# Polynomial hulls of rectifiable curves.

The material in this paper grew out of an attempt to generalize some work of Alexander on the structure of polynomial hulls of 1-rectifiable subsets of

[C.sup.n.] Specifically, Alexander has shown that if X C [C.sup.n] is a connected compact set of finite length, then X\ X is a one-dimensional analytic variety [A11]. Here X refers to the polynomial hull of the compact set X C [C.sup.n.] In [A14] he showed that if IF is a rectifiable simple closed curve, then [gamma] \ [gamma] has at most one global branch, and [gamma] \ [gamma] has finite area. In addition, there is an isoperimetric inequality that establishes a bound on the area in terms of the length. We have generalized both of these results and have proved that Stokes's theorem holds for these sets. In the course of this research, we also extended a theorem [GS2] on the derivatives of proper maps that have cluster sets of finite length. The paper is organized into four sections: in the first the number of branches of [gamma] \ [gamma] is counted; in the second the theorem on proper maps is proved; in the third Stokes's theorem is proved; we end the paper with a discussion of density at the boundary and harmonic measure on [gamma].

For a compact set X C [C.sup.n], we denote by P(X) the uniform closure of the algebra of polynomials on X.

1. Counting the branches of [gamma] \ [gamma]. We prove first that if F has finitely generated cohomology, then [gamma] \ [gamma] has finitely many global branches. We denote by [H.sup.1]([gamma, Z) the first Cech cohomology of F with integer coefficients and by

[rkz[H.sup.1] ([gamma], Z) its rank as an abelian group. By a theorem of Besicovitch [Fa] we know that connected sets of finite length are arc-wise connected.

Theorem 1. Let [gamma] be a compact connected subset of [C.sup.n] with

[^.sup.1] < [infinity]. If rkZ[H.sup.1] ([gamma],Z)] 1 < [infinity] has at most 1 global branches.

Proof. Since [gamma] has topological dimension 1, the map [H.sup.1] ([gamma],Z)] [right arrow] [H.sup.1] (K,Z) is surjective for every closed K [subset] X [HW, Th. VIII 3]. Letting K run over the collection of subsets of [gamma] that are unions of simple closed curves in [gamma], we see that [gamma] contains only finitely many simple closed curves; otherwise [H.sup.1] ([gamma], Z) would not be finitely generated. Let [gamma] be the union of all the simple closed curves in [gamma]; clearly rkZ[H.sup.1]) ([gamma, Z)] [less than or equal to] 1-Write [gamma] = {[p.sub.1],.., [p.sub.m1]} [union] [union] ([[union].sup.m2.sub.j=1] [[gamma].sub.j]), where the p's are the points of intersection of the simple closed curves in [gamma] and the [gamma.sub.j]'s are the leftover open arcs. The set [gamma] [gamma] can be written in the form [gamma] \ [gamma] = [union][[infinity].sub.i=1] [F.sub.i] where the [F.sub.i's] are the connected components of [gamma] \ [gamma]. (Each of these has positive length, so there cannot be uncountably many of them.) Since the set [gamma] \ [gamma'] contains no simple closed curves, the [F.sub.i's] satisfy

(1) [F.sub.i] [[intersection] [F.sub.j] = 0 if i j (2) [F.sub.i] does not contain a simple closed curve and (3) [F.sub.i] [intersection] ([gamma \ [F.sub.i]) = qi,

a single point. We now show that the [F.sub.i's] are removable sets for [gamma] \ [gamma], i.e., that [gamma] \ [gamma] is analytic through [F.sub.i]. We present two proofs of this fact: the first purely geometrical and the second relying on the general version of Stokes's theorem for the hull of a rectifiable curve which is proved in section 2.3.

Choose [phi] [element of] C[infinity] ([C.sup.n]\{[q.sub.i]}) nonnegative with [phi] < 1/2 on [F.sub.i] and [phi] > 1 on [gamma]/[F.sub.i], and [phi] [right arrow] [delta]} is smooth and {[phi] = [infinity]}. By Sard's theorem, for some [delta], 1/2 < [delta < 1, the surface {[phi] = [infinity]} is smooth and {[phi] = [infinity]} [union] {[q.sub.i]} cuts off a neighborhood of [F.sub.i] from the rest of [gamma.]. By the maximum principle, [gamma] [intersection] {[q.sub.i]] [less than or equal to] [delta]} [subset] ([gamma] [[intersection] ([gamma] [intersection] ({[phi] = [delta] [union] [F.sub.i])). To show that [F.sub.i] is removable, is suffices to show that ([gamma] [intersection] ({phi]= [delta]} [union] [F.sub.i])). ^ = ([gamma] [intersection] {[phi] = [delta]})^. Let us denote the set [gamma] [intersection] {[phi] = [delta}] by X To simplify the proof, we can remove from X all the curves which are removable in X \ X. We can do this because if a part of a curve has two branches meeting on it, then by transversality, the same must be true of the whole curve: and in this case X \ X continues analytically across the curve [GS1 theorem 5]. Call the new, reduced set X'. Let C be a branch of (X'[union] [F.sub.i]) ^ X'[union] [F.sub.i]). The set (C \ C) [intersection X must be nonempty and nonpolynomially convex [Al1]. Let C' be a branch of (X [intersection] C \ C)). Then C \ C' intersects X [intersection] C \ C) along some arc [beta]. By the transversality of {[phi] = [delta]} to [gamma] \ [gamma], we obtain that C' = C or else C' comes in from the other side [beta]. The second possibility is ruled out by construction. This proves that each [F.sub.i] is removable, so we now have shown that [gamma] \ T, [subset] [gamma] [gamma]. of the [F.sub.i]. Let U be an open set such that U n IF = [F.sub.i] and let T [element of] [D.sup.2](U) be the current defined by integration over ([gamma] \ [gamma]) [intersection] U. Theorem 3, applied to T gives that 9T is closed 1-dimensional integral current whose support is contained in [F.sub.i]. (See section 2.3 for relevant definitions pertaining to currents.) Applying the result of 4.2.25 of [Fe], we can write T = [sigma][[infinity].sub.j=1][R.sub.j], Since T indecomposable 1-dimensional integral current and N(T) = [sigma][[infinity].sub.j=1]N([R.sub.j] Since T is closed, N( T) = M(T), which implies that N([R.sub.j]) = M([R.sub.j]) for each j, i.e., each [R.sub.j] is closed. The only closed indecomposable 1-dimensional integral currents are those which consist of integration around a simple closed curve. Since [F.sub.i] contains no simple closed curves, each [R.sub.j] is equal to zero. Thus T = 0 holds and we can apply theorem 5.2.1 of [Ki] to show that U [intersection of] )[gamma] \ [gamma]) is analytic.

To show that the number of branches of [gamma] \ [gamma] [less than or equal to] l, we argue by contradiction. First, we note that the decomposition theorem for 1-dimensional integral currents cited above implies that a closed integral current supported on [gamma]' consists of integration around some of the simple closed curves in [gamma]' a finite number of times; in particular, such a current gives an integral homology class on [gamma]'. Now suppose that [V.sub.k], k = 1, 2,... m are distinct irreducible branches of [gamma]' \ [gamma]', with m > 1. Then letting [T.sub.k] denote the collection of boundaries of the [V.sub.k's] considered as currents, we have that each [T.sub.k] represents a nonzero homology class on [gamma]'. Since m > l there exist integers [C.sub.k], k = 1, 2 ... m not all equal to zero, such that [[sigma].sub.k=1.sub.m]=1 [c.sub.k][T.sub.k] = 0. This implies that the current [[sigma].sub.k=1.sup.m][c.sub.k] [[V.sub.k]] is closed on [C.sup.n], which by theorem 5.2.1 of [Ki] implies that [[union].sup.m.sub.k=1] [V.sub.k] is a subvariety of [C.sub.n]. This is clearly impossible, thus giving the desired contradiction.

2. Proper maps from [union] into [C.sup.n] \[gamma]. Denote by U the open units disc in C. We prove here a generalization of the theorem that a conformal map f : U [right arrow] D, where [[lambda].sub.1] ( D) < [infinity], has f' [element of] [H.sup.1]. This classical theorem is due to F. and M. Riesz [Du]. The theorem we prove was conjectured by Globevnik and Stout, who proved in the paper [GS2] the result in the case that [gamma] is a simple closed curve.

Theorem 2. If [gamma] is a compact subset of [C.sup.n] of finite length, and if f : U [right arrow] [C.sup.n]\[gamma] is a bounded, proper, holomorphic map, then f' [element of] [H.sup.1]

Proof. We can assume without loss of generality f is one-to-one off of a countable set [St]; also, by [A13] and [P2] we know that f extends continuously to U. By a theorem of Hardy [Du, 3.10], it suffices to show that f has bounded variation on U = [S.sup.1] We will show that almost every point in f([S.sup.1) [intersection] [gamma] has one or two preimages in [S.sup.1] This being the case, we have

[Mathematical Expression Omitted]

for any partition 0 = [[theta].sub.1] < ... < [[theta].sub.m+1] = 2[pi] of [0, 2[pi] considered as [S.sup.1].

We need to make a definition. For z [element of] [gamma] let mr(z) denote the number of irreducible components of [gamma] \ ([gamma] [intersection] B(z,r)) whose closure contains z. The function [m.sub.r](z) is decreasing in r; let m(z) = [lim.sub.r] [arrow right] 0+ [m.sub.r](z). Note that any neighborhood basis of z could be used in this definition without changing the value of m(z). Hence if [[gamma].sub.n] is an arbitrary finite covering of [gamma] by balls of radius 1/n, together with a measurable assignment of each point of [gamma] to one ball containing it, we could define m[[gamma].sub.n](z) as the number of irreducible branches of B [intersection] ([gamma] [gamma]) adjacent to z, where B [element of] [[gamma].sub.n] is the ball assigned to z. We see that m[[gamma.sub.n] is measurable and since m = [lim.sub.n] [right arrow] [infinity] m[[gamma].sub.n], so is m.

We are now in a position to state the following lemma, which is the key to our result.

Lemma. With m, [gamma] as above, m [less than or equal to] 2 a. e. with respect to [[lambda].sup.1] [lambda].sup.1] [vertical bar] [gamma].

Proof. Suppose m = k, an integer (or +[infinity]) on a compact set C of positive length. C is countably 1-rectifiable [Fe, 3.2.29], so we can find a C-linear projection [pi] onto C with the following properties. [[lambda].sup.1] ([pi](C)) > 0 and [pi] is not constant on any component of [gamma] \ [gamma]. Given a compact planar subset X of finite linear measure, we say that two components [[omega].sub.1], and [[omega].sub.2] of the complement of X are amply adjacent if there exists a rectangle R = [a, b] x [c, d] in the plane such that the bottom edge lies in [[omega].sub.1], and the top edge lies in [[omega.sub.2] and such there exists a compact subset [K.sub.1] of [a,b] of positive measure such that vertical lines x = t for t in [K.sub.1], meet R [intersection] X in exactly one point. As t varies over [K.sub.1], these points of R [intersection of] X give a compact subset K of R which is mapped homeomorphically onto [K.sub.1] by projection to the x-axis. This construction is due to Alexander [All]: we have borrowed from a shortened version of this appearing in [A13].

In the case that X is also connected, we can also find two rectifiable arcs [J.sub.1], [J.sub.2] such that K [subset] [J.sub.i] [J.sub.i] \ K is a union of straight line segments and [J.sub.1] and [J.sub.2] connect {a} x [c, d] to {b} x [c, d]. This procedure is generic so that given any Borel set F of the form F = [pi] (C) such that [[lambda].sup.1] (F) > 0, we can find a compact subset E of F with [[lambda].sup.1] (E) > 0 such that E is the K for some rectangle R as described above. Also, R is chosen so that iRr n7r(T) consists of one point each in {a} x [c, d] and {b} x [c, d]; [J.sub.1], and [J.sub.2] are then chosen so that [J.sub.1] lies a bove [J.sub.2] and [pi](T) [intersection of] R lies between [J.sub.1] and [J.sub.2]. Denote by [[omega].sub.1] the region in R lying above [J.sub.1], and by [[omega].sub.2] the region lying below [J.sub.2]. Each [[omega].sub.i] is Jordan domain with rectifiable boundary. The usefulness of this construction lies in the fact that harmonic measure on [[omega].sub.1] is not singular to harmonic measure on [[omega].sub.2].

The map [pi] has multiplicity [l.sub.i] over [[omega].sub.i]. By a theorem of Bishop [All; We, Th. 10.7] over almost every point of [J.sub.i] lie exactly [l.sub.i] points of [[pi].sup.-1] ([[omega].sub.i)] [intersection] [gamma]. (This also follows from looking at [pi][vertical] ([gamma \ [gamma]) as a covering map over [[omega.sub.i] and looking at the Weierstrass polynomials.) Choose a point z [element of] K such that [[pi].sup.-1] contains [l.sub.i] points in [[pi].dup.-1] ([omega.sub.i]) [intersection] [gamma]; we choose z so that [[pi].sup.-1] (z) [intersection] [gamma] is finite.

Let [gamma] [intersection] [[pi].sup-1] (z) = {wi,..ws}, a certain finite set. Choose [delta] < 1/2 min [vertical] wi - wj]). Then for each wi the sets [[pi].sup.-1] B(z,r) [intersection] B ([w.sub.1], [delta])) [intersection] ([gamma] \ [gamma]).borhood basis for wi in [gamma]. One of the wi's, say wi's, say w1 is in C. For small r, let V be an irreducible component of ([[pi].sup.1] B (z,r) [intersection] B (wi, [delta])) ([gamma] \ [gamma]. V projects by [pi] to a connected open subset of B(z,r) whose boundary is contained in { [vertical] z - w [vertical] = r} [union] ([pi](E) [union] B (z,r)). If wi [element of] V, then z [element of] [pi](V). This forces [pi](V) [contains or equal to] [[omega].sub.2] [intersectio] B (z,r), since [pi](V) is connected. By construction B(w1, [delta]) intersects at most one sheet of [[pi].sup.1] ([[omega].sub.i]) for i = 1 or 2 and since r was arbitrary, we have m([w.sub.1]) = 0, 1 or 2. This proves the lemma.

Now for the proof of the theorem, we will show that m(z) = 1 or 2 implies {[f.sup.-1](z)} has 1 or 2 elements, respectively. (Clearly if m(z) {[f.sup.-1](z)} is empty.) Suppose m(z) = 1 and [f.sup.-1] (z) [contains or equal to] {z1, z2] z1 [is not equal to] z2. Choose r < 1/2 [vertical bar] z1 - z2 [vertical bar] so that f(w) [is not equal to] z for [vertical bar] zi - w [vertical bar] = r, w [element of] [S.sup.1]. Then f({ [vertical bar] zi - w [vertical bar] = r} [intersection] U) is at positive distance pi from z. Let p = min (P1, P2) and let [V.sub.p] be the single irreducible component of ([gamma] [gamma]) [intersection] B(z, p) whose closure contains z. Then f({[vertical bar] zi - w [vertical bar] < r} [intersection] U) [contains or equal to] V for i = 1, 2, which is a contradiction of the fact that f is one-to-one off a discrete set. The proof for m(z) = 2 is analogous. This completes the proof of Theorem 2.

3. Stokes's theorem. In [A14], Alexander proved that if [gamma] [subset or equal to] [C.sup.n] is a simple closed rectifiable curve, then [[lambda].sup.3] ([gamma] [gamma] < [infinity] . He then asked whether d[[gamma] \ [gamma] = [gamma], where [XI denotes the current defined by integration over X. We have answered this question affirmatively in the most general situation.

Theorem 3. If F [gamma] [subset or equal to] [C.sup.n] is compact, connnected and [[lambda].sup.1]([gamma]) < [infinity], then d [gamma] \ [gamma] is a rectifiable 1 -dimensional current with multiplicity 0 or l a. e. d[A 1 1 F. If F is also a simple closed curve, then d [[gamma]] \ [[gamma]] = [[gamma]] where [gamma] receives the orientation given in [Al4].

Before proving this theorem, some words of explanation are in order. The orientation of [gamma] \ [gamma] comes from its complex structure, while the orientation of F given by Alexander [A14] satisfies

[Mathematically expression omitted]

for every polynomial f and every p [element of] [C.sup.n] such that f(p) f([gamma]).

The proof of Stokes's theorem is an exercise in the theory of flat currents; see [Fe] for a thorough treatment of these. We denote the space of m-dimensional currents on an open set U [subset or equal to] [R.sup.n] by [D.sub.m] (U). The space of m-vectors (covectors) on [R.sup.n] will be denoted by [[lambda][.sub.m] ([R.sup.n]) ([[lambda].sup.m]([R.sup.n]). Recall that a k-rectifiable set M [subset or equal to] [R.sup.n] has a k-dimensional approximate tangent plane at [[lambda].sup.k] almost every point of M. Let w(x) be a k-vectorfield on M such that [vertical bar] w(x) [vertical bar] = 1 a.e. d[[lambda.sup.k]. A rectifiable current T supported on M is a current defined by integration over M with integer density, i.e., T(cts) = fmf(x) cts(x), w(x))da'(x) where f is an integer valued function. An integral current is a rectifiable current whose boundary is also rectifiable.

We also need the definition of a flat current and the slice of a flat current by a smooth map. If T is an m-dimensional current defined on an open set U [subset or equal to] [R.sup.n] then the mass of T, denoted by M(T) is defined by

M(T) = sup {[vertical]T ([phi])T[vertical bar] [phi] [element of] [D.sup.m](U), [parallel] [phi] ([chi]) [parallel] [less than or equal to] 1, [chi] [element of] U}.

We use here the comass norm on forms,

[parallel] [phi] [parallel] = sup{(xi, [phi]); [xi] is simple, [vertical bar] xi] [vertical bar] [less than or equal to] 1}

where [vertical bar] [vertical bar] is the norm on [[lambda].sub.m][R.sup.n] obtained by extension of the Euclidean inner product. If M(T) and M( T) are both finite, then T is said to be normal and we write T [element of] [N.sub.m,K](U). The norm of an element T [element of] [N.sub.m,K](U) is given by N(T) = M(T) + M( T). Now if K [sebset or equal to] U is compact and [phi] [element of] [D.sup.m](U), define the flat seminorm by

[Mathematically expression omitted]

For T [element of] [D.sub.m](U) we let [F.sub.K] [phi] [less than or equal to] 1) be the dual flat seminorm and note that if [F.sub.K](T) < [infinity], then supp(T) [subset or equal to] K. Next we define the set of flat m-dimensional currents with support in K by

[F.sub.m,K](U) = the [F.sub.K] closure of [N.sub.m,K](U) in [D.sup.m](U).

The following fact is useful, allowing us to show that integration over [gamma] \ [gamma] < defines a flat current once we know that [[lambda.sup.2] ([gamma] \ [gamma] < [infinity]:

[F.sub.m,K](U) [intersection] {T:M(T) < [infinity]} = the M closure of [N.sub.m,K](U) in [D.sup.m](U).

We also need to know that if T is flat and M(T) < [infinity] , then T L [phi] is flat for any form [phi] with bounded [parallel] T [parallel] measurable coefficients [Fe, p. 374].

We need the theory of slicing. If T [element of] [F.sub.m,K][U), U [subset or equal to] [R.sup.q] and f: U [arrow right] [R.sup.n] is a smooth map, m [greater than or equal to] n, then for almost all y [element of] [R.sup.n], the slice of T in [f.sup.1] (y) exists as an m - n dimensional flat current with the following properties:

(1) For every T [psi] [element of] [D.sup.m-n](U), [phi] [element of] C[[infity].sub.c] ([R.sup.n], [phi](y) <T,f,y> ([psi])d[[pound].sup.n](y)= [TL [f.sup.#]([phi] [conjuncion] [omega]) where [omega] is the volume form on [R.sup.n]. In other words, the action of TL [f.sup.#]([phi] [conjuction] [omega]) on m - n forms can be recovered by letting the slices of T act on forms, and then integrating with respect to the slicing variable.

(2) If m > n, then <T,f, y> = ( - 1)[.sup.n] <T,f, y> for almost all y [element of] [R.sup.n].

(3) Of course supp <T,f, y> [subset or equal to] supp (T [intersection] ([f.sup.-1](y)) where the slice exists.

The first step in the proof of Theorem 3 is to show that [gamma] [gamma] has finite area, so that integration over [gamma] \ [gamma] makes sense. It suffices to show that the area with multiplicities of the projections of f \ F onto C are finite [Sz, ch. 1]. Let [pi]: [C.sup.n] [right arrow] C be such a projection. We wish to show that

[Mathematically expression omitted] where the [omega]'s are the bounded components of [C.sup.n]\[pi](T) and [m.sub.j] is the multiplicity of [pi] [vertical bar] gamma]\[gamma] over [[omega].sub.j]. Let [T.sub.s], be the current [[sigma][infinity].sub.j=1] [m.sub.js]([[omega].sub.j] where [m.sub.js] = min ([m.sub.j],s). Then M([T.sub.s]) < [infinity]. We have

[Mathematical Expressions Omitted]

as s [right arrow] [infinity]. Alexander has proved in [A15] that for a connected planar set X of finite length, every point of X off a countable set lies in the boundary of exactly two components of the complement of X. Thus, we have that

[Mathematical Expressions Omitted]

so [alpha][T.sub.s] = [[sigma][infinity].sub.j=1] [m.sub.js][[alpha][omega].sub.j] with convergence in the mass topology.

At almost every point of [element of] [pi]([gamma]) with two adjacent domains [[omega].sub.j and [[omega].sub.k], the multiplicity

[Mathematical Expressions Omitted]

The last inequality is the crossing over the edges principle proved in [All]. For all n the isoperimetric inequality [Fe, p. 4081 gives

[Mathematical Expressions Omitted]

Therefore, setting T = [[pi].sub.#[[gamma]\[gamma]], we have

[Mathematical Expressions Omitted]

We prove Stokes's theorem by identifying the slices of [alpha]T, where T = [[gamma]\[gamma]]. We use this information to show that M([alpha]T) < [infinity], when we can use an argument of King to show that [alpha] is rectifiable.

Let V = [gamma]\[gamma]. Let l : [C.sup.n] [arrow right] R be a real linear functional on [C.sup.n] Without loss of generality we may assume that l(z) = [x.sub.1]. For almost all y [element of] R, [l.sup.-1] (y) [intersection] V consists of a finite union of real analytic arcs. To better understand what is happening, we look at the local picture in the projection of [gamma] into the [z.sub.1] plane. Let [pi] be that projection. Almost every point of [pi]([gamma]) has a rectangle R containing it with the properties listed in section 2.2. We retain the notation from that section. Let p [element of] K be such that [[pi].sup.-1](p) [intersection] [[pi].sup.-1] ([[omega].sub.i] [intersection] ([gamma]\[gamma] has exactly [m.sub.i] elements.

Near [[pi].sup.-1](p), the slice of V by l consists of s disjoint real analytic arcs (of finite length). If such an arc ends in [[pi].sup.-1] (p), then it lies over the segment above p or below p, but not both at once. For w [element of] [[pi].sup.-1](p), there are three possibilities: one arc of [l.sup.-1](x(p)) [intersection] could end at w; two arcs could end at w but their projections come at p from opposite directions; or no arcs could end at w. In the first case <[alpha]T,l,x(p))> [L.sub.x{w}] = [+ or -][{w}], in the other two cases, <[alpha]T,l,x(p)> [L.sub.x{w}] = 0.

To prove that [alpha]T has finite mass, we show that each [alpha]TL[dx.sub.i] has finite mass, i = 1,..., 2n. If [phi] [element of] [D.sup.0](U), then [alpha]T[Ldx.sub.i]([phi]) = <[alpha]T,l,y>[phi]dy [less than or equal to] [sup.sub.[gamma][absolute value][phi] [#l.sup.-1] (y) [intersection] [gamma] [less than or equal to] [[lambda].sub.1]([gamma]sup.[[phi]].

Thus [alpha]T has finite mass. The basic tool for proving Stokes's theorem is the support theorem [Fe, p. 372-373] which says that an m-dimensional flat current whose support is contained in an m-dimensional [C.sup.1] manifold (or m-dimesional Lipschitz retract) is given by integration against an [L.sup.1] weight. The support theorem does not apply directly here because a rectifiable set need not be a Lipschitz retract; however, using an argument of King [Ki, p. 218] we can overcome this difficulty once we know the mass of T is finite. Following King, we can write [alpha]T = [[element of][infinity].sub.i=1] [alpha]T L [K.sub.i], where the [K.sub.i]'s are disjoint compacta, each of which lies on a [C.sup.1] curve [[lambda].sup.1]([gamma]\[[union][infinity].sub.i=1] [K.sub.i))] = 0. Each [alpha]T L [K.sub.i] is flat [Fe, 4.1.17], therefore by the support theorem [Fe, p. 374], [alpha]TL[K.sub.i] = [[lambda].sup.1]L [K.sub.i][lambda] [eta](x), where [eta] n(x) is tangent to [K.sub.i] at x. We have proved now that [alpha]T = [[lambda].sup.1] L[gamma] [lambda] n(x), where n is a 1-vectorfield on F. Theorem 4.3.8 of [Fe] says that we can recover n from the slices of [alpha]T - since those slices have multiplicity 0 or 1 at points of [gamma], we are done.

If [gamma] is a simple closed curve, the proof consists of two steps: first, we show that m = 1 a.e. d[[lambda].sup.1] if [gamma] is a simple closed curve, and then we use a theorem on indecomposable 1-dimensional currents to show that [alpha][V] = [gamma]. We state the first step as a lemma.

Lemma. If [gamma] is a simple closed rectifiable curve and [Mathematical Expressions Omitted], then for every p [element of] [gamma] and for each neighborhood U of p, there is exactly one component [V.sub.0] of U [intersection] ([gamma]\[gamma]) such that [V.sub.0][intersection][gamma] contains an arc C with p [element of] C; for every other component W of U [intersection] [gamma]\[gamma], [[lambda].sup.1] (dW [intersection] C) = 0.

Proof. For any [epsilon] > 0, there is a [epsilon]', 0 < [epsilon]' < [epsilon], such that [gamma][intersection] [B.sub.[epsilon](p) consists of a finite number of components, one of which contains p. Thus, shrinking U if necessary, we may assume that U [intersection] [gamma] is an arc, which we shall call C. The first assertion then follows from the analytic continuation theorem proved below, which only depends on the general form of Stokes's theorem. The analytic continuation theorem we prove states that if the multiplicity of a 1-dimensional variety is two at almost every point of a rectifiable curve, then the variety is analytic across the curve. If there were two components of U [intersection][gamma]\[gamma] whose closure coincided along an arc, then the muitiplicity of V[intersection]U would be two along that arc, implying that [gamma] continued analytically through C, which is impossible. To finish the proof of the lemma, we need only show that if W is another component of U [intersection][gamma]\[gamma], [[lambda].sup.1] (W [intersection][gamma]) = 0. First, we observe that since W[intersection][gamma] is totally disconnected, theorem 3 of [Al4] applies; therefore, W [intersection] U is a variety. This being the case, we can easily see that the multiplicity of W at almost every point of C [intersection] W is equal to two. This implies (because [V.sub.0] [contains or equal to] C) that the multiplicity of [gamma] [subset or equal to] [gamma] is equal to three at almost every point of W [intersection] [gamma]. Since m = 2 a.e. d[[lambda].sup.1], this forces [[lambda].sup.1] (W [intersection] [gamma] = 0. This ends the proof of the lemma.

Fix an arbitrary orientation for [gamma]. We want to show that [alpha][V] = [+ or -][[gamma]]. Suppose that this is not the case. Consider the integral current I defined by the equation

[Mathematical Expressions Omitted]

Then I is a closed nonzero integral current whose mass is equal to the [[lambda].sup.1] measure of the set of points on F where its density is equal to one. The same is also true of I - [[gamma]]; thus

M([[gamma]]) = M(I) + M(I - [[gamma]]).

This contradicts the fact that integration around a rectifiable simple closed curve defines an indecomposable integral current [Fe, p. 420-421]. An integral current T is said to be indecomposable if there is no integral current S of the same dimension such that S [not equal to] 0 [not equal to] T - S and N(T) = N(S) + N(T - S). Since I and [[gamma]] are closed, N(I) = M(I) and N([[gamma]]) = M([[gamma]]). We have therefore shown that [alpha][V] = [[gamma]] for some orientation of [gamma].

We can use Stokes's theorem to prove an analytic continuation theorem for varieties defined near rectifiable curves. First we need a generalization of the proof that [[lambda].sup.2] ([gamma]\([gamma] < [infinity].

Lemma. Suppose U [subset or equal to] [C.sup.n] is an open set, [gamma] [subset or equal to] U is a rectifiable curve and V [subset or equal to] U \ [gamma] is a 1-dimensional variety. Then V has locally finite area in U.

Proof. We need only consider what happens near [gamma]. Let p [element of] [gamma] and choose a polysisc [delta], p [element of] [delta] [subset or equal to] U. We need only show that the area with multiplicity of [pi]([delta]intersection]V) is finite for each complex coordinate projection [pi]. If [pi]([delta])=[vertical bar]z-[pi](p)[vertical bar] < r we may assume, since [[lambda].sup.1] [pi]([gamma]) < [infinity] that {[vertical bar]z - [pi](p)[vertical bar] = r} [intersection] [pi]([gamma] is finite. As in the case of the polynomial hull, we look at the formal current T = [element of][m.sub.j][[omega].sub.j] whose formal boundary is [pi]([[gamma]]L[delta] + [T.sub.0] where [T.sub.0] is the part of [delta]T which lies on [vertical bar]z - [pi](p)[vertical bar] = r. We need only show that [T.sub.0] has finite mass; this corresponds to knowing that there are only finitely many branches of [delta] [intersection] V which miss [gamma]. Since {[vertical bar]z - [pi](p)[vertical bar] = r} [intersection] [pi][gamma] has finitely many components, [T.sub.0],hence also T, has finite mass and the proof concludes as before.

If C [subset or equal to] [C.sup.n] is a smooth curve and V is a 1-dimensional subvariety of [C.sup.n] which is two-sided near C, in the sense that near any point of C there are two components of V, then V is analytic across C [GS1]. This result motivates the following corollary.

Corollary 1. Suppose [gamma], V and U are as in the previous lemma and suppose also that m(z) = 2 a. e. on [gamma]F. Then V [intersection] U is a 1-dimensional subvariety of U.

(A weaker version of this was previously proved in [Xu].)

Proof. Since this is a local question, we may assume that [[lambda].sup.2](V) is finite because of the lemma. The proof of Stokes's theorem is the same here as in the case of the polynomial hull. Let T be the current defined in U by integration over V. Then since m = 2 a.e. on [gamma], [alpha]T = 0. By theorem [5.2.1] from of [Ki], we have that V [intersection] is a variety.

4. Density at the boundary and harmonic measure. We obtain here a partial result about the two dimensional density at the boundary of [gamma]; we end the chapter with a brief discussion of the problem of harmonic measure on [gamma].

Theorem 4. If [gamma] [C.sup.n] is a connected rectifiable set of finite length, then the density of [gamma] at a point p [element of] [gamma] is equal to 1/2 m(p) a. e. d[[lambda].sup.1].

By the density we mean the of the two-dimensional density-

[Mathematical Expressions Omitted]

We also will use the upper density, which is the lim sup of the above quantity. m(p) is the multiplicity function defined in section 2.

The answer we obtain is the best possible; it is what we would get if the hull were a smooth surface with boundary.

Proof. We can estimate most of the density by Wirtinger's inequality; we use the isoperimetric inequality to estimate the remainder. We tackle the estimation of the remainder first.

Let [pi] : [C.sup.n] [arrow right] C be a projection. Let T = [pi][[gamma]\[gamma] and choose a rectangle R as in section 2.2, with the same notation as in that section. We showed in the proof of Stokes's theorem that the density of [alpha]T is [l.sub.1] - [l.sub.2] at almost all points of K. Let p be such a point, and assume as well that K has an approximate tangent line at p and that p is a point of density for K. Since the density of [alpha]TLK is [l.sub.1] - [l.sub.2] at p, the density of aT L [K.sup.c] is 0 there. Choose [delta] small enough so that m([alpha]L K is [k.sup.c] [intersection] B(p,[epsilon]))) < [epsilon] for [epsilon] < [delta]. Then if U is a component of B(p,[epsilon]) [intersection] (C\[pi]([gamma]), U must be contained in a component of B(p,[epsilon]) [intersection] (C\([gamma]), U must be contained in a component of C \ [pi]([gamma] which is contained in B(p, 2[epsilon]). This fact, combined with the isoperimetric inequality, gives that M(T L (([[omega].sub.1] [intersection] [[omega].sub.2].sup.c] [intersection] B(p,[epsilon]))) [less than or equal to] CM([alpha]T L ([K.sup.c] [intersection] B(p,2[[epsilon))).sup.2] = 0([epsilon]).

Before we can find the density, we need to bound the upper density a.e. We retain the assumptions of the preceding paragraph and section 2.2. If the rectangle R is small enough, [[pi].sup.-1][[omega].sub.i]) [intersection] [gamma] consists of [l.sub.i] separate sheets. Recall that for the generic point z [element of] [gamma], with m(z) = t, [gamma] projects to only the upper or the lower half of the rectangle R if t = 1, and onto both upper and lower halves if t = 2. (If t = 0, then the density is obviously 0.) Let w be a point in [[pi].sup.-1]{p}[intersection] [gamma]. If [epsilon] is small enough, then [gamma]\[gamma] [intersection] [B.sub.[epsilon]](w) projects by [pi] in a one-to-one fashion over [[omega].sub.1] [intersection] [[omega].sub.2]. Hence the area with multiplicity of the projection of [gamma]\[gamma] [intersection] [B.sub.[epsilon]](w) is dominated by [[pi][epsilon].sup.2] plus the area with multiplicity of the part of the projection which does not lie in [[omega].sub.1] [union] [[omega].sub.2]. This contribution is negligible, by the preceding paragraph. By projecting onto the n coordinate axes, we see that the upper density is bounded by n a.e. d[[lambda].sup.1][vertical bar][gamma].

We have now shown that the upper density at the generic point of r is bounded; we now show that it is equal to 1/2 a.e. d[[lambda].sup.1]. Let us assume that p = 0. We denote by [T.sub.[delta]] the dilation of T by [delta]-i.e., T[delta]([phi]) = [1.[delta].sub.2T([phi][omicron][S.sub.[delta]], where S[delta](z) = [delta]z. By the compactness theorem (Th.2.2. 1, [Ki]), for every sequence [delta.sub.k], [delta.sub.k] [right arrow] 0, there is a subsequence [delta.sub.kj] such that [lim.sub.j,] [right arrow] [infinity] [T.sub.[delta]kj] [LB.sub.1] (0) exists as an integral current of dimension 2. By Bishop's theorem ( ) the support of each limit T' so obtained will be a 1-dimensional subvariety of [C.sup.n]\ l where l is the tangent to [gamma] at p. In each case, the boundary of T' will be [+ or -][l]. This implies that T' is the union of a half-line and a finite number of complex lines. Since p is assumed to be a generic point of multiplicity 1, the projection of [gamma]\[gamma] misses [[omega].sub.1] [[omega].sub.2]. Since [[omega].sub.1] and [[omega].sub.2] share a common tangent at 0 , it is clear that the support of T' does not contain a complex line. Hence [lim.sub.[delta]] [right arrow] 0 exists and is a half-line with multiplicity 1. This implies the desired result about the density of [gamma] at p.

It seems difficult to give interesting explicit examples of polynomial hulls of rectifiable curves, but we do have the following example: There is a rectifiable curve which is simple except for one self-intersection and whose hull has infinite genus. Let [r.sub.i] i = 1, 2,. . be an increasing sequence of positive numbers such that [lim.sub.i] [right arrow] [delta] [r.sub.i] = 1 and [r.sub.i] satisfies the Blaschke condition - [[sigma][infinity].sub.i=1(1-[r.sub.i] < [infinity]. Let B be the Blaschke product with zeros at the [r.sub.i]. We know that [vertical bar]B'(z)[vertical bar] < C/1(1 - [vertical bar]Z[vertical bar]).sup.2]). We define [gamma] = {(z,w) [element of] [C.sup.2] : [w.sup.2] = (1 - z).sup.4]B(z), [vertical bar]z[vertical bar] = 1}. The curve [gamma] has finite length; its polynomial hull must therefore be a variety. Since the boundary of the set {(z,w) [element of ] [C.sup.2]\[W.sub.2] = [(1 - z).sup.4]B(z), [vertical bar]z[vertical bar] < 1} is [gamma] it is clear that it must be the polynomial hull of [gamma]. The set [gamma]\[gamma] is a Riemann surface, which can be seen to have infinite genus because of the presence of infinitely many branch points. The hull of [gamma] has a nice property which is not known to be true in general: harmonic measure for a point in [gamma]\[gamma] is absolutely continuous with respect to [[lambda].sup.1] on [gamma]. This is easy to show: let [pi] denote projection onto the first coordinate in [C.sup.2]. Then if [omega] is harmonic measure for some point p [element of] [gamma]\[gamma], [pi]([omega]) is harmonic measure for [pi](p) on the disc. Since harmonic measure for a point in the disc is absolutely continuous with respect to [lambda.sup.1] on the circle and since [pi] [vertical value of] [gamma] is a two-to-one mapping over the circle, the absolute continuity of [omega] follows.

Given a rectifiable curve [gamma] whose hull is a Riemann surface which is regular for the Dirichlet problem, we would like to say that harmonic measure for a point p [element of ] [gamma]\[gamma] is absolutely continuous with respect to arclength on [gamma]; furthermore, it would be natural to hope that harmonic measure be given by integration of the normal derivative of the Green's function with respect to arclength on [gamma]. We are unable to prove this conjecture. We are able to say something about the fundamental domain which may shed some light on the problem.

Proposition. Let [gamma] be a rectifiable curve such that [gamma]\[gamma] is a Riemann surface which is regular for the Dirichlet problem. Let [phi] : [right arrow] [gamma]\[gamma] be a uniformizing map for [gamma]\[gamma] such that [phi](0) = p. Here U is the unit disc. Then the boundary of the normal fundamental domain for [phi] has positive length. Furthermore, if g denotes the Green's function for [gamma]\[gamma], then G = o [phi] has a radial derivative at almost every point of the fundamental domain.

See [Ts] for relevant definitions.

Proof. We know from the preceding sections that we can find a Jordan domain [omega] [subset or equal to] [gamma]\[gamma] and a projection [pi] such that [pi] projects [omega] in a one-to-one fashion onto a Jordan region [omega]' with rectifiable boundary in the plane so that [alpha] [omega] [intersection] [gamma] projects to a set of positive length in [omega]'. (Any [[omega].sub.i] from the construction outlined in section 2.2. will suffice.) This implies that [gamma] [intersection] [alpha][omegal has positive harmonic measure with respect to [omega]. Let D be any component of [phi.sup.D-1] ([omega]). Then D is a simply connected domain contained in U, and [alpha]D [intersection] [alpha]D [intersection] [alpha]U has positive harmonic measure with respect to D. The desired results follow immediately from Theorems 1 and 2 of [P2].

REFERENCES

[AL1] H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in [C.sup.n], Amer. J. Math. 93 (1971), 65-74.

[A12] _____, The polynomial hull of a set of finite linear measure in [C.sup.n], J. Analyse Math. 47, (1986), 238-242.

[A13] _____, Polynomial hulls and linear measure, Complex Analysis, II, (College Park, MD 1985-86), Lecture Notes in Math., vol. 1276, Springer-Verlag, New York, 1987, pp. 1-11.

[A14] _____, The polynomial hull of a rectifiable curve in [C.sup.n], Amer J. Math. 110 (1988), 629-641.

[A15] _____, Linear measure on plane continua of finite linear measure, Ark. Mat. 27 1989), no. 2, 169-177.

[Du] Peter Duren, Theory of Hp Spaces, Academic Press, New York, 1970.

[Fa] K. Falconer, The Geometry of Fractal Sets, Cambridge U. Press, 1985.

[Fe] H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.

[Fr] O. Frostman, Sur les produits de Blaschke, Kungl. Fysiogr Sallsk. i Lund Forh. 12 (1942), 169-182.

[Ga] T. Gamelin, Uniform Algebras, Prentice Hall, Englewood Cliffs, NJ, 1969.

[GS1] J. Globevnik and E. L. Stout, The ends of discs, Bull. Soc. Math. France, 114 (1986), 175-195.

[GS2] _____, Analytic discs with rectifiable simple curves as ends, Ann. Math. 127 (1988), 389-401.

[GR] R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice Hall, Englewood Cliffs, NJ, 1965.

[HW] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton, 1948.

[Ki] J. King, The currents defined by analytic varieties, Acta Math. 127 (1971), 185-220.

[P1] Ch. Pommerenke, On the Green's function of Fuchsian groups, Ann. Acad. Sci. Fenn. Ser AI Math. 2 1976), 409-427.

[P2] On analytic functions with cluster sets of finite linear measure, Michigan Math. J. 34 (1987), 93-99.

[Sz] G. Stolzenberg, Volumes, limits, and extensions of analytic varieties, Springer-Verlag, New York, 1966.

[St] E. L. Stout, Bounded extension: The case of discs in polydiscs, J. Analyse Math. 27 (1975), 239-254. ITs] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen Co., Tokyo, 1959.

[Xu] Y. Xu, Continuation of analytic varieties across rectifiable sets, preprint.

[We] J. Wermer, Banach Algebras and Several Complex Variables, Springer-Verlag, New York, 1976.
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