (2M − 3)-dimensional sphere of radius ε; its volume is (2π M−1/(M − 2)!)ε 2M−3 , and the coefficient of each term in the form ω 0(ϕ) ¯ is one of the coordinates of ϕ and hence is O(ε). Consequently, the only term in the sum under the last integral that (in the limit) can make a nonzero contribution is the one corresponding to r = 0. Thus, by using the equality cM = ((−1)M−1(M − 1)/2πi)cM−1, we reach Z Z Z ¯ − GdG) ¯ cM (−1)M |G|2M−4 (GdG ∗ 8 βM = lim+ ε→0 2(M − 1)(ε 2 + |G|2 )M−1 0f {ζ ∈ C M−1 :|ζ|=ε} 0f ∩ϕ −1(ζ) ¯ ∧ ω(ζ) × ε −2M+2 ω 0(ζ) Z ¯ 1 dG dG − . = ¯ 4πi 0∩ϕ −1(0) G G R Observe that the quantity (1/2πi) ϕ −1(0)(dG/G) is the degree d of the mapping R ¯ G) ¯ z 7 → G(z, f (z)) from 6 ∩ bD to C \ {0}. The quantity (1/2πi) ϕ −1(0)(dG/ is −d. We have established that Z Z 1 dQ ∗ 8 βM = . 2πi 6∩bD Q 0f By hypothesis, the former number is nonnegative. Consequently, the latter degree is, too. We may conclude that the restriction f (6∩bD) extends holomorphically into the slice 6 ∩ D.


364

This has been done under the assumption that the curve 6 is a complete intersection. In dimension ≥ 3, this condition is not restrictive, as follows from work of Forster and Ramspott [5; 6]. That work is not simple, and we prefer to proceed without appeal to it. Moreover, it is not true that every nonsingular curve is a complete intersection in the 2-dimensional case: There are unsolvable Cousin II problems on certain 2-dimensional Stein manifolds. However, it is true that, in every dimension ≥ 2, a nonsingular curve can be approximated by a collection of irreducible branches of nonsingular complete intersections. This device allows us to obtain the desired 1-dimensional extension property in the general case. To do this, fix a nonsingular analytic curve 6 in a Stein neighborhood W of the compact set D¯ in M . The first observation is that, although 6 may not be a complete intersection in W, there is a neighborhood V of 6 (V ⊂ W ) in which 6 is a complete intersection. This follows because a neighborhood of 6 in W is biholomorphically equivalent to a neighborhood of the zero section of the normal bundle of the embedding 6 ,→ W. (See [10, p. 257].) This normal bundle is a holomorphic vector bundle over the noncompact Riemann surface 6 and so is trivial. (See [9, p. 303].) Thus, if V is a sufficiently small neighborhood of 6 in W, then 6 is a complete intersection in V. Let F be a holomorphic map from a neighborhood of V¯ to C M−1 that defines 6 as an analytic set. (We may have to shrink V ¯ Let W1 b W be a Stein domain whose cloa little to have F be defined on V.) sure is O (W )-convex. The set 6 ∩ W¯ 1 is then O (W )-convex. Consequently, if V is chosen to be sufficiently thin and to be a Stein domain, then there will exist a sequence {Fn } n=1,2, ... of holomorphic maps from W to C M−1 that converges uniformly on V¯ ∩ W¯ 1 to f. The maps Fn can be chosen such that (a) for each n, 0 is a regular value of Fn , and (b) 6 n = Fn−1(0) is transverse to bD. For each n, the restriction f (6 n ∩bD) extends holomorphically into the slice 6 n ∩D. We are to deduce from this that f (6∩bD) extends holomorphically through the slice 6 ∩ D. For this, since V is a Stein domain R it will suffice to show that, for every holomorphic 1-form α on V, the integralR 6∩bD∩V fα vanishes. (See [11].) We already know that, for each n, the integral 6 n ∩bD∩V fα vanishes. Hence, we must prove that Z Z fα = lim fα. lim n→∞ F −1(0)∩bD∩V n

n→∞ F −1(0)∩bD∩V

By a device already used several times, it entails no loss of generality for us to suppose that the function f is smooth and defined on W. Then, by Stokes’s theorem, Z Z lim

n→∞ F −1(0)∩bD∩V n

and also

fα = lim

n→∞ F −1(0)∩D n

Z

Z F −1(0)∩bD∩V

df ∧ α

fα =

F −1(0)∩D

df ∧ α.

We shall show that the right-hand sides of the last two equations are the same.


365

In this, we need the following remark. Let ω denote the fundamental area form P ∧ d z¯j . Thus, the area of a 1-dimensional variety E on C N : ω = (−i/2) N j =1 dz Rj N in an open subset of C is E ω. Let T and T 0 be thin tubes in C N over the system of curves 6 ∩ bD with T 0 b T . Let χ be a nonnegative function of class C ∞ on C N with χ = 1 on a neighborhood of T¯ 0 and with χ = 0 on a neighborhood of C N \ T 0. Then Z Z χ df ∧ α = χ df ∧ α. lim n→∞ T 0∩6 n

T 0∩6

If T 0 is sufficiently small, then the latter quantity is small. It follows that if T 0 is chosen so that the area of the part of F −1(0) in T 0 is small, then in Z Z df ∧ α − df ∧ α Fn−1(0)∩D

F −1(0)∩D

Z

=

Fn−1(0)∩D

χ df ∧ α −

F −1(0)∩D

χ df ∧ α

Z

Z

+

Z

(1 − χ) df ∧ α −

Fn−1(0)∩D

(1 − χ) df ∧ α

F −1(0)∩D

we have that the first summand on the right is small uniformly in n for sufficiently large n while the second summand tends to zero as n → ∞. It follows that, as desired, Z Z lim

n→∞ F −1(0)∩D n

df ∧ α =

F −1(0)∩D

df ∧ α,

which completes the proof.

5. Two Open Questions In this work we have repeatedly imposed the hypothesis that the closure of our domain D have a Stein neighborhood basis. It is not at all obvious that this hypothesis is necessary for the conclusion, and we ask whether it can be replaced by something weaker. In particular, might it suffice for the boundary of the domain to be connected? (The hypothesis that the closure of the domain has a Stein neighborhood basis implies that the boundary of the domain is connected.) The Stein neighborhood basis hypothesis is stable under passage to intersections by lower-dimensional hypersurfaces; the hypothesis of having connected boundary is not. The second open question arises in connection with the classical description of the boundary values of holomorpic functions on the unit disc U. From classical through function theory, a continuous function f on bU extends holomorphically Rπ U if and only if the Fourier coefficients fˆ(n) = (1/2π) −π f (e iϑ )e−inϑ dϑ vanish for n = −1, −2, . . . . The vanishing of these integrals for all n = −1, −2, . . . is equivalent to the vanishing of the integral


366

1 2πi

Z

f (z) dz |z|=1 z − w

for all w ∈ C with |w| > 1. Accordingly, we ask for a simple, direct proof of the equivalence—in the case of functions on the unit circle—of these classical conditions to the positivity conditions given by the Main Theorem and its corollary.

6. Appendix: A Primitive for the Bochner–Martinelli Kernel In this appendix we shall determine an explicit primitive for the Bochner–Martinelli kernel in the complement of a complex line through the origin in C N. In order to avoid dealing with constants in some calculations, we shall not work initially with the Bochner–Martinelli kernel βN itself but rather with the form k = |z|−2Nω 0( z¯ ) ∧ ω(z), defined and of bidegree (N, N −1) on C N \{0}. The kernel k differs from βN only by the constant factor cN = (−1)(1/2)N(N−1)(N − 1)!/(2πi)N. Denote by λ the complex line λ = {z ∈ C N : z1 = · · · = zN −1 = 0}, and define H : (C N \ λ) × [0, 1] → C N \ λ by H(z, t) = H(z1, . . . , zN −1, zN , t) = (z1, . . . , zN −1, tzN ). The map H is a homotopy in C N \ λ between the identity map on C N \ λ and the projection z 7 → (z1, . . . , zN −1, 0). The form k is smooth on C N \ λ and so (since k is a closed form) the homotopy formula for forms [14, p. 8-53] provides the formula k = d2, where 2 denotes the (2N − 2)-form obtained as follows. Write H ∗ k = dt ∧ ϑ + η with η a (2N − 1)-form on (C N \ λ) × [0, 1] that does not have a factor dt and with ϑ a (2N − 2)-form on the same manifold that does not have a factor dt. (To be sure, the coefficients of ϑ and η will depend on t.) Then 2 is the form given by R1 2 = 0 ϑ dt. (In connection with the homotopy formula, note that the range of the map z 7 → H(z, 0) is of complex dimension N − 1, so that the form k induces on it the zero form.) We determine 2 explicitly as follows. In this it will be convenient to write z 0 for (z1, . . . , zN −1) ∈ C N −1 if z = (z1, . . . , zN −1, zN ) ∈ C N. We have H ∗ k = (|z 0 |2 + t 2 |zN |2 )−Nω 0( z¯ 1, . . . , z¯ N −1, t z¯ N ) ∧ ω(z1, . . . , zN −1, tzN ). Then

ω(z1, . . . , zN −1, tzN ) = tω(z) + zN ω(z 0 ) ∧ dt

and ω 0( z¯ 1, . . . , z¯ N −1, t z¯ N ) =

N −1 X

(−1) j −1z¯j d z¯ 1 ∧ · · · ∧ [j ] ∧ · · · ∧ d(t z¯ N )

j =1

+ (−1)N −1t z¯ N ω( z¯ 1, . . . , z¯ N −1).


367

From this it follows (after some computation) that N −2 X N −1 1 2N −2r−4 0 2r−2N +2 |z | |zN | 2= 2(N − 1)(|z 0 |2 + |zN |2 )N −1 r=0 r × [(−1)N(zN d z¯ N − z¯ N dzN ) ∧ ω 0( z¯ 0 ) ∧ ω(z 0 ) + (−1)N −1|zN |2 ω( z¯ 0 ) ∧ ω(z 0 )]. This is the desired primitive—a d-primitive—for k on C N \ λ. We then have that βN = d [cN 2] N

on C \ λ. (In this formula, cN denotes the constant introduced in Section 1 in connection with βN .)

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Department of Mathematics University of Washington Seattle, WA 98195-4350 [email protected]