Analytic continuation in

several complex variables

An M.S. Thesis

Submitted, in partial fulfillment

of the requirements for the award of the

degree of

Master of Science

in the Faculty of Science,

by

Chandan Biswas

Department of MathematicsIndian Institute of Science

Bangalore - 560012

April, 2012

DeclarationI hereby declare that the work in this thesis has been carried out by me in the Integrated

Ph.D. program under the supervision of Professor Gautam Bharali, and in partial fulfillment ofthe requirements of the Master of Science degree of the Indian Institute of Science, Bangalore.I further declare that this work has not been the basis for the award of any degree, diploma,fellowship or any other title elsewhere.

Chandan BiswasS. R. No. 9310-310-091-07278

Indian Institute of Science,Bangalore,April, 2012.

Professor Gautam Bharali(Thesis adviser)

AcknowledgementsIt is my pleasure to thank Professor Gautam Bharali for many valuable suggestions, interest-

ing discussions and his help in the preparation of this thesis, especially for the infinite patiencethat he has shown in correcting the mistakes in my writings. I must also thank Professor KaushalVerma for the many discussions he had with me during my coursework. Finally, I wish to thankPurvi Gupta and Jonathan Fernandes for friendly advice and tips in the course of typing thesis.

AbstractWe wish to study those domains in Cn, for n ≥ 2, the so-called domains of holomorphy,

which are in some sense the maximal domains of existence of the holomorphic functions definedon them. We demonstrate that this study is radically different from that of domains in C bydiscussing some examples of special types of domains in Cn, n ≥ 2, such that every functionholomorphic on them extends to strictly larger domains. Given a domain in Cn, n ≥ 2, we wishto construct the maximal domain of existence for the holomorphic functions defined on the givendomain. This leads to Thullen’s construction of a domain (not necessarily in Cn) spread over Cn,the so-called envelope of holomorphy, which fulfills our criteria. Unfortunately this turns out tobe a very abstract space, far from giving us a sense in general how a domain sitting in Cn canbe constructed which is strictly larger than the given domain and such that all the holomorphicfunctions defined on the given domain extend to it. But with the help of this abstract approachwe can give a characterization of the domains of holomorphy in Cn, n ≥ 2.

The aforementioned characterization is as follows: a domain inCn is a domain of holomorphyif and only if it is holomorphically convex. However, holomorphic convexity is a very difficultproperty to check. This calls for other (equivalent) criteria for a domain in Cn, n ≥ 2, to be adomain of holomorphy. We survey these criteria. The proof of the equivalence of several ofthese criteria are very technical – requiring methods coming from partial differential equations.We provide those proofs that rely on the first part of our survey: namely, on analytic continuationtheorems.

If a domain Ω ⊂ Cn, n ≥ 2, is not a domain of holomorphy, we would still like to explic-itly describe a domain strictly larger than Ω to which all functions holomorphic on Ω continueanalytically. Aspects of Thullen’s approach are also useful in the quest to construct an explicitstrictly larger domain in Cn with the property stated above. The tool used most often in such con-structions is called “Kontinuitatssatz”. It has been invoked, without a clear statement, in manyworks on analytic continuation. The basic (unstated) principle that seems to be in use in theseworks appears to be a folk theorem. We provide a precise statement of this folk Kontinuitatssatzand give a proof of it.

7

Contents

1 Introduction and basic results 11

2 The Hartogs phenomenon 172.1 Differentiating under the integral sign . . . . . . . . . . . . . . . . . . . . . . . 172.2 The ∂-problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 The Hartogs continuation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Envelopes and domains of holomorphy 233.1 Envelopes of holomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Domains of holomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 More characterizations of domains of holomorphy in Cn 314.1 Further characterizations of domains of holomorphy . . . . . . . . . . . . . . . . 314.2 Domains of holomorphy that have smooth boundary . . . . . . . . . . . . . . . . 35

5 A schlichtness theorem for envelopes of holomorphy 385.1 A schlichtness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 A simple Kontinuitatssatz 426.1 Recalling notations and terminology . . . . . . . . . . . . . . . . . . . . . . . . 446.2 A fundamental proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.3 The proof of the main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

9

Chapter 1

Introduction and basic results

The initial part of this chapter will be devoted to understanding what it means for a functionf : Ω → C, where Ω is an open subset of Cn, n ≥ 2, to be holomorphic. For this purpose, let usintroduce some convenient notations (which we shall follow in the later chapters also).

• We write z ∈ Cn as z := (z1, z2, . . . , zn) and further, z j = x j + iy j , j = 1, 2, . . . , n.

• Given α := (α1, α2, . . . , αn) ∈ Nn, and for any z ∈ Cn, we set

|α| := α1 + · · · + αn, α! := α1! . . . αn!, zα := zα11 zα2

2 . . . zαnn .

• We define the differential operators

∂

∂z j:=

12

(∂

∂x j− i

∂

∂y j

),

∂

∂z j:=

12

(∂

∂x j+ i

∂

∂y j

),

Dα :=∂|α|

∂z1α1 . . . ∂zn

αn.

• Given a point a ∈ Cn and r = (r1, . . . , rn) ∈ Rn+, the set

D(a, r) := z ∈ Cn : |z j − a j| < r j, j = 1, . . . , n

is called a polydisc with centre at a and polyradius r. We will have the occasion where allthe r j’s are equal. In that case we will use the notation Dn(a, r) := D(a, (r, . . . , r)), wherer > 0.

Here are four plausible definitions for holomorphicity, each of which gives us the usual defi-nition of holomorphicity when n = 1. Let Ω below be a domain (open connected set) in Cn.

11

Definition 1.1. A function f : Ω → C is holomorphic if for each j = 1, . . . , n, and each fixedz1, . . . , z j−1, z j+1, . . . , zn, the function

ζ 7−→ f (z1, . . . , z j−1, ζ, z j+1, . . . , zn)

is holomorphic in the classical one variable sense on the set

Ω(z1, . . . , z j−1, z j+1, . . . , zn) := ζ ∈ C : (z1, . . . , z j−1, ζ, z j+1, . . . , zn) ∈ Ω.

Definition 1.2. A function f : Ω → C that is continuously (real) differentiable with respect toeach pair of variables (x j, y j) separately on Ω is said to be holomorphic if f satisfies the Cauchy-Riemann equations in each variable separately.

Definition 1.3. Let f : Ω→ C be continuous in each variable separately and locally bounded.Thefunction f is said to be holomorphic if for each w ∈ Ω there is an r = r(w) > 0 such thatDn(w, r) ⊂ Ω and

f (z) =1

(2πi)n

∮|ζn−wn |=r

. . .

∮|ζ1−w1 |=r

f (ζ1, . . . , ζn)(ζ1 − z1) . . . (ζn − zn)

dζ1 . . . dζn

for all z ∈ Dn(w, r).

Definition 1.4. A function f : Ω → C is holomorphic if for each a ∈ Ω there is an r = r(a) > 0such that Dn(a, r) ⊂ Ω and f can be written as an absolutely convergent power series

f (z) =∑α∈Nn

aα(z − a)α for all z ∈ Dn(a, r),

that converges uniformly on compact subsets of Dn(a, r).

It turns out that all four definitions are equivalent.To see this let us provisionally assume thatf : Ω → C is continuous. That 1.1 ⇒ 1.2 is trivial. Showing that 1.2 ⇒ 1.3 involves iteratingthe one variable Cauchy Integral Formula. To carry out this argument we must worry about theorder of integration. Also we need to know, for instance that the functions

t ∈ C : |t − wk| = r 3 t 7−→ f (ζ(k), t, zk+1, . . . , zn)

are integrable for each fixed ζ(k) ∈∏

1≤ j≤k−1

∂D1(w j, r) and (zk+1, . . . , zn) ∈ Dn−k((wk+1, . . . ,wn), r),

k = 1, 2, . . . , n − 1. We do not have to worry about these points because we have taken f to becontinuous on Ω. Now 1.3 ⇒ 1.4 using the same trick of expanding out a geometric series thatwe use in one complex variable. Finally, 1.4 ⇒ 1.1 because uniform convergence allows us toevaluate the C-derivative with respect to a given z j, j = 1, 2, . . . , n, term by term.

Just as Goursat showed that a univariate function admitting all first order partial derivatives(not necessarily continuous) and satisfying the Cauchy-Riemann equation is automatically C-differentiable, it can be shown that the a priori assumption that f is continuous is superflu-ous.This was shown by Hartogs.We direct the reader to [ [10]][chapter 3] for a proof of Hartogs’result.

12

We must mention that many fundamental results from the one-variable theory, e.g. theCauchy Integral Formula, Liouville’s Theorem, the Open Mapping Theorem, Weierstrass’ Theo-rem, Montel’s Theorem, etc., have obvious generalizations to several variables. Their proofs, inmost cases, involve subscripts by multi-indices in suitable places and iterating the one-variableargument, so we will skip these proofs. One theorem that has a slightly different form in higherdimensions is the principle of analytic continuation (which we shall use in proving the theoremspresented below).

Theorem 1.5 (Principle of analytic continuation). Let Ω be a domain in Cn and let f be holo-morphic on Ω. If f vanishes on a non-empty open subset of Ω then f ≡ 0.

The second part of this chapter focuses on how, in contrast to the above-named theorems,remarkably different the behaviour of holomorphic functions in multiple variables can be fromthe one-variable case. For instance, it is easy to describe domains with the property that everyfunction holomorphic on them extend holomorphically to a strictly larger domain. It is less easyto describe completely the larger domain to which the above functions extend holomorphically.But we can provide a complete description for an important special class of domains. These arethe motivations for the following results.

Theorem 1.6. Let V be a neighbourhood of ∂Dn(0, 1), n ≥ 2 such that V∩Dn(0, 1) is connected.Then, for any holomorphic function f on V, there is a holomorphic function F on Dn(0, 1)∪V sothat F|V ≡ f .

Proof. Let ε > 0 be such that if

A := z ∈ Cn : 1 − ε < |z1| < 1, |z j| < 1, j ≥ 2 ∪ z ∈ Cn : 1 − ε < |z2| < 1, |z j| < 1, j , 2,

A ⊆ V. Existence of such a ε is guaranteed by the compactness of ∂Dn(0, 1). Let z′ = (z2, . . . , zn).We note here that in this proof, and elsewhere in this report, if v = (v1, . . . , vn) is a vector,then we shall, by a slight abuse of notation, denote |v| := max j≤n |v j| (to distinguish it from theEuclidean norm ‖ · ‖). If |z′| < 1, the function z1 7−→ f (z1, z′) is holomorphic on the annulusz1 ∈ C : 1 − ε < |z1| < 1, so that

f (z1, z′) =∑ν∈Z

aν(z′)zν1,

for any z′ ∈ Dn−1(0, 1). For each ν ∈ Z

aν(z′) =1

2πi

∫|z1 |=1

f (z1, z′)zν+1

1

dz1, z′ ∈ Dn−1(0, 1).

Differentiating under the integral sign, we get

∂aν∂z j

(z′) =1

2πi

∫|z1 |=1

1zν+1

1

∂ f∂z j

(z1, z′) dz1 = 0, j = 2, 3, . . . , n,

13

for each z′ ∈ Dn−1(0, 1). Of course one must justify whether we can validly differentiate underthe integral sign above. We shall give a criterion for being able to do this in the first part ofChapter 2. In view of Definition 1.2, aν(z′) is holomorphic in Dn−1(0, 1) for each ν ∈ Z. Ifnow 1 − ε < |z2| < 1, |z3| < 1, . . . , |zn| < 1, the function z1 7−→ f (z1, z′) is holomorphic inthe disc |z1| < 1, so that its Laurent series contains no negative powers of z1; i.e. aν(z′) = 0for ν < 0. As Dn−1(0, 1) is connected, by the principle of analytic continuation aν(z′) = 0 forν < 0, z′ ∈ Dn−1(0, 1). We define

F(z) :=

f (z), z ∈ V,∑ν∈N aν(z′)zν1, z ∈ Dn(0, 1).

This latter series converges uniformly on compact subsets of Dn(0, 1) and so is holomorphicthere; further, it coincides with f on z ∈ Cn : 1 − ε < |z1| < 1, |z j| < 1, j ≥ 2, a nonempty opensubset of V ∩Dn(0, 1) and so on the whole of V ∩Dn(0, 1), as this set is open and connected.

We need a definition in order to formulate our next theorem.

Definition 1.7. Let Ω be a domain in Cn. We say that Ω is a Reinhardt domain if wheneverz = (z1, z2, . . . , zn) ∈ Ω and θ1, θ2, . . . , θn ∈ R, we have (eiθ1z1, eiθ2z2, . . . , eiθnzn) ∈ Ω.

Theorem 1.8. Let Ω be a Reinhardt domain in Cn. Then for any holomorphic function f on Ω,there is a “Laurent series” ∑

α∈Zn

aαzα

which converges uniformly to f on compact subsets of Ω. Moreover, the aα’s are uniquely deter-mined by f .

Proof. We begin by proving the uniqueness. Let w ∈ Ω be a point with coordinates (w1, . . . ,wn),w j ,0 for all j. Then, since the series converges uniformly to f on compact subsets of Ω, we may setz j = w jeiθ j , multiply by e−i(α1θ1+α2θ2+···+αnθn) and integrate term by term. Now we observe thatπ∫−π

einθ dθ =

0, if n , 0,2π, if n = 0. This gives, for α ∈ Zn,

aα = w−α(2π)−n

π∫−π

· · ·

π∫−π

f (w1eiθ1 ,w2eiθ2 , . . . ,wneiθn)e−i(α1θ1+α2θ2+···+αnθn)dθ1 . . . dθn.

It seems that aα depends on w, so that the coefficient aα is not unique. So suppose f (z) =∑α∈Nn bαzα be a different representation of f in Ω. Then 0 =

∑α∈Zn(aα − bα)zα, z ∈ Ω. Then,

by the previous formula :

aα − bα = w−α(2π)−n

π∫−π

· · ·

π∫−π

0 e−i(α1θ1+α2θ2+···+αnθn)dθ1 . . . dθn = 0.

To prove the existence of an expansion as above, we first remark that if D = z ∈ Cn : r j < |z j| <R j ,−∞ < r j < R j , j = 1, 2, . . . , n and f is holomorphic on D, then, by iteration of the Laurent

14

expansion of one complex variable functions, it follows that f has an expansion in a Laurentseries. Let w ∈ Ω. If ε(w) > 0 is small enough, since Ω is a Reinhardt domain, the set

D(w, ε(w)) = z ∈ Cn : |w j| − ε(w) < |z j| < |w j| + ε(w)

is contained in Ω. Since this is a set of the form D above, there is a Laurent expansion∑α∈Zn

aα(w)zα = f (z), z ∈ D(w, ε(w)),

converges to f uniformly in a neighbourhood of w. Now, if w1 ∈ D(w, ε(w)) and∑α∈Zn aα(w1)zα

is the expansion corresponding to w1 in a set D(w1, ε(w1)) ⊆ Ω, then the uniqueness assertionabove shows that aα(w) = aα(w1). Hence the function w 7−→ aα(w) is locally constant on Ω forany α ∈ Zn and since Ω is connected, aα(w) ≡ aα, independent of w. Hence any point z ∈ Ω hasa neighbourhood N(z) ⊆ Ω so that the series∑

α∈Zn

aαzα

converges uniformly to f (z) on N(z), hence uniformly to f (z) for z in any compact subset ofΩ.

Corollary 1.9. Let Ω be a Reinhardt domain such that for each j , 1 ≤ j ≤ n, there is a pointz ∈ Ω whose j-th coordinate is 0. Then any holomorphic function f on Ω admits an expansion

f (z) =∑α∈Nn

aαzα

which converges uniformly to f on compact subsets of Ω.

Proof. For n = 1 as the domain Ω is a Reinhardt domain in C containing 0, Ω = D1(0, r) forsome r > 0 or Ω = C. Therefore in the Laurent series expansion of f all the coefficients withnegative index vanish. Hence we get the result. For n ≥ 2, let w1 represent an element in Ω

whose first coordinate is 0. So w1 = (0,w2 . . . ,wn) ∈ Ω. As Ω is a Reinhardt domain, there is aε > 0 such that D(w1, ε) ⊆ Ω, where

D(w1, ε) := D1(0, ε) × Ann(0, |w2| − ε, |w2| + ε) × · · · × Ann(0, |wn| − ε, |wn| + ε).

Fix z′ = (z2, z3, . . . , zn) ∈ Ann(0, |w2|−ε, |w2|+ε)×· · ·×Ann(0, |wn|−ε, |wn|+ε). Then the functionz1 7−→ f (z1, z′) is holomorphic on D1(0, ε). Using Theorem 1.8

f (z) = f (z1, z′) =∑α1∈Z

∑α′∈Zn−1

a(α1,α′)z′α′

z1α1 , z1 ∈ D1(0, ε).

But as z1 7−→ f (z1, z′) is holomorphic on D1(0, ε),∑α′∈Zn−1

a(α1,α′) z′α′

= 0, α1 < 0 and z′ ∈ Ann(0, |w2| − ε, |w2| + ε) × · · · × Ann(0, |wn| − ε, |wn| + ε).

15

Ann(|w2| − ε, |w2| + ε) × · · · × Ann(|wn| − ε, |wn| + ε) is a Reinhardt domain in Cn−1. Hence fromthe uniqueness assertion in Theorem 1.8

a(α1, α′) = 0, for all α ∈ Zn : α1 < 0.

Similarly we can show that a(α1,α2,...,αn) = 0, if α j < 0 for some j . Hence f (z) =∑α∈Nn aαzα.

Corollary 1.10. Let Ω be a Reinhardt domain such that for each j , 1 ≤ j ≤ n, there is apoint z ∈ Ω whose j-th coordinate is 0. Then any holomorphic function f on Ω has an uniqueholomorphic extension F to the set Ω′ := (λ1z1, λ2z2, . . . , λnzn) : 0 ≤ λ j ≤ 1, (z1, z2, . . . , zn) ∈ Ω.

Proof. By Corollary 1.9, f (z) =∑α∈Nn aαzα for z ∈ Ω. Now let z = (λ1z1, λ2z2, . . . , λnzn) ∈ Ω′,

where for all j, λ j ∈ [0, 1] and (z1, z2, . . . , zn) ∈ Ω.As the series∑α∈Nn aαzα converges at the point

(z1, z2, . . . , zn), by Abel’s Lemma the series also converges at the point z = (λ1z1, λ2z2, . . . , λnzn).Therefore

F(z) :=∑α∈Nn

aαzα, z ∈ Ω′

is a well defined holomorphic function on Ω′. Clearly F extends f since Ω ⊆ Ω′. Now 0 ∈ Ω′.Let z = (z1, z2, . . . , zn) ∈ Ω′. Then λz = (λz1, λz2, . . . , λzn) ∈ Ω′ for all λ ∈ [0, 1] by definition ofΩ′. Therefore Ω′ is connected. Now Ω is a nonempty open subset of Ω′. Hence by the principleof analytic continuation if there exists another holomorphic function G on Ω′ such that it alsoextends f then F(z) = G(z) for all z ∈ Ω′.

16

Chapter 2

The Hartogs phenomenon

We saw in Chapter 1 that we can easily give examples of domains in Cn, n ≥ 2, with the propertythat every function holomorphic on them extends holomorphically to a strictly larger domain.However, the proof of the above fact depended on the domains having a lot of symmetry. All thedomains in Theorem 1.8 of Chapter 1, and in its corollaries, are Reinhardt. However, Hartogsshowed that the phenomenon demonstrated by Theorem 1.6 holds true for arbitrary domains inCn, n ≥ 2. This will be demonstrated in Section 2.3 below. But first, we will need to go througha few technicalities.

2.1 Differentiating under the integral sign

Many theorems in Complex Analysis involve differentiating under the integral sign. However,this needs a justification. In many cases, the classical Leibniz Theorem does not suffice. Thefollowing proposition provides sufficient conditions that are general enough to be usable in manycontexts.

Proposition 2.1. Let K : Rn \ 0 → C be a measurable function that satisfies limx→0|K(x)| =+∞, but which diverges sufficiently slowly at 0 ∈ Rn that K ∈ L1

loc(Rn).

1 Suppose f : Rn × Rn → R is a continuous function having the following two properties:

• There exists R > 0 such that, for each x ∈ Rn, f (x, y) = 0 for all y ∈ Rn \ B(x,R);

• The family f (., y) : y ∈ Rn is equicontinuous at each x ∈ Rn.

Then, the function

g(x) :=∫Rn

K(y) f (x, y) dV(y) (2.1)

is well-defined and continuous.

17

2 Now suppose that, given N ≥ 1, ∂α f∂xα exists for all α ∈ Nn such that |α| ≤ N. Assume,

furthermore, that the families∂α f∂xα (., y) : y ∈ Rn

are equicontinuous at each x ∈ Rn, |α| ≤

N.

Then, g ∈ CN(Rn), and we can, in fact, differentiate under integral sign.

Proof. Observe that, by hypothesis:∫Rn

K(y) f (x, y) dV(y) =

∫B(x,R)

K(y) f (x, y) dV(y).

The integral on the right-hand side is convergent and hence g is well defined.Now note; for any x0 ∈ R

n:

|g(x) − g(x0)| ≤∫Rn|K(y)| | f (x, y) − f (x0, y)| dV(y). (2.2)

By the equicontinuity hypothesis, given ε > 0, there exists δ1 ≡ δ1(x0, ε) > 0 such that |x−x0| < δ1

implies | f (x, y) − f (x0, y)| < ε for all y ∈ Rn. Hence, by (2.2),

|g(x) − g(x0)| ≤∫B(x0,R+1)

|K(y)| | f (x, y) − f (x0, y)| dV(y) ≤ ε ‖K‖L1(B(x0,R+1)),

whenever |x − x0| < min(1, δ1(x0, ε)).This establishes continuity and hence part 1.For clarity of explanation, we shall carry out the proof of part 2 in the following distinct steps:Step 1: Illustrating a special case.Pick a j ≤ n. Then, from the first condition in part 1, we see that the analogue of that condition

is true for ∂ f∂x j

also. Hence, the integral

I j (x) :=∫Rn

K(y)∂ f∂x j

(x, y) dV(y) (2.3)

is well defined. For any t ∈ R \ 0,∣∣∣∣∣ g(x0 + te j) − g(x0)t

− I j (x0)∣∣∣∣∣ =

∣∣∣∣∣ ∫Rn

K(y)

f (x0 + te j, y) − f (x0, y)t

−∂ f∂x j

(x0, y)

dV(y)∣∣∣∣∣

=

∣∣∣∣∣ ∫RnK(y)

∫ 1

0

[∂ f∂x j

(x0 + ste j, y) −∂ f∂x j

(x0, y)]

ds

dV(y)∣∣∣∣∣.(2.4)

Here e j denotes the j-th vector of the standard basis of Rn. The last equation is obtained byapplying the Fundamental Theorem of Calculus to the univariate functions

s 7→ f (x0 + ste j, y).

18

Once again, by the equicontinuity hypothesis, given any ε > 0 we can find a δ2 ≡ δ2(x0, ε) such

that |t| < δ2 =⇒

∣∣∣∣∣ ∂ f∂x j

(x0 + ste j, y) − ∂ f∂x j

(x0, y)∣∣∣∣∣ < ε for all y ∈ Rn and for all s ∈ [0, 1].

Applying the above inequality to (2.4) gives us∣∣∣∣∣ g(x0 + te j) − g(x0)t

− I j(x0)∣∣∣∣∣ ≤ ε ‖K‖ L1(B(x0,R+1)),

whenever 0 < |t| < min(1, δ2(x0, ε)).We have established that, for each j ≤ n :

∂g∂x j

(x0) =∂

∂x j

[∫Rn

K(y) f (x0, y) dV(y)]

=

∫Rn

K(y)∂ f∂x j

(x0, y) dV(y).

The continuity of ∂g∂x j

follows by repeating the proof of part 1 with ∂g∂x j

replacing f in (2.1). Henceg ∈ C1(Rn).

Step 2: Completing the proof.We use induction to complete the proof. We are given N ≥ 1. Assume we have the desired

result for all α ∈ Nn such that |α| ≤ m for some m < N. Now we notice that step 1 was the basecase of the induction. Given an α ∈ Nn such that |α| = m + 1, there exists β such that |β| = m anda j ≤ n such that

(α1,α2, . . . ,αn) = (β1, . . . , β j−1, β j + 1, β j+1, . . . , βn) =: 〈 β, j 〉.

We now repeat the argument in step 1 with the following replacements:

•∂β f∂xβ replacing f in the relevant places ;

• ∂∂x j

(∂β f∂xβ

)replacing ∂ f

∂x jin the definition of I j in (2.3) ;

which establishes the result for all α ∈ Nn such that |α| = m + 1. By induction, g ∈ CN(Rn).

2.2 The ∂-problem

Let D be a domain in Cn. For n ≥ 2, the ∂ -problem denotes the following system of PDE’s:

∂u∂z j

= f j on D j = 1, . . . , n, (2.5)

subject to:

∂ f j

∂zk=∂ fk

∂z jfor all j , k, (2.6)

19

where f1, . . . , fn ∈ L1loc(D). When n = 1, the compatibility condition (2.6) is vacuous, and the ∂

-problem is just the PDE (2.5). We note that if the f j’s are not smooth, then (2.6) is interpreted inthe sense of distributions. We must demand (2.6) when n ≥ 2 because it is a necessary conditionfor (2.5) to have a smooth solution. To see this, first assume that the f j’s are of class C1. If asmooth solution u exists, then as it satisfies the PDE (2.5) we have for each j, ∂u

∂z j= f j. As u is a

smooth function, therefore all the mixed second-order partial derivatives of u exist and they areindependent of the order of the variables. In particular as f j ∈ C1, (2.6) is satisfied. The sameargument can be made in the weak sense if the f j’s are not smooth.

Let us provide a solution to the ∂-problem when n = 1, D = C and the right-hand side is inCk(C) and has compact support. For this, we need an elementary fact from Advanced Calculus.

Theorem 2.2 ( The Generalized Cauchy Integral Formula). Let Ω be a bounded domain in Csuch that ∂Ω is a disjoint union of finitely many piecewise smooth simple closed curves. Let Ube a neighbourhood of Ω and let f : U → C be of class C1(U). Then for any w ∈ Ω,

f (w) =1

2πi

∮∂Ω

f (z)z − w

dz −1π

"Ω

∂ f∂z

1z − w

dA(z) .

Theorem 2.3. Let k ≥ 1 and let φ ∈ Ck(C), and suppose φ is compactly supported. Then , thefunction

u(z) := −1π

"C

φ(w)w − z

dA(w)

satisfies the PDE∂u∂z

= φ.

Furthermore, u ∈ Ck(C).

Proof. Since φ is compactly supported, there exists r0 > 0 such that supp(φ) ⊂ B(0; r0). Now,given z ∈ C, let R > r0 and such that z ∈ B(0,R). By Theorem 2.2

φ(z) = −1π

"B(0,R)

1w − z

∂φ

∂z(w) dA(w) = −

1π

"C

1w − z

∂φ

∂z(w) dA(w). (2.7)

Now, from (2.7), and the fact that R can be arbitrarily large, the last integral represents φ(z) forall z ∈ C. Now we observe that, by a change of variable

u(z) = −1π

"C

1wφ(w + z) dA(w).

We observe that if we set

K(w) : =1w, w ∈ C \ 0,

f (z,w) : = φ(w + z),

20

then the hypotheses of Proposition 2.1, are satisfied. Thus, u ∈ Ck(C), and we can differentiateunder the integral sign to get

∂u∂z

(z) = −1π

"C

1w∂ f∂z

(z,w) dA(w)

= −1π

"C

1w − z

∂φ

∂z(w) dA(w) [ change of variable ]

= φ(z) for all z ∈ C. [ by (2.7) ]

The above plays a very important part in the solution of the ∂ -problem in Cn, n ≥ 2, withcompactly-supported data.

Theorem 2.4. Let f j ∈ C0k(Cn), j=1,. . . ,n, where k > 0, and assume that (2.6) is fulfilled (n > 1).

Then there is a function u ∈ C0k(Cn) satisfying (2.5).

Now note that the part about u having compact support in this theorem is false for n = 1 (takean arbitrary f1 ∈ C0

∞ with Lebesgue integral different from 0).

Proof. We set

u(z) := −1π

"C

f1(w, z2, . . . , zn)w − z1

dA(w) =1π

"C

f1(z1 − w, z2, . . . , zn)w

dA(w).

The second form of the definition shows that we can apply Proposition 2.1 to get that u ∈ Ck(Cn).Now u(z) = 0 if |z2| + · · · + |zn| is large enough. By Theorem 2.3 it follows that ∂u

∂z1= f1. If k > 1,

by differentiating under the sign of integration and using the fact that ∂ f1∂zk

=∂ fk∂z1, we obtain

∂u∂zk

(z) = −1π

"1

w − z1

∂ fk

∂z1(w, z2, . . . , zn) dA(w) = fk(z).

Hence u satisfies all the equations in (2.5), which means in particular u is analytic outside acompact set. As this complement is connected, and u(z) = 0 when |z2| + · · · + |zn| is large, wededuce from the principle of analytic continuation, that u has compact support.

2.3 The Hartogs continuation theoremNow we have sufficient background to prove a theorem that is nowadays called “Hartogs phe-nomenon.” Clearly, as mentioned in the beginning of Chapter 2, this theorem gives a generaliza-tion of Theorem 1.6.

Notation: H(Ω) denotes the set of all holomorphic functions from Ω into C.

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Theorem 2.5. Let Ω be an open set in Cn, n > 1, and let K be a compact subset of Ω such thatΩ \ K is connected. For every u ∈ H

(Ω \ K

)one can then find U ∈ H(Ω) so that u = U in Ω \ K.

Proof. Using the partition of unity we construct a function ϕ ∈ C∞0 (Ω), such that ϕ is identicallyequal to 1 in a neighbourhood of K. Now we define

u0(z) :=

0 if z ∈ K,(1 − ϕ(z))u(z) if z ∈ Ω \ K.

Then u0 ∈ C∞(Ω), and we want to find v ∈ C∞(Cn), so that

U := u0 − v

is holomorphic on Ω. Now the U will be analytic if ∂U = 0, i.e.

∂v = ∂u0 = −u ∂ϕ = f , (2.8)

where f is defined by

f (z) :=

0 if z ∈ K,−u(z) ∂ϕ(z) if z ∈ Ω \ K,0 if z ∈ Cn \Ω .

(2.9)

Now we note that as ϕ is compactly supported the components of f are in C∞0 (Cn). Hence, if wewrite f =

∑nj=1 f j dz j, by (2.9) we have

∂ f j

∂zk=∂ fk

∂z jfor all j , k.

Therefore by Theorem 2.4, (2.8) has a C∞0 (Cn) solution. Furthermore, a careful look at the proofof Theorem 2.4 tells us that the solution given by the formula in Theorem 2.4 vanishes in theunbounded component of the complement of the support of ϕ. Now the boundary of this setbelongs to Ω \ K, so there exists an open set in Ω \ K where v = 0 and u = u0. Hence theanalytic function U in Ω which coincides with u on some open subset of Ω \ K, and since this isa connected set, we have u ≡ U on Ω \ K.

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Chapter 3

Envelopes and domains of holomorphy

In view of the phenomenon seen in Chapters 1 and 2, we would now like to discuss, givena domain Ω, the “largest” domain, in the appropriate sense, to which all functions in H(Ω)simultaneously extend. As our experience with the logarithm in 1-variable complex analysisshows, we need to take the help of germs to construct this “largest” domain. But because we willneed to construct such objects for the simultaneous extension of a whole family of functions, wemust introduce the idea of S -germs.

But first, we provide a definition.

Definition 3.1. Let X be a Hausdorff topological space. If there exists a map p : X → Cn suchthat p is a local homeomorphism then (X, p,Cn) is called an (unramified) domain over Cn.

Let S be some fixed set. We want to define the sheaf O(S ), the sheaf of S -germs of holo-morphic functions on Cn. If U is a non-empty open set, then the pair (U, fss∈S ) will denotean arbitrary family of functions holomorphic on U indexed by S . For a ∈ Cn and (U, fss∈S ),(V, gss∈S ) with a ∈ U ∩ V, we say that these two pairs are equivalent if there exists a neighbour-hood W of a,W ⊂ U∩V such that, for all s ∈ S , fs|W ≡ gs|W .An equivalence class with respect tothis relation is called an S -germ of holomorphic functions at a. We denote the set of all S -germsat a by Oa(S ). We set O(S ) :=

⊔a∈Cn Oa(S ). We have a natural projection p = ps : OS → Cn

defined by p(ga) := a if ga ∈ Oa(S ). We define a topology on O(S ) as follows: Let ga ∈ Oa(S )and let (U, fss∈S ) be a representative of ga. Let [U, fs : s ∈ S ]b denote the germ at b determinedby the above pair for any b ∈ U. Now we write

N(U, fss∈S ) :=⋃b∈U

[U, fs : s ∈ S ]b.

We topologize O(S ) by demanding that the sets N(U, fss∈S ) form a fundamental system ofneighbourhoods of ga. The proof of the following proposition is very routine and mechanical, sowe shall skip its proof.

Proposition 3.2. The map p : O(S ) → Cn is continuous and is a local homeomorphism ofO(S ) onto Cn. Further, O(S ) is a Hausdorff topological space. The triple (O(S ), p,Cn) is an(unramified) domain over Cn.

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3.1 Envelopes of holomorphyLet p : X → Cn be a domain over Cn.A continuous function f : X → C is said to be holomorphicif for each open set U ⊆ X such that p|U is a homeomorphism, f (p|U)−1 ∈ H(p(U)). We canextend the notion of holomorphicity to maps between domains in an analogous fashion. Now letp : X → Cn, p′ : X′ → Cn be domains over Cn. A continuous map u : X → X′ is called a localisomorphism if every point a ∈ X has a neighbourhood U such that u|U is a homeomorphism ontoan open set U′ ⊆ X′ and u|U , (u|U)−1 are holomorphic (on U and U′ respectively). If, in addition,u is a homeomorphism of X onto X′, we say that u is an isomorphism.

If u : X → X′ is a continuous map such that p′ u = p, then u is automatically a localisomorphism.

Definition 3.3. Let p0 : Ω → Cn be a connected domain and S ⊆ H(Ω). Let p : X → Cn bea connected domain and ϕ : Ω → X a continuous map with p ϕ = p0. We say (X, p, ϕ) is anS -extension of p0 : Ω→ Cn if, to every f ∈ S , there exists F f ∈ H(X) such that F f ϕ = f .

Note that F f is uniquely determined (first on Ω since F f ϕ = f , hence on X by the principleof analytic continuation). F f is called the extension (or continuation) of f to X.

Definition 3.4. Let p0 : Ω → Cn be a connected domain and S ⊆ H(Ω). An S -envelope ofholomorphy is an S -extension (X, p, ϕ) such that the following holds:

(∗) For any S -extension (X′, p′, ϕ′) of p0 : Ω → Cn, there is a holomorphic map u : X′ → Xsuch that p u = p′, u ϕ′ = ϕ and F′f = F f u for all f ∈ S , where F f , F′f are theextensions of f ∈ S to X, X′ respectively.

Note that u in (∗) is unique (since it is determined on ϕ′(Ω) by the equation u ϕ′ = ϕ).

Remark 3.5. The S -envelope of holomorphy, if it exists, is unique up to “isomorphism”. In fact,let (X, p, ϕ), (X′, p′, ϕ′) be two S -envelopes of holomorphy. Then, by (∗) of Definition 3.4 thereare holomorphic maps u : X′ → X, v : X → X′ such that p = p′v, p′ = pu, ϕ = uϕ′, ϕ′ = vϕ.Then u v ϕ = u ϕ′ = ϕ, so that u v is the identity on ϕ(Ω) which is open in X. Hence, by theprinciple of analytic continuation u v = identity on X. Similarly, v u = identity on X′. Thus, uis an isomorphism of X′ onto X with p′ = p u, ϕ = u ϕ′.

Theorem 3.6 (Thullen). The S -envelope of holomorphy of any S ⊆ H(Ω) exists.

Proof. For any p0 : Ω → Cn and S ⊆ H(Ω), we define a map ϕ = ϕ(po, S ) of Ω into O(S ) asfollows. Let a ∈ Ω and a0 = p0(a) ∈ Cn. Let U be an open neighbourhood of a such that p0|U isan isomorphism onto an open set U0 ⊂ C

n. Let [U0, fs : s ∈ S ]a0 be the S -germ at a0 defined bythe pair (U0, fss∈S ), where fs := s (p0|U)−1, s ∈ S . We set ϕ(a) := [U0, fs : s ∈ S ]a0 .

Easy to verify that ϕ is a continuous and that p ϕ = p0, where p : O(S )→ Cn is the naturalprojection. In particular, ϕ is a local isomorphism.

Since Ω is connected, so is ϕ(Ω). Let X be the connected component of O(S ) containingϕ(Ω), and denote again by p the restriction to X of the map p : O(S )→ Cn.

We claim that (X, p, ϕ) is an S -envelope of holomorphy of Ω.

24

First, we observe that, for all s ∈ S , we have a holomorphic function Fs on O(S ) defined asfollows. We set Fs([U0, fs : s ∈ S ]a0) := fs(a0). Easy to verify that Fs is holomorphic on O(S ).We denote the restriction of Fs to X again by Fs. Now, by the very definition of ϕ, it follows thatFs ϕ = s for all s ∈ S .

Let p′ : X′ → Cn, ϕ′ : Ω → X′ be given with p′ ϕ′ = p0, and suppose that for all s ∈ S ,there exists F′s ∈ H(X′) so that s = F′s ϕ

′. Let S ′ = F′s : s ∈ S . Let u : X′ → O(S ) be the mapϕ(p′, S ′) (defined at the beginning of this proof). Since F′s ϕ

′ = s and p′ ϕ′ = p0, we haveϕ = u ϕ′(locally, F′s p′−1 = F′s ϕ

′ ϕ′−1 p′−1 = s p−10 ). Clearly, p′ = p u.

Definition 3.7. Let p0 : Ω → Cn be a connected domain over Cn. If S = H(Ω), the S -envelopeof holomorphy of Ω is called the envelope of holomorphy of Ω.

Proposition 3.8. Let p0 : Ω → Cn be a connected domain over Cn and f ∈ H(Ω). let F beits extension to the envelope of holomorphy (X, p, ϕ). Then f (Ω) = F(X). In particular, if f isbounded, | f (x)| ≤ M for all x ∈ Ω, then F is bounded and |F(x)| ≤ M for all x ∈ X.

Proof. Since f = F ϕ, we have f (Ω) ⊆ F(X). Suppose there is a c ∈ F(X) \ f (Ω). Then1

f−c ∈ H(Ω). Let G be its extension to X. Then G.(F − c) is the extension of 1 = ( f − c)−1( f − c)to X, so that G.(F − c) ≡ 1 on X. This implies that F(x) , c for all x ∈ X, a contradiction.

At this stage, one might like to know how two envelopes of holomorphy are related to eachother if the underlying domains are related by a holomorphic mapping. To answer such questions,we require two lemmas. Their proofs are routine and so we shall skip their proofs.

Lemma 3.9. Let p0 : Ω −→ Cn be a domain and Ψ : Ω −→ Cn be a holomorphic map such thatdet(dΨ)(a) , 0 for some a ∈ Ω. Then, there exists a neighbourhood U 3 a such that Ψ|U is ananalytic isomorphism.

Lemma 3.10. Let p0 : Ω → Cn, p′0 : Ω′ → Cn be connected domains over Cn and (X′, p′, ϕ′)be the T-envelope of holomorphy of p′0 : Ω′ → Cn,T ⊆ H(Ω′). Let u : Ω → Ω′ be a localisomorphism, and let S = f u : f ∈ T . Then (X′, p′, ϕ′ u) is the S -envelope of qo : Ω → Cn

where qo = p′0 u.

Proposition 3.11. Let p0 : Ω → Cn, p′0 : Ω′ → Cn be connected domains over Cn, and(X, p, ϕ), (X′, p′, ϕ′) their envelopes of holomorphy. Let u : Ω → Ω′ be a local isomorphism.Then there exists a holomorphic map u : X → X′ such that ϕ′ u = u ϕ.

Proof. Let v := ϕ′ u : Ω → X′. Then v is holomorphic and a local isomorphism. We have toshow that there is u : X → X′ so that uϕ = v. Consider the map ψ := p′ v : Ω→ Cn. Then ψ isagain a local isomorphism. If ψ = (ψ1, ψ2, . . . , ψn), the ψ j’s are holomorphic. Let η : Ω −→ C begiven by η(x) := det(dψ(x)). Then, since ψ is a local isomorphism, η(x) , 0 for all x ∈ Ω. Let Ψ j

be the extension of ψ j to X, and let Ψ = (Ψ1,Ψ2, . . . ,Ψn). Let H be the extension of η to X. Then,by the principle of analytic continuation, H = det(dΨ). Moreover, by Proposition 3.8 we haveH(x) , 0 for all x ∈ X. Hence by Lemma 3.9, Ψ : X → Cn is a local isomorphism. Moreover,Ψ ϕ = ψ.

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Consider now the domains ψ : Ω → Cn, p′ : X′ → Cn, and v : Ω → X′. Let S = f u :f ∈ H(Ω′) = F v : F ∈ H(X′). By Lemma 3.10 (X′, p′, v) is the S -envelope of holomorphyof ψ : Ω → Cn. Now any holomorphic function on Ω can be extended to X, so that (X, p,Ψ) isan S -extension of ψ : Ω → Cn. Therefore there exists a holomorphic map u : X → X′ such thatp′ u = Ψ and u ϕ = v.

Definition 3.12. Let p0 : Ω→ Cn be a connected domain over Cn and S ⊆ H(Ω). Ω is called anS -domain of holomorphy if the natural map of Ω

ϕ : x 7→ [p0(Ux), f (p0|Ux)−1]p0(x)

into its S -envelope of holomorphy is an analytic isomorphism. If S = H(Ω),Ω is called a domainof holomorphy.

The reader would intuit that the envelope of holomorphy of a connected domain p0 : Ω→ Cn

is a domain of holomorphy. We now give the proof behind this intuition.

Corollary 3.13. If (X, p, ϕ) is the envelope of holomorphy of p0 : Ω → Cn then p : X → Cn is adomain of holomorphy.

Proof. In Lemma 3.10 if we replace Ω′ by X, T by F f : f ∈ H(Ω) and u by ϕ then we getS = H(Ω). Therefore by Lemma 3.10 we get the desired result.

Consider a domain Ω Cn (in which case p0 is just the inclusion map). If such an Ω

is a domain of holomorphy, then combining the above corollary with the details of Thullen’sconstruction in Theorem 3.6, it is easy to deduce the following equivalent definition for Ω Cn

to be a domain of holomorphy.

Definition 3.14. Let Ω ⊂ Cn.We say Ω is a domain of holomorphy if there do not exist nonemptyopen sets U1,U2, with U2 connected, U2 * Ω,U1 ⊂ U2 ∩Ω such that to every f ∈ H(Ω), thereexists F f ∈ H(U2) satisfying F f |U1 = f |U1 .

3.2 Domains of holomorphyNow that we have introduced the concept of a domain of holomorphy, it would be interesting tosee if one can characterize when a connected domain p0 : Ω −→ Cn is a domain of holomorphy.This is quite technical, even when Ω ⊂ Cn. In the latter case, there are several equivalent char-acterizations. A characterization that is easy to check would require more space than is availablein this report. But we shall aim to provide one characterization.

Definition 3.15. Let p0 : Ω → Cn be a connected domain over Cn, f ∈ H(Ω) and A ⊆ Ω. Wewrite

‖ f ‖A := supx∈A| f (x)|.

• If A ⊆ Ω, and S ⊆ H(Ω), we set

AS := x ∈ Ω : | f (x)| ≤ ‖ f ‖A for all f ∈ S .

26

• If S = H(Ω), we simply write A := AS .

We shall need the following technical, but elementary, lemma.

Lemma 3.16. Let A ⊆ Ω such that ‖ f ‖A < ∞ for any f ∈ H(Ω). Then, there is a compact setK ⊂ Ω such that A ⊆ K.

Lemma 3.17. The following two statements are equivalent.

(a) For any K ⊂ Ω, K compact, K is also compact.

(b) For any (infinite) sequence (xν) ⊂ Ω which has no limit point in Ω there exists f ∈ H(Ω)such that f (xν) is unbounded.

Proof. (a)⇒ (b). Let (xν) be a sequence without limit point in Ω. Then xν 1 K for any compactset K. By Lemma 3.16 there exists f ∈ H(Ω) such that f (xν) is not bounded.

(b) ⇒ (a). If K is not compact, there exists a sequence xν ⊂ K, which has no limit pointin k and therefore in Ω as K is closed in Ω. Let f ∈ H(Ω) such that f (xν) is unbounded. Then‖ f ‖K = ∞. But, it follows from the definition of K that ‖ f ‖K = ‖ f ‖K < ∞.

If the conditions of Lemma 3.17 are satisfied we say that Ω is holomorphically convex.We now extend the notion of a polydisc to arbitrary domains over (i.e. not necessarily con-

tained in) Cn. Such a notion would enable us to define, for any point a ∈ Ω, a notion of “distancefrom the boundary” even when Ω is not a domain contained in Cn. Readers will note that theextended notion of a polydisc will result in the polydiscs Dn(a, r) when Ω ⊆ Cn, and that thenotion of “distance from the boundary” will coincide with the distance from ∂Ω with respect tothe | · |-norm when Ω Cn.

To distinguish these new objects from the classical polydiscs, we shall use the notationP(a, r). To be precise, we make a definition.

Definition 3.18. Let p0 : Ω → Cn be a connected domain over Cn, a ∈ Ω. A polydisc of radiusr about a is a connected open set U containing a such that p0|U is an analytic isomorphism ontothe set z ∈ Cn : |z j − b j| < r, where p0(a) = (b1, b2, . . . , bn). We denote the set U by P(a, r). Themaximal polydisc around P(a, r0) is the union of all polydiscs about a.

Lemma 3.19. P(a, r0) is a polydisc about a of radius

r0 = sup r

where the supremum is over all polydiscs P(a, r) about a.

Proof. It suffices to show that the map

p0 : P(a, r0)→ Dn(b, r0)

is bijective. By definition p0(P(a, r0)) ⊂ Dn(b, r0).p0 injective: If x, x′ ∈ P(a, r0), there is a polydisc P(a, r) containing both x, x′, so that p0(x) ,

p0(x′).p0 is surjective: If z ∈ Dn(b, r0), then max j|z j−b j| < r0, hence there is a polydisc P(a, r), such

that |z j − b j| < r ≤ r0, so that there is a point x ∈ P(a, r) with p0(x) = z.

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Definition 3.20. The radius of the maximal polydisc about a is called the distance of a from theboundary of Ω and is denoted by d(a).

Lemma 3.21. If there is a point a ∈ Ω with d(a) = ∞, then p0 is an isomorphism of Ω onto Cn.

Proof. Let S = x ∈ Ω : d(x) = ∞. By assumption S , ∅. By definition S is open as d(a) = ∞

implies that there is an open set U containing a such that po|U is an isomorphism onto Cn. Letxn ∈ S , xn → x. Let P(x, r) be a polydisc about x. Now we consider xn0 such that xn0 ∈ P(x, r).Now as xn0 ∈ S there exists U containing xn0 such that po|U is isomorphism onto Cn. ThereforeP(x, r) ⊂ U, and hence x ∈ U. Therefore x ∈ S , and hence S is closed. As Ω is a domain we haveS = Ω. Therefore p0 is a covering. As Cn is simply connected we have that p0 : Ω → Cn is anisomorphism.

Definition 3.22. If A ⊂ Ω, we set

d(A) := infa∈Ad(a).

Remark 3.23. Since d is continuous (which follows from definition), if K is a compact subsetof Ω then d(K) > 0.

The following is a very important result that we shall make use of in this and the next chapter.

Proposition 3.24. Let po : Ω → Cn be a domain. Let K be a compact subset of Ω and x0 ∈ K.Let a = p0(x0), and let V be a polydisc about x0, and D = p0(V). Then, for any f ∈ H(Ω), ifg := f

(pO|V

)−1 , the series ∑α∈Nn

Dαg(a)α!

(z − a)α

converges in the polydisc Dn(a, d(K)).

Proof. Note that clearly g ∈ H(D). Let 0 < r < d(K). For any x ∈ K,we consider the compact set,K′ =

⋃x∈K P(x, r). Let M = ‖ f ‖K′ . By Cauchy’s inequality applied to f |P(x,r), we have |Dα f (x)| ≤

M.α!.r−|α|, x ∈ K, so that ‖Dα f ‖K ≤ M.α!.r−|α|. Hence, by definition of K, we have

‖Dα f ‖k ≤ M.α!.r−|α|.

Since x0 ∈ K, this implies that |Dαg(a)| ≤ M.α!.r−|α|. Therefore the series∑α∈Nn

Dαg(a)α! (z − a)α

converges on Dn(a, r). Since r < d(K) is arbitrary, the result follows.

Before we prove the main result of this section, we state a simple

Lemma 3.25. Let p0 : Ω −→ Cn be a domain and let S ⊆ H(Ω) has the property that for eachf ∈ S and α ∈ Nn, Dα f ∈ S . Let (X, p, ϕ) be the S -envelope of holomorphy of p0 : Ω −→ Cn.Then, ϕ is injective if and only if S separates points of p−1

0 p0(a) for every a ∈ Ω. In particular,if p0 : Ω −→ Cn is a domain of holomorphy, then H(Ω) separates points of Ω.

The above is an elementary consequence of identifying X with a special connected componentof O(S ) as done in Thullen’s construction.

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Theorem 3.26 (H. Cartan-P. Thullen). Suppose that p0 : Ω → Cn is a domain of holomorphy.Then, for any compact set K ⊂ Ω, we have

d(K) = d(K).

Proof. Clearly d(K) ≥ d(K) as K ⊆ K′. Suppose that strict inequality holds. Then, there isx0 ∈ K such that d(x0) = r0 < ρ = d(K). Let a = p0(x0) and let D0 = z ∈ Cn : |z−a| < d(K) = ρ.Let Ω0 be the connected component of p−1

0 (D0) containing x0. We assert that p0|Ω0 is injective.In fact, it follows from Proposition 3.24 that for any f ∈ H(Ω), there is g f ∈ H(D0) such thatf |Ω0 = g f p0. On the other hand, since p0 : Ω → Cn is a domain of holomorphy, by Lemma3.25, since g f p0 takes the same value at any two point x, y ∈ Ω0 with p0(x) = p0(y), it followsthat p0|Ω0 is injective.

Let r0 < r < ρ and Ω1 = x ∈ Ω0 : |p0(x)− a| < r, and let Y be the disjoint union of Ω and D,where D = z ∈ Cn : |z − a| < r. We define an equivalence relation on Y by the requirement thatz ∈ D is equivalent to at most one point x ∈ Ω, and that, if and only if x ∈ Ω1 and p0(x) = z. LetX be the quotient of Y by this equivalence relation. Then X is Hausdorff, and the map of Y intoCn which is p0 on Ω and the inclusion of D in Cn induces a local homeomorphism p : X → Cn.Moreover, the inclusion of Ω in Y induces a map ϕ : Ω→ X such that p ϕ = p0. We claim thatfor any f ∈ H(Ω), there is F f ∈ H(X) such F f ϕ = f . In fact, since there is g f ∈ H(D0) suchthat f |Ω = g f po by Proposition 3.24, we define a function G f on Y by G f |Ω = f , G f |D = g f |D.This induces F f ∈ H(X) and we clearly have F f ϕ = f . Hence (x, p, ϕ) is an H(Ω)-extensionof p0 : Ω → Cn. Since p0 : Ω → Cn is, by assumption, a domain of holomorphy, it follows thatϕ is an analytic isomorphism. In particular, since X contains a polydisc of radius r about x0, Ω

contains a polydisc of radius r about x0, contradicting our assumption that r > r0 = d(x0).

The same reasoning can be used to prove the following:

Theorem 3.27 (Cartan-Thullen). Let p0 : Ω→ Cn be a domain. Let S ⊆ H(Ω) be a subalgebraof H(Ω) containing the functions p1, ..., pn, (p0 = (p1, ..., pn)) and closed under differentiation(i.e., f ∈ S implies Dα f ∈ S for all α ∈ Nn). Then, if the natural map of Ω into its S -envelope ofholomorphy is an isomorphism, we have d(K) = d(KS ) for any compact K ⊂ Ω.

Corollary 3.28. If Ω is an open set in Cn which is a domain of holomorphy and p0 is the inclusionof Ω in Cn, then for any compact set K ⊂ Ω, K is also compact.

Proof. K is clearly closed in Ω. Moreover, since d(K) = d(K), it follows that K is closed inCn (since the closure of K in Cn cannot meet ∂Ω). Moreover, K is contained in the polydiscz ∈ Cn : |z| ≤ ρ where ρ = max

j|z j|K , and so is bounded. Hence K is compact.

We are now quite close to characterizing domains of holomorphy when the domains lie in Cn.

Proposition 3.29. Suppose Ω Cn is a connected open subset and has the property that for anycompact set K ⊂ Ω, K is also compact. Then Ω is a domain of holomorphy.

29

Proof. Since Ω Cn it is possible to construct a countable subset Z = xν : ν ∈ Z+ of Ω thatcontains no limit points in Ω and such that ∂Ω ⊂ Z. Now by Lemma 3.17 there exists F ∈ H(Ω)such that F is unbounded on Z. Now, for each z ∈ ∂Ω, there is a subsequence xνkk∈N such thatxνk → z. Thus, as z ∈ ∂Ω was arbitrary, we conclude that for any domain U such that U∩∂Ω , ∅,F is unbounded on U ∩ Ω. Now If U2 is a domain in Cn such that U2 ∩ Ω , ∅ and U2 * Ω thenU2 ∩ ∂Ω , ∅. Therefore by the last two assertions and by Definition 3.14, Ω is a domain ofholomorphy.

Summarizing from Corollary 3.28 and the above proposition:

Theorem 3.30. Let Ω Cn be a connected open subset. Then, Ω is a domain of holomorphy ifand only if it is holomorphically convex.

5This is because of (ii) above. This allows us to use Proposition

30

Chapter 4

More characterizations of domains ofholomorphy in Cn

Chapter 3 ended with a characterization of a proper subdomain of Cn to be a domain of holomor-phy. Unfortunately, when n ≥ 2, the condition of holomorphic convexity is very hard to check.It would thus be desirable to have other (equivalent) characterizations that are more amenableto computation. It is with this aim that we present a long theorem in Section 4.1 that presentsseveral equivalent characterizations for a domain Ω ⊂ Cn to be a domain of holomorphy.

The proof of this theorem, i.e. Theorem 4.8, is not easy. The proof of one of the implica-tions requires the technical and very sophisticated result of Hormander that the ∂-problem has asolution, with not necessarily compactly-supported data, on a pseudoconvex domain. This willbe clarified in greater detail just after the statement of Theorem 4.8. Hormander’s work is a bitbeyond the scope of this survey. Therefore, we shall present proofs of only some of the parts ofTheorem 4.8.

4.1 Further characterizations of domains of holomorphyWe begin this section with an application of Theorem 3.30.

Theorem 4.1. Let Ω be a convex domain in Cn. Then, Ω is a domain of holomorphy.

Proof. Without loss of generality, assume 0 ∈ Ω. Define:

Kν := (1 −1ν

) Ω ∩ Bn(0, ν)

for each ν (≥ 2) ∈ Z+. Then Kν’s are convex sets. Now pick a compact subset L ⊂ Ω. Then thereexists ν(L) ∈ Z+ such that L ⊂ Kν for all ν ≥ ν(L). Let z0 < Kν(L). Then, there exists a real linearfunctional on R2n, Λ, such that

Λ(z0) = 1, supz∈Kν(L)Λ(z) = α < 1. (4.1)

31

Now write z j = x j + iy j for j = 1, . . . , n. We can write Λ(z) =∑n

j=1 a jx j + b jy j for somea j, b j ∈ R. Now let

λ(z) :=n∑

j=1

(a j − ib j)z j.

Note that Re λ(z) = Λ(z). Now define F(z) := eλ(z), z ∈ Cn. By (4.1) we have

|F(z0)| = e > eα = supKν(L)|F| ≥ supL|F|.

Hence if z0 < Kν(L), then z0 < LΩ. Therefore Ω c Kν(L) ⊇ LΩ. This implies LΩ is compact. Hence,by Theorem 3.30, Ω is a domain of holomorphy as L is an arbitrary compact subset of Ω.

Definition 4.2. A function µ : Cn −→ R is called a distance functional if:

i) µ ≥ 0 and µ(z) = 0 if and only if z = 0;

ii) µ(tz) = |t| µ(z) for all t ∈ C and for all z ∈ Cn; and

ii) µ is continuous.

Given a distance functional µ, we can talk about the “distance from ∂Ω” of a point z ∈ Ω

measured in terms of µ. This motivates the next definition.

Definition 4.3. Let Ω be a domain in Cn, and let µ : Cn −→ R be a distance functional. For eachz ∈ Ω we define

µΩ(z) := infµ(z − w) : w ∈ ΩC. (4.2)

If X is a subset of Ω, then we write

µΩ(X) := infµΩ(x) : x ∈ X.

In the above definition, we do not distinguish between proper subdomains and the case whenΩ = Cn. From the right-hand side of (4.2), we see that for some z ∈ Ω, µΩ(z) = ∞ ⇐⇒ Ω = Cn.This is reminiscent of Lemma 3.21. In fact, if the Ω of Lemma 3.21 were a domain in Cn (withp0 just being the inclusion map), then the entire discussion in Chapter 3 could be viewed as onewhere the l∞-norm on Cn is the distance functional of choice. In a sense, parts of the theorembelow arise from the flexibility we get by considering various distance functionals and not justthe l∞-norm.

Perhaps the most important concept in studying domains of holomorphy is that of plurisub-harmonic functions.

We recall that a C2-smooth function h in an open set Ω(⊂ C) is called harmonic if ∆h =

4∂2h/∂z∂z = 0 in Ω.

Definition 4.4. A function u defined in an open set Ω ⊂ C and with values in [−∞,+∞) is calledsubharmonic if

i) u is upper semicontinuous, that is, z : z ∈ Ω, u(z) < s is open for every real number s;

32

ii) For every compact set K ⊂ Ω and every continuous function h on K which is harmonic inthe interior of K and is ≥ u on the boundary of K, we have u ≤ h in K.

Definition 4.5. A function u defined in an open set Ω ⊂ Cn and with values in [−∞,+∞) iscalled plurisubharmonic if

i) u is upper semicontinuous;

ii) For every arbitrary z and w ∈ Cn, the function τ 7→ u(z + τw) is subharmonic in its domainof definition.

We need a couple of final definitions before we can state Theorem 4.8. The definitions givetwo notions of “convexity” for domains in Cn. One of the outcomes of Theorem 4.8 is that thetwo notions coincide for domains with C2-smooth boundaries.

Definition 4.6. Let Ω be a domain in Cn. We say that Ω is Hartogs pseudoconvex if there existsa distance functional µ : Cn → [0,∞) such that the function −logµΩ is plurisubharmonic on Ω.

Definition 4.7. Let Ω Cn be a domain having C2-smooth boundary and let ρ be a definingfunction for Ω, i.e. ρ : V −→ R is a C2-smooth function, where V is an open neighbourhood ofΩ, such that

• Ω = z ∈ V : ρ(z) < 0; and

• ∇ρ(z) , 0 ∀z ∈ ∂Ω.

We say that Ω is Levi pseudoconvex if

n∑j,k=1

∂2ρ

∂z j∂zk(z)v jvk ≥ 0 for all V = (v1, . . . , vn) ∈ Tz(∂Ω) ∩ iTz(∂Ω), and for all z ∈ ∂Ω. (4.3)

Note that, in (4.3), we are viewing the tangent spaces to ∂Ω extrinsically as real subspaces inCn R2n. Thus, Tz(∂Ω) ∩ iTz(∂Ω) denotes the largest R-linear subspace of Tz(∂Ω) that is C-linear in the ambient Cn. We shall abbreviate Tz(∂Ω) ∩ iTz(∂Ω) := Hz(∂Ω).

We now have all the definitions needed to state the main theorem of this chapter.

Theorem 4.8. Let Ω be a domain in Cn. Then, the following statements are equivalent.

1) Ω is a domain of holomorphy;

2) For any distance functional µ : Cn → [0,∞), the function −logµΩ is plurisubharmonic onΩ;

3) Ω is Hartogs pseudoconvex;

4) Ω admits a continuous exhaustion function u, i.e. given any c ∈ R, z ∈ Ω : u(z) < c is arelatively compact subset of Ω;

33

5) Ω is holomorphically convex.

Furthermore if Ω has C2-smooth boundary, then the following is equivalent to the five state-ments above:

(6) Ω is Levi pseudoconvex.

A sketch of the proof. The scheme of the proof of the equivalance of the first five statementsabove is summarized by the following diagram of implications:

-

6

?HHHH

HY

HHH

HHY

(4)

(1)

(3)

(2)

(5)

The proofs of some of the above implications are very technical. Of these, the proof that(3)⇒ (5) actually relies on the very sophisticated result of Hormander that if Ω is Hartogs pseu-doconvex, then for every smooth (0, q)-form F on Ω (F not necessarily compactly supported),q = 1, . . . , n − 1, such that ∂F = 0, there exists a u ∈ C∞(0,q−1)(Ω) such that ∂u = F. The detailsare given in [7, Theorem 4.2.8-9]; to the best of my knowledge, there is no other approach toproving (3) ⇒ (5). As stated above, the solvability of the ∂-problem in this generality is a deepresult, which is beyond the scope of this survey. Hence, we shall present proofs of only a fewselected implications.We shall discuss some aspects of the equivalence of Levi pseudoconvexity, in case Ω has C2-smooth boundary, with the statements (1)–(5) in greater detail in Section 4.2. There are severaldifferent proofs of this last equivalence. We will present a proof that is in keeping with the theme— i.e. analytic continuation — of this survey.(2)⇒ (3): Given (2), (3) follows from the definition of Hartogs pseudoconvexity.(3) ⇒ (4): By (3) we know that there exists a distance functional µ such that −logµΩ(z) isplurisubharmonic on Ω. Now if Ω is unbounded then it is possible that −logµΩ(z) is not a ex-haustion function (although, by hypothesis, it is plurisubharmonic). Note that if zνν∈Z+

is asequence that approaches ∂Ω, then −logµΩ(zν) → ∞ as ν → ∞, but, if Ω is unbounded, then thesets z ∈ Ω : −logµΩ(z) < c could be unbounded. Therefore if we set u(z) := ||z||2 − log µΩ(z)where µ is given by (3) we get the desired implication (it is easy to check that ‖z‖2 is plurisub-harmonic).(5)⇒ (1): We already have proved this in Theorem 3.30.

34

4.2 Domains of holomorphy that have smooth boundaryBefore we embark upon the proof of the last stage of Theorem 4.8, we need to answer a questionabout whether Levi pseudoconvexity is, as per Definition 4.7, well defined. In other words, wemust ascertain whether (4.3) holds true independently of the choice of the defining function. Weestablish this in the following proposition.

Proposition 4.9. Let Ω ⊂ Cn be an open set with C2-smooth boundary having a defining func-tion ρ. Let ρ1 be another defining function; let Ω = z : ρ1(z) < 0 where ρ1 is in C2 in aneighbourhood of Ω and ∇ρ1 , 0 on ∂Ω. If Ω is Levi pseudoconvex, then

n∑j,k=1

∂2ρ1

∂z j∂zk(z)v jvk ≥ 0 for all V = (v1, . . . , vn) ∈ Hz(∂Ω), and for all z ∈ ∂Ω. (4.4)

Proof. Let us pick a point p ∈ ∂Ω. Then, there is a small ball Bp around p such that (ρ1/ρ)(z) > 0for all z ∈ Bp. Now, using a partition-of-unity argument, we may assume that if ρ1 is as stated inProposition 4.9 then ρ1 = hρ with h > 0 in a neighbourhood of Ω. Hence for all z ∈ ∂Ω,

n∑j,k=1

∂2ρ1

∂z j∂zk(z)v jvk = h(z)

n∑j,k=1

∂2ρ

∂z j∂zk(z)v jvk, provided V = (v1, . . . , vn) ∈ Hz(∂Ω).

This is because ρ|∂Ω ≡ 0 and V = (v1, . . . , vn) ∈ Hz(∂Ω) ⇐⇒∑n

j=1 ∂z jρ(z)v j = 0. Hence theproposition.

We will now look at one of the two implications that establish the final stage of Theorem 4.8.Under the hypothesis that ∂Ω is C2-smooth, one may achieve this by showing that (2) ⇒ (6) ⇒(3). But this proof has several intermediate steps of a technical nature that make the proof quitelengthy and obscure the main idea behind the machinery. We would like to bring out this clearly.Hence, we shall prove (1) ⇒ (6) directly under the slightly stronger hypothesis that ∂Ω is C3-smooth.

Theorem 4.10. Let Ω be a domain with C3-smooth boundary. If Ω is a domain of holomorphy,then Ω is Levi pseudoconvex.

Proof. Suppose there is a point p ∈ ∂Ω such that there exists a vector V ∈ Hp(∂Ω) and

n∑j,k=1

∂2ρ

∂z j∂zk

(p)v jvk < 0,

where V = (v1, . . . , vn), and ρ is some choice of defining function for Ω. This proof depends onmaking several holomorphic change of coordinates. We shall carefully argue what happens withthe first of these changes of coordinates. This will serve as a model for the assertions followingsubsequent coordinate changes.

35

Without loss of generality, we may assume that ‖V‖ = 1 = ‖∇ρ(p)‖. Now, let 〈. , .〉 denotethe standard Hermitian inner-product on Cn and let T be a unitary transformation with respect to〈. , .〉 such that T (∇ρ(p)) = (0, . . . , i), T (V) = (1, 0, . . . , 0) and T (Hp(∂Ω)) = zn = 0. Now if

(z′1, z′2, . . . , z

′n) =T (z − p),

ρ′(z′) = ρ(T−1(z′) + p), (4.5)

then the above represents a holomorphic change of coordinate, say ψ, and ρ′ is the definingfunction of ψ(Ω). By Taylor’s theorem:

ρ′(z′) =2Re

n∑j=1

∂ρ′

∂z′j(0)z′j

+ Re

n∑

j,k=1

∂2ρ′

∂z′j∂z′k(0)z′jz

′k

+

n∑j,k=1

∂2ρ′

∂z′j∂z′k(0)z′jz

′

k + O(||z′||3). (4.6)

If we view vectors U,W ∈ Cn as vectors in R2n, then define

U ·W =

n∑j=1

Reu jRew j + Imu jImw j = Re[〈U,W〉].

From this, and applying the chain rule to (4.5), we get

ρ′(z′) = ∇ρ(p) · T−1(z′) + Q(z′) + O(||z′||3),

where Q is the quadratic form seen in (4.6). If we write z′j = x′j + i y′j, j = 1, . . . , n, then, by theproperties of T ,

∇ρ(p) · T−1(z′) = Re[〈∇ρ(p),−iz′n∇ρ(p)〉] = y′n.

This argument shows that, to simplify notations, we may assume without loss of generality that

• p = 0;

• ρ(z) = yn + Q(z) + O(||z||3), where Q is the quadratic form given by

Q(z) = Re

n∑

j,k=1

∂2ρ

∂z j∂zk(0)z jzk

+

n∑j,k=1

∂2ρ

∂z j∂zk(0)z jzk;

• The V occuring in our assumption is (1, 0, . . . , 0).

Note that these imply that:

∂2ρ

∂z1∂z1(0) < 0. (4.7)

36

Let us now make another change of variable

(z′1, . . . , z′n) = Ψ(z) =

z1, . . . , zn−1, in∑

j,k=1

∂2ρ

∂z j∂zk(0)z jzk

.Let Ψ = (Ψ1, . . . ,Ψn). Then we observe that

∂Ψi

∂z j(0) = δi j for i, j = 1, . . . , n − 1 and

∂Ψn

∂zn(0) = 1.

Therefore JacC(Ψ)(0) = 1. By inverse function theorem for holomorphic functions we have thatthere exists ω, a neighbourhood of 0 in Cn, such that Ψ|ω is a biholomorphism.

After this change of variables the Taylor expansion of ρ assumes the simpler form

ρ = Imz′n +

n∑j,k=1

∂2ρ

∂z j∂zk(0)z′jz

′

k + O(||z′||3).

Again, to simplify notations we may therefore, without loss of generality, assume that the originalcoordinates were choosen so that

ρ = Imzn +

n∑j,k=1

A jkz jzk + O(||z||3), (4.8)

where A jk is a Hermitian symmetric matrix. Note that, By (4.7), A11 < 0. By (4.8) we have

ρ(z1, 0, . . . , 0) = A11|z1|2 + O(|z1|

3). (4.9)

Now, since A11 < 0, we can first choose δ > 0, then ε > 0, and finally r << δ such that

• D(0, δ)×Dn−1((0, . . . , 0, −iε/2), ε/2)∪Ann(0, δ− r, δ+ r)×Dn−1((0, . . . , 0, −iε/2), ε) =: H ⊂ ω;and

• ρ(z) < 0 for all z ∈ H.

Therefore H ⊂ Ω. Now note that the set H′ := D(0, δ)×Dn−1(0, ε/2)∪Ann(0, δ−r, δ+r)×Dn−1(0, ε)is a Reinhardt domain and it contains the point 0, whose each coordinate is 0. By Corollary 1.10every holomorphic function on H′ extends holomorphically to the set D(0, δ) × Dn−1(0, ε). Thisimplies that for every f ∈ H(Ω), f |H extends holomorphically to D(0, δ)×Dn−1((0, . . . , 0, −iε/2), ε).Since Ω is a domain of holomorphy, we must have D(0, δ) × Dn−1((0, . . . , 0, −iε/2), ε) ⊂ Ω. Inparticular 0 ∈ Ω. This is a contradiction. Hence, our initial assumption cannot be sustained, andΩ is Levi pseudoconvex.

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Chapter 5

A schlichtness theorem for envelopes ofholomorphy

In the previous chapters we discussed many equivalent conditions for a domain in Cn to be adomain of holomorphy. Also we saw that for a domain in Cn there exists a maximal domain ofexistance for every holomorphic function on the given domain, namely the envelope of holomor-phy of the given domain. However we note that, the envelope of holomorphy is not necessarilysitting inside Cn. Therefore it is apparently very difficult to obtain information about the enve-lope of holomorphy from information about the given domain in Cn. By “information” we meanhere any statement about the global geometry of the envelope of holomorphy (its local geometryis completely understood from the material in Chapter 3). In this chapter we present a result thatgives us a very strong insight into the envelope of holomorphy based on information about theportion of the envelope that is “spread over” the given domain Ω.

The theorem presented in this chapter is by Jupiter. We shall follow the notation used byJupiter. To begin with we need some definitions.

Definition 5.1. Let Ω be a domain in Cn and let (Ω, p, j) denote the envelope of holomorphy ofΩ. Let X1 ⊂ X2 ⊂ Ω be unramified domains spread over Cn such that (X j, p|X j ,C

n) are H(Ω)-extensions of Ω. We say that X2 is schlicht over X1 if p|p−1(p(X1))∩X2 is injective. We say that X1 isschlicht over Ω if X1 is schlicht over j(Ω). Finally, we say that X1 is schlicht if p|X1 is injective.

Definition 5.2 (Jupiter, [9]). Let Ω be a domain in C2. Let z ∈ Ω. We say that z can be reachedfrom Ω by pushing discs if there is a neighbourhood U of z in Ω such that p|U is a biholomorphismand the following holds: there is a biholomorphism, F, of D2(0, 1) into U such that

• z ∈ F(D2(0, 1)); and

• F(H) b U ∩ j(Ω),

where H is the Hartogs figure,

H := (D × 1/2 < |w| < 1) ∪ (D(0, 1/2) × D).

38

Remark 5.3. We should note that:

i) Without loss of generality we can take the neighbourhood U to be biholomorphic to anopen ball in C2.

ii) By Corollary 1.10 every holomorphic function on F(H) has a single valued holomorphicextension into F(D2(0, 1)).

iii) By rotating and scaling, if necessary, without loss of generality we can assume that if z ∈ Ω

can be reached from Ω by pushing discs, then z ∈ F([0, 1] × D).

In other words, we have a continuous family of holomorphic discs, contained Ω, such thatthe “bottom” disc is contained in Ω, but the “top” disc does not lie in Ω.

5.1 A schlichtness theoremWe now have sufficient background to state the main theorem of this chapter.

Theorem 5.4 (Jupiter, [9]). Let Ω be a domain in C2, and (Ω, p, j) be its envelope of holomorphy.If Ω is schlicht over Ω, then Ω is schlicht.

Before we proceed to prove Theorem 5.4 we need two important lemmas.Define Ωn inductively as follows:

Ω0 := j(Ω),

Ωn+1 := z ∈ Ω : z can be reached from Ωn by pushing discs.

Remark 5.5. Note that if z ∈ Ω can be reached from Ωn by pushing discs, then, by definitionof pushing discs, every point in F(D2(0, 1)) 3 z, can be reached from Ωn by pushing discs.Therefore for each n ∈ Z+, Ωn is an open subset of Ω.

Proposition 5.6. Let Ω′ := ∪∞n=0Ωn. Then Ω′ = Ω.

Proof. Let, if possible, Ω′ be a proper subset of Ω. Then there exists z ∈ Ω such that z ∈ ∂Ω′

since Ω is connected. Now, let U be a neighbourhood of z in Ω such that p|U is a biholomorphism,p(U) is an open ball in C2 and U ∩ Ω′ is connected . If U ∩ Ω′ is not a domain of holomorphy,then there exists a biholomorphism, F, of D2(0, 1) into U such that

i) F(H) b Ω′; and

ii) F(D2(0, 1)) is not contained in Ω′.

This follows from the fact that we can view U ∩ Ω′ as an open set in C2, since p|U is a biholo-morphism, and from the argument in the final paragraph of our proof of Theorem 4.10.

Since F(H) is contained in Ω′, F(H) ⊂ Ωn0 for some n0 ∈ N. Therefore, by definition ofpushing discs, F(D2(0, 1)) = U ⊂ Ωn0 , which contradicts the fact that Ω′ ⊇ Ωn0 .

Therefore by one of the many equivalent notions of domain of holomorphy, given in Chap-ter 4, we see that Ω′ is a domain of holomorphy and Ω′ ⊃ j(Ω). This contradicts the fact thatΩ′ Ω. We conclude that Ω′ = Ω.

39

To prove Theorem 5.4 we need the following lemmas. The hypothesis that Ω ⊂ C2 is used ina major way in the following lemma. In fact, in [9] Jupiter constructs an example of an Ω ⊂ C3

for which Lemma 5.7 fails, and shows that Theorem 5.4 can not be true for the latter Ω ⊂ C3.

Lemma 5.7. Let us assume that for some n ∈ N, Ωn is schlicht. If Ωn+1 is schlicht over Ωn thenΩn+1 is schlicht.

Proof. Assume, if possible, that Ωn+1 is not schlicht. Then there exists two points z1, z2 ∈ Ωn+1

such that p(z1) = p(z2) = a (say). By the definition of Ωn+1 and the Remark 5.3, for each pointz j there is a neighbourhood U j of z j in Ω and biholomorphic maps F j from D2(0, 1) into U j suchthat

i) z j ∈ F j([0, 1] × D); and

ii) F j(H) b U j ∩Ωn.

Define the setT := (0 × D) ∪ ([0, 1] × ∂D).

We have a ∈ p(F1([0, 1] × D)) ∩ p(F2([0, 1] × D)). Let X be the connected component ofp(F1([0, 1]×D))∩p(F2([0, 1]×D)) that contains a. Note that as p|U j is, by definition, a biholomor-phism, p(F j([0, 1]×D)) is a smooth manifold with boundary. Also, all of the previous statementsremain true if we replace F j by a small perturbation — with respect to the C∞(D2(0, 1); U j)-metric — of F j, j = 1, 2. Thus, by transversality, we may assume that p(F j([0, 1]×D)), j = 1, 2,intersect in general position. Hence, we may assume that X is a two-dimensional manifold.Moreover, for dimensional reasons, there exists a t j ∈ [0, 1] such that M j := (p|U j)

−1(X) ∩F j(t j × D) is a one-dimensional real-analytic submanifold of F j(t j × D), j = 1, 2. By theproperties of real-analytic submanifolds, M j ∩ F j(t j × ∂D) , ∅. Hence, there is a point b ∈ Xthat lies either in p(F1(T )) or p(F2(T )). We then have that b = p(F1(ζ1)) for some ζ1 ∈ T in thefirst case, or b = p(F2(ζ2)) for some ζ2 ∈ T in the second case. In other words, there exist pointsζ j ∈ T such that in the first case β1 = F1(ζ1) ∈ Ωn and, in the second case β2 = F2(ζ2) ∈ Ωn.

We now claim that both the points β1 and β2 cannot lie in Ωn. If not, then since Ωn is schlichtand p(β1) = p(β2) = b, it follows that β1 = β2. Now take any path in X joining b to a. Thenby the uniqueness of lifting (for unramified domains spread over open sets in Euclidean space),z1 = z2. This is a contradiction.

Now we examine the case when β1 ∈ Ωn. Since β2 = F2(ζ2) does not belong to Ωn, β1 , β2.But p(β1) = p(β2) = b. This implies that Ωn+1 is not schlicht over Ωn which is a contradiction toour hypothesis.

The case when b ∈ p(F2(T )) but is not contained in p(F1(T )) can be dealt in similar way.Therefore Ωn+1 is schlicht.

Lemma 5.8. Assume that Ωn is schlicht. If Ωn+1 is not schlicht over Ωk, for some k ≤ n then Ωn+1

is not schlicht over Ωk−1.

Proof. Since Ωn+1 is not schlicht over Ωk there exist z1 ∈ Ωn+1 and z2 ∈ Ωk such that p(z1) =

p(z2) = a (say). By the definition of Ωn+1 and the Remark 5.3, for each point z j there is aneighbourhood U j of z j in Ω and biholomorphic maps F j from D2(0, 1) into U j such that

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i) z j ∈ F j([0, 1] × D)); and

ii) F1(H) b U1 ∩Ωn and F2(H) b U2 ∩Ωk−1.

As in the proof of Lemma 5.7, let X be the connected component of p(F1([0, 1]×D))∩p(F2([0, 1]×D)) which contains a. By a similar argument as in the proof of Lemma 5.7, X is a connected man-ifold and contains a point b which lies either in p(F1(T )) or in p(F2(T )). So, either b = p(F1(ζ1))for some ζ1 in T or b = p(F2(ζ2)) for some ζ2 in T .

Now, it is not possible that β1 = F1(ζ1) lies in Ωn. If not, then since β2 = F2(ζ2) lies inΩk ⊂ Ωn and Ωn is schlicht we shall have that β1 = β2. Similarly as in the previous proof, anypath in X connecting a and b will give a contradiction.

Therefore β1 lies in Ωn+1 but does not lie in Ωn and β2 ∈ F2(T ) ⊂ Ωk−1. This means that Ωn+1

is not schlicht over Ωk−1.

5.2 Proof of the main theoremWe now prove Theorem 5.4.

Proof. Suppose we knew that for each n ∈ N, Ωn is schlicht. Then since Ωn ⊇ Ωn−1 for eachn ∈ N and Ω = ∪∞n=0Ωn, p is injective on Ω. In other words Ω is schlicht. Therefore it is sufficientto prove that each Ωn is schlicht.

Suppose Ω1 is not schlicht. By Lemma 5.7, Ω1 is not schlicht over Ω0 = j(Ω). Thereforethere exists two points z1, z2 ∈ Ω1 such that p(z1) = p(z2) ∈ Ω. This contradicts the fact that Ω isschlicht over Ω as Ω ⊇ Ω1. Therefore Ω1 is schlicht.

Let Ωn be schlicht. If Ωn+1 is not schlicht then by Lemma 5.7, Ωn+1 is not schlicht over Ωn.By repeating Lemma 5.8 (n−1) times we get that Ωn+1 is not schlicht over Ω0, which contradictsthe fact that Ω is schlicht over Ω.

Therefore if Ωn is schlicht then Ωn+1 is schlicht. As Ω1 is schlicht, by induction we have Ωn

is schlicht for every n ∈ Z+.

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Chapter 6

A simple Kontinuitatssatz

In this chapter, we explore another tool for obtaining information about the envelope of holomor-phy from information about the given domain in Cn.

In several recent papers on the subject of analytic continuation — see [1,4,5], for instance —one comes across versions of an informal lemma referred as “Kontinuitatssatz” or “the continuityprinciple”. The various conclusions deduced from it are obtainable from the following generalprinciple:

(∗) Let Ω be a domain in Cn, n ≥ 2, let Dtt∈[0,1] be an indexed family of smoothly boundedopen sets in C, and let Ψt : Dt → Cn−1 be maps such that Ψt ∈ H(Dt) ∩ C(Dt) for eacht ∈ [0, 1]. Assume that the Dt’s vary continuously with t in some appropriate sense suchthat Γ := ∪t∈[0,1](Dt × t) forms a “nice” compact body: such that (C × (0, 1)) ∩ ∂Γ is asmooth submanifold with boundary, for instance. Also assume that F : Γ→ Cn−1 given byF(ζ, t) := Ψt(ζ) is continuous. Suppose we know that:

i) (ζ,Ψt(ζ)) : ζ ∈ ∂Dt ⊂ K, for some compact set K ⊂ Ω for each t ∈ [0, 1];

ii) Graph(Ψ0) ⊂ Ω and Graph(Ψ1) * Ω.

Then there exists a neighbourhood V of Graph(Ψ1) ∪ K and a neighbourhood W of K,K b W ⊂ Ω ∩ V , such that for each f ∈ H(Ω) there exists a G f ∈ H(V) and G f |W ≡ f |W .

Here, if D is an open set then H(D) denotes the class of holomorphic functions or maps on D.To the best of my knowledge, there does not seem to be a proof of (∗) in the above generality

in the literature. If Graph(Ψt) ∩ Graph(Ψs) = ∅ for s , t, then (∗) is classical. If Dt = D, a fixedplanar region, for each t ∈ [0, 1], then the above can be deduced by analytically continuing anH(Ω)-germ (see the first page of Chapter 3 for a definition) along the path t 7→ (ζ,Ψt(ζ)) for afixed ζ ∈ D. In more generality, but still assuming Graph(Ψt) ∩ Graph(Ψs) = ∅ for s , t, (∗)follows from Behnke’s work [2].

When Graph(Ψt) ∩ Graph(Ψs) , ∅ for some t , s, we must worry about multivaluedness.This forces us to ask whether the maps idDt

×Ψt can be “lifted” into the envelope of holomorphyof Ω continuously in the parameter t ∈ [0, 1]. With Dt = D, the open unit disc in C, for all

42

t ∈ [0, 1], and with maps more general than the form idDt×Ψt (to be precise: continuously varying

holomorphic immersions into Cn), Joricke has shown that such “lifting” is possible [8, Lemma1]. With some care, the conclusion of (∗) can now be deduced in this special case — but it wouldbe nice to have some record of the latter argument in the literature.

As for (∗) in the above generality: it appears to be a folk therem (also see the remark followingTheorem 6.1). The aim of this chapter is to give a proof of (∗) after making precise some of theloosely-stated assumptions in (∗). The two theorems presented below are from the article [3],currently submitted for publication.

Some notation: if S ⊂ C × [0, 1], we shall use tS to denote the set ζ ∈ C : (ζ, t) ∈ S .

Theorem 6.1. Let Ω be a domain in Cn, n ≥ 2. Let K be a compact subset of Ω. Let Γ be acompact body in C × [0, 1] such that ∂∗Γ := (C × (0, 1)) ∩ ∂Γ is a (not necessarily connected)C1-smooth submanifold with boundary. Assume that the sets

Dt :=

t(Γ), if t ∈ (0, 1),( tΓ) ∩

(C \ t(∂∗Γ)

), if t = 0, 1,

are domains in C with C1-smooth boundaries for each t ∈ [0, 1], and that the set-valued function[0, 1] 3 t 7−→ Dt is continuous relative to the Hausdorff metric (on the space of compact subsetsof C). Suppose there is a continuous map Ψ : Γ→ Cn with the following properties:

1) For each t ∈ [0, 1], Ψt := Ψ(·, t) ∈ H(Dt) ∩ C(Dt);

2) Ψt(∂Dt) ⊆ K for all t ∈ [0, 1];

3) Ψ0(D0) ⊂ Ω and Ψ1(D1) * Ω;

4) For each t ∈ [0, 1], Ψt is a continuous imbedding and Ψt|Dt is a holomorphic imbedding.

Then there exists a neighbourhood V of Ψ1(D1) ∪ K and a neighbourhood W of K satisfyingK b W ⊂ Ω ∩ V such that for any f ∈ H(Ω), there exists G f ∈ H(V) such that G f |W ≡ f |W .

Remark 6.2. Theorem 6.1 would follow as a special case of a Kontinuitatssatz-type result byChirka and Stout [6]. However, there are gaps in their proof. L. Nobel has shown in his thesis [11]that the Chirka–Stout Kontinuitatssatz is erroneous. Now, Theorem 6.1 does not directly followfrom Nobel’s (unpublished) alternative to Chirka–Stout’s statement. Moreover, the approach ofChirka–Stout and Nobel requires much machinery, such as currents, volume inequalities, etc.Thus, an elementary proof in the set-up of Theorem 6.1 would be desirable.

In this proof, we shall continue with the notation introduced in Chapter 5. So, (Ω, p, j) willdenote the envelope of holomorphy of Ω. We recall: p is the canonical local homeomorphisminto Cn that gives Ω the structure of an unramified domain spread over Cn and j : Ω → Ω is animbedding such that p j ≡ idΩ. The only machinery that our proof of Theorem 6.1 requires isThullen’s construction of the envelope of holomorphy of Ω — see Theorem 3.6 above. All otherelements of our proof are elementary and developed from scratch.

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Now, the usefulness of the Kontinuitatssatz is that it gives us information about p(Ω). Gen-erally, we have very little information about how Ω looks. But, if we have explicit informationabout the pair (Ω,K) occuring in Theorem 6.1, then we gain information about p(Ω). In ourproof of Theorem 6.1, an explicit description of V , in terms of Ω,K and Ψ1(D1), is obtained.

If we have slightly more information about (Ω, p, j) than is available in Theorem 6.1, then itshould be possible to relax some of the conditions on the family Ψtt∈[0,1]. For example: couldwe drop the requirement that Ψt, t ∈ [0, 1], be imbeddings ? This leads to our next theorem.

Theorem 6.3. Let Ω be a domain in C2, and let (Ω, p, j) denote the envelope of holomorphy of Ω.Let K be a compact subset of Ω. Let Γ be a compact body having exactly the same properties asin Theorem 6.1 and let the domains Dt, t ∈ [0, 1], be the same as in Theorem 6.1. Suppose thereis a continuous map Ψ : Γ → Cn having the properties (1), (2) and (3) stated in Theorem 6.1.Let us assume that p is injective on p−1Ω. Then there exists an open set V ⊃ Ψ1(D1) ∪ Ω suchthat for each f ∈ H(Ω), there exists G f ∈ H(V) such that G f |Ω ≡ f .

Both theorems require a description of Thullen’s construction of the envelope of holomorphy.Salient features from Chapter 3 are recalled in Section 6.1. Also, we require a lifting result forthe family Ψtt∈[0,1]. We state and proof this result in Section 6.2.

6.1 Recalling notations and terminologyRecall that Thullen’s construction depends on the sheaf of S -germs on Cn, denoted here as O(S ).Precise definitions have been given in Chapter 3. The case that is important for us here is whenS = H(Ω).

We shall use the notation evolved in Chapter 3. Thus, throughout this paper, if (U, φ f : f ∈H(Ω)) is a representative of an H(Ω)-germ at a, we shall denote that germ as [U, φ f : f ∈ H(Ω)]a.We shall denote the canonical extension of f to Ω, f ∈ H(Ω), as F f . For the reader’s convenience,let us also recapitulate/summarize:

(i) Ω is realised as the connected component of O(H(Ω)) (i.e. the sheaf of H(Ω)-germs ofholomorphic functions) containing the (connected) set [Ω, f : f ∈ H(Ω)]a : a ∈ Ω.

(ii) Given any f ∈ H(Ω), the function F f is defined on Ω as follows. Each point x ∈ Ω isan H(Ω)-germ, so there exists a neighbourhood U, containing a = p(x) ∈ Cn, such thatx =

[U, φg : g ∈ H(Ω)

]a. Then F f (x) := φ f (a).

Consider an unramified domain (X, p0,Cn) spread over Cn. Let a ∈ X. Refer to Defini-

tion 3.18 for the definition of a polydisc of radius r about a. We denote this object by P(a, r).The maximal polydisc about a is the union of all polydiscs about a. As demonstrated in Sec-tion 3.2, the maximal polydisc is itself a polydisc about a, and its radius

r0 = supr > 0 : P(a, r) is a polydisc about a.

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Finally — see Definition 3.20 — the radius of the maximal polydisc about a is called the distanceof a from the boundary of X and is denoted by dX(a). If A ⊂ X, we set

dX(A) := infa∈AdX(a)

6.2 A fundamental propositionIn this section, we shall prove a proposition that allows us to tackle both Theorem 6.1 and Theo-rem 6.3 within the same framework. We point out to the reader that an essential part of the proofof the proposition below depends on the structure of Thullen’s construction of the envelope ofholomorphy (Ω, p, j). The other important fact that we use is the Cartan–Thullen characterizationof a domain of holomorphy — see Theorem 3.26.

Recall that, given a compact K ⊂ X, the holomorphically convex hull of K is the set

KX := x ∈ X : | f (x)| ≤ supK | f | for all f ∈ H(X).

We make one small notational point. In this section, if z = (z1, . . . , zn) ∈ Cn, then

‖z‖ := max|z1|, . . . , |zn|.

Proposition 6.4. Let Ω be a domain in Cn, n ≥ 2, and let (Ω, p, j) denote the envelope ofholomorphy of Ω. Let K be compact subset of Ω. Let Γ be a compact body in C × [0, 1] havingexactly the same properties as in Theorem 6.1. Let Ψ : Γ → Cn be a continuous map satisfyingconditions (1), (2) and (3) from Theorem 6.1. Then, for each t ∈ [0, 1], there exists a continuousmap Ψt : Dt → Ω such that p Ψt ≡ Ψt and satisfies Ψt(ζ) = [Ω, f : f ∈ H(Ω)]Ψt(ζ) for everyζ ∈ ∂Dt.

Proof. Define the set

S :=t ∈ [0, 1] : ∃ Ψt′ : Dt′ → Ω, Ψt′ ∈ C(Dt′), p Ψt′ ≡ Ψt′ and

Ψt′(ζ) = [Ω, f : f ∈ H(Ω)]Ψt′ (ζ) for all ζ ∈ ∂Dt′ and for all t′ ∈ [0, t].

We shall show that S is both open and closed in [0, 1] and S , ∅. To begin, define Ψ0 : D0 → Ω

by

Ψ0(ζ) := [Ω, f : f ∈ H(Ω)]Ψ0(ζ) for all ζ ∈ D0. (6.1)

Since Ψ0(D0) ⊂ Ω, (6.1) is well-defined. Continuity is easy. Therefore 0 ∈ S .Let τ = sup S . By the definition of S we have that [0, τ) ⊂ S . So to show that S is closed

it is therefore enough to show that τ ∈ S . For this purpose, let us consider the envelope ofholomorphy, (Ω, p, j). We recall an important aspect of the construction of Ω (given at thebeginning of Section 6.1). For each f ∈ H(Ω), its canonical extension to Ω is described asfollows:

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(∗∗) For each point x ∈ Ω, which is an H(Ω)-germ, let a = p(x) ∈ Cn and let U be a neighbour-hood of a such that x = [U, φg : g ∈ H(Ω)]a. Then F f (x) = φ f (a) = φ f (p(x)).

Claim: If t ∈ S , then Ψt(Dt) ⊂ (K∗)Ω where K∗ = j(K).Proof of claim: Let F ∈ H(Ω). As p Ψt = Ψt, both p and Ψt are holomorphic, and p is alocal isomorphism (biholomorphism) we have Ψt ∈ H(Dt) (by the definition of holomorphicityof maps with values in an unramified domain spread over Cn). Therefore F Ψt ∈ H(Dt)∩C(Dt).Let ζ ∈ Dt. By the maximum modulus principle

|F(Ψt(ζ))| = |F Ψt(ζ)| ≤ sup∂Dt|F Ψt| = supΨt(∂Dt) |F| ≤ supK∗ |F|, (6.2)

because Ψt(∂Dt) ⊂ K∗ (this is a consequence of property (i), stated in Section 6.1, of the envelopeof holomorphy). Thus Ψt(Dt) ⊂ (K∗)Ω. Hence the claim.

For any ζ ∈ Dt and t ∈ S , let us denote

Ψt(ζ) = [U t,ζ , φt,ζf : f ∈ H(Ω)]Ψt(ζ) (6.3)

where U t,ζ is some neighbourhood of Ψt(ζ) and φt,ζf ∈ H(U t,ζ).

In what follows, we shall abbreviate dΩ(K∗) to d(K∗). There exists an ε > 0, a priori ε =

ε(t, ζ), such that all the H(Ω)-germs in the definition below makes sense and the open set

N(ε, t, ζ) := [U t,ζ , φt,ζf : f ∈ H(Ω)]a : a ∈ Dn(Ψt(ζ), ε) (6.4)

is contained in Ω. As Ω is a domain of holomorphy, it follows from the above claim and fromthe Cartan–Thullen Theorem that P(Ψt(ζ), d(K∗)) ⊂ Ω for all t ∈ S and for all ζ ∈ Dt. HereP(Ψt(ζ), d(K∗)) is as explained in Definition 3.18. It follows from the observations followingDefinition 3.18 that p|P(Ψt(ζ), d(K∗)) is invertible. Thus, the functions

Φt,ζf := F f (p|P(Ψt(ζ), d(K∗)))

−1 (6.5)

are holomorphic functions on Dn(Ψt(ζ), d(K∗)). Given our description (∗∗) above of the con-struction of F f , the above statement gives us the following equality of H(Ω) germs:

[U t,ζ , φt,ζf : f ∈ H(Ω)]Ψt(ζ) = [Dn(Ψt(ζ), d(K∗)), Φ

t,ζf : f ∈ H(Ω)]Ψt(ζ).

We summarise this as the following:

Fact 1: For any t ∈ S and ζ ∈ Dt, (Dn(Ψt(ζ), d(K∗)), Φt,ζf : f ∈ H(Ω)) is a representative of

the H(Ω)-germ Ψt(ζ), where Φt,ζf is the function given by (6.5).

Given this fact, the open set N(ε, t, ζ) makes sense for any number ε such that 0 < ε ≤ d(K∗),and is contained in Ω for all ζ ∈ Dt, for all t ∈ S . Furthermore, it will be understood that, forthe remainder of this proof, U t,ζ = Dn(Ψt(ζ), d(K∗)) and φt,ζ

f = Φt,ζf whenever referring to the

representation (6.3) for Ψt for t ∈ S .

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As Ψ : Γ → Cn is a continuous map and Γ is compact, there exists δ > 0 such that for(ζ, t1), (η, t2) ∈ Γ,

|(ζ, t1) − (η, t2)| < δ ⇒ ‖Ψt1(ζ) − Ψt2(η)‖ < d(K∗), (6.6)

where, for any (ζ, t) ∈ C × [0, 1], |(ζ, t)| :=√|ζ |2 + t2.

In the following argument D(a,R) will denote the disc in C with centre a and radius R. Givenour hypotheses:

• By the smoothness of ∂Dt, for each t ∈ [0, 1], there exists an ε∗ := ε∗(t) > 0 such that foreach z ∈ At := ζ ∈ C : dist[ζ, ∂Dt] < ε∗ there is a unique closest point in ∂Dt, and suchthat each connected component of ∂Dt is a strong deformation retract of the component ofAt containing it.

• By the properties of ∂∗Γ, there exists a δ∗ := δ∗(t) > 0 such that ∂∗Γ divides each connectedcomponent ofAt × ([0, 1] ∩ [t − δ∗, t + δ∗]) into two connected components.

It now follows from an elementary argument, using compactness of [0, 1], that there exists aconstant r, 0 < r ≤ δ, such that for each t ∈ [0, 1] and each ζ ∈ Dt, D(ζ, r)∩Dt is connected. Letus now fix a t ∈ S . By continuity, there exists a constant δt, 0 < δt ≤ r, depending only on t, suchthat

η ∈ D(ζ, δt) ∩ Dt ⇒ Ψt(η) ∈ N(d(K∗), t, ζ).

Therefore

η ∈ D(ζ, δt) ∩ Dt ⇒ Ψt(η) = [U t,ζ , φt,ζf : f ∈ H(Ω)]Ψt(η). (6.7)

Let us now fix ζ ∈ Dt. Set A(t, ζ) := D(ζ, r) ∩ Dt. Due to connectedness, for any x ∈ A(t, ζ),there is a finite chain of discs ∆1,∆2, . . . ,∆N(x) of radius δt and points ζ = y0, y1, . . . , yN(x) = xsuch that

y j = centre of ∆ j+1, y j+1 ∈ ∆ j+1 for j = 0, 1, . . . ,N(x) − 1.

Suppose we have been able to show that for any x ∈ A(t, ζ) that can be linked to ζ by a chain ofat most M discs in the above manner, we have

Ψt(x) = [U t,ζ , φt,ζf : f ∈ H(Ω)]Ψt(x).

Now let z ∈ A(t, ζ) be a point that is linked to ζ by a chain ∆1,∆2, . . . ,∆M+1 of discs of radius δt.Let yM be the centre of ∆M+1. By our inductive hypothesis

Ψt(yM) = [U t,ζ , φt,ζf : f ∈ H(Ω)]Ψt(yM).

By the representation (6.3) and the fact that U := U t,ζ ∩ U t,yM is convex,

φt,ζf |U ≡ φ

t,yMf |U for all f ∈ H(Ω). (6.8)

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Now applying (6.7) to the pair (yM, z), we get

Ψt(z) = [U t,yM , φt,yMf : f ∈ H(Ω)]Ψt(z)

= [U, φt,ζf : f ∈ H(Ω)]Ψt(z),

which follows from (6.8) and the fact that, as |z− yM | < δt ≤ δ, Ψt(z) ∈ U. From the last equality,we get Ψt(z) = [U t,ζ , φ

t,ζf : f ∈ H(Ω)]Ψt(z). Given (6.7), mathematical induction tells us that we

have actually established the following:

Fact 2: For any t ∈ S and ζ ∈ Dt,

φt,ζf |X(ζ,η) ≡ φ

t,ηf |X(ζ,η) for all f ∈ H(Ω), (6.9)

and for any η ∈ Dt such that |η − ζ | < r, where X(ζ, η) = U t,ζ ∩ U t,η.

Now we have that Dtt∈[0,1] is a continuous family (hence a uniformly continuous family)with respect to the Hausdorff metric. From uniform continuity, the fact that [0, τ) ⊂ S , and fromthe properties of ∂∗Γ, we can find a t0 ∈ S such that (τ − t0) ≤ r/4 and such that

Dτ ⊂ B(Dt0 ,r/4), ∂Dτ ⊂ B(∂Dt0 ,

r/4),

where, given a set A ⊂ C and C > 0, B(A,C) := ∪ζ∈AD(ζ,C). Let ζ ∈ Dτ. Then there existsx(ζ) ∈ Dt0 such that |ζ − x(ζ)| < r/4. Define

Ψτ(ζ) := [U t0,x(ζ), φt0,x(ζ)f : f ∈ H(Ω)]Ψτ(ζ).

As r ≤ δ, it follows from Fact 1 and (6.6) that Ψτ(ζ) ∈ Ω.Claim: The function Ψτ : Dτ → Ω is well-defined and continuous.Proof of claim: Let ζ ∈ Dτ. Suppose x(ζ), y(ζ) are two different points in Dt0 such that |ζ−x(ζ)| <r/4, |ζ − y(ζ)| < r/4. Then |x(ζ) − y(ζ)| < r/2. As (τ − t0) ≤ r/4, it follows from (6.6) thatΨτ(ζ) ∈ U t0,x(ζ) ∩ U t0,y(ζ). Therefore, by (6.9)

[U t0,x(ζ), φt0,x(ζ)f : f ∈ H(Ω)]Ψτ(ζ) = [U t0,y(ζ), φ

t0,y(ζ)f : f ∈ H(Ω)]Ψτ(ζ).

Therefore Ψτ is well-defined.We now pick a ζ ∈ Dτ and fix it. By the construction of the topology on Ω, given any neigh-

bourhood G of Ψτ(ζ) in Ω, there exists ε ∈ (0, d(K∗)), sufficiently small, such thatN(ε, τ, ζ) ⊂ G.Now let 0 < ε < d(K∗). Since Ψτ : Dτ → Cn is continuous there exists σ = σ(ε), withσ(ε) ∈ (0, r/4), such that if |η − ζ | < σ, η ∈ Dτ, then Ψτ(η) ∈ Dn(Ψτ(ζ), ε). Let η ∈ Dτ such that|η− ζ | < σ. There exists x(η) ∈ Dt0 such that |η− x(η)| < r/4. Therefore |x(ζ)− x(η)| < 3r/4. Recallthat (τ − t0) ≤ r/4. Therefore, applying (6.6) and (6.9) we get [U t0,x(ζ), φ

t0,x(ζ)f : f ∈ H(Ω)]Ψτ(η) =

[U t0,x(η), φt0,x(η)f : f ∈ H(Ω)]Ψτ(η). Therefore Ψτ(η) ∈ N(ε, τ, ζ) whenever |ζ − η| < σ and η ∈ Dτ.

By the remarks at the begining of this paragraph, Ψτ is continuous at ζ. As ζ is an arbitrary pointof Dτ we have that Ψτ is continuous. Hence the claim.

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By definition of Ψτ, p Ψτ ≡ Ψτ. For all x ∈ ∂Dt0 , Ψt0(x) = [Ω, f : f ∈ H(Ω)]Ψt0 (x). Since∂Dτ ⊂ B(∂Dt0 , r/4), for any ζ ∈ ∂Dτ, there exists an x(ζ) ∈ ∂Dt0 such that ζ ∈ D(x(ζ), r/4). By theargument in the first part of the previous claim, Ψτ(ζ) = [Ω, f : f ∈ H(Ω)]Ψτ(ζ) for all ζ ∈ ∂Dτ.This implies that τ ∈ S .

We have established that S is closed.The above method for showing that τ ∈ S tells us more. As [0, 1] 3 t 7−→ Dt is uniformly

continuous relative to the Hausdorff metric, and as ∂∗Γ is smooth, there exists a constant δ′ > 0such that

|t − s| < δ′ ⇒ H(Dt,Ds) < r/4,

0 < (t − s) < δ′ ⇒ ∂Dt ∈ B(∂Ds, r/4),

whereH denotes the Hausdorff metric and r is exactly as fixed above. Let δ∗ := min(δ′, r/4). Thesame argument, with appropriate replacements where necessary, shows that if t0 ∈ S , then τ ∈ Sfor each τ ∈ [t0,min(1, t0 +δ∗)). Therefore S is both open and closed. We have shown that 0 ∈ S .Hence S = [0, 1], which completes the proof.

6.3 The proof of the main theorems

The proof of Theorem 6.1. By Proposition 6.4 we have that there exists Ψ1 : D1 → Ω, a con-tinuous map such that p Ψ1 ≡ Ψ1 and Ψ1(∂D1) ⊂ K∗. The condition p Ψ1 ≡ Ψ1 actuallyimplies that Ψ1 ∈ H(D1), by the definition of holomorphicity of maps with values in an unram-ified domain spread over Cn. Thus, as argued in the proof of Proposition 6.4, Ψ1(D1) ⊆ (K∗)Ω.Therefore, by Theorem 3.26, P(Ψ1(ζ), d(K∗)) ⊆ Ω for all ζ ∈ D1. Let us define for any r > 0,

ω(r) := ζ ∈ D1 : dist[ζ, ∂D1] > r.

Claim: There exists ε0 ∈ (0, dΩ(K)/2) such that if ζ, η ∈ D1, and Dn(Ψ1(ζ), ε0) ∩ Dn(Ψ1(η), ε0) ,∅, then

either Dn(Ψ1(ζ), ε0) ∪ Dn(Ψ1(η), ε0) ⊆ Ω,

or Dn(Ψ1(ζ), 2ε0) ∩ Dn(Ψ1(η), 2ε0) ∩ Ψ1(D1) , ∅ and

Dn(Ψ1(x), 2ε0) ∩ Ψ1(D1), x = ζ, η, is path-connected. (6.10)

Proof of claim: K is a compact subset of Ω. Therefore there exists an ε ∈ (0, dΩ(K)) such thatDn(z, ε) ⊆ Ω for all z ∈ K. Therefore, as Ψ1(∂D1) ⊆ K and Ψ1 is continuous on D1, which is acompact set, there exists δ > 0 such that Ψ1(ω(2δ)C) ⊂ ∪z∈KDn(z, ε/4). By construction we seethat for any ε ∈ (0, ε/4], if ζ ∈ D1 \ ω(2δ) and η ∈ D1, then

Dn(Ψ1(ζ), ε) ∩ Dn(Ψ1(η), ε) , ∅ =⇒ Dn(Ψ1(ζ), ε) ∪ Dn(Ψ1(η), ε) ⊆ Ω. (6.11)

The above statement remains true when ζ and η are interchanged.

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Let us viewCn as a Hermitian manifold with T (Cn) as the holomorphic tangent space equippedwith the standard Hermitian inner product 〈. , .〉std on each TpC

n, p ∈ Cn. Let

NΨ1 := the normal bundle of Ψ1(D1) with respect to 〈. , .〉std.

Let π denote the bundle projection. It is well known that for any relatively compact subdomain∆ b D1, there exists r∆ > 0, such that, if we define:

N(Ψ1, r∆) :=⋃

p∈Ψ1(∆)v ∈ NΨ1

∣∣∣p

: 〈v, v〉std < r2∆,

then the map Θ(v) := π(v) + v is a holomorphic imbedding of N(Ψ1, r∆) into Cn.Write ∆ := ω(δ/2) and ω := ω(δ). Since Ψ1(ω) is a compact subset of Cn, there exists an

ε0 ∈ (0,min(dΩ(K)/2, ε/4)) so small that:

• Dn(Ψ1(z), 2ε0) ∩ Ψ1(D1) is path-connected for all z ∈ Ψ1(∆);

• Dn(z, 2ε0) ∩ Ψ1(∆) b Ψ1(∆) (with respect to the relative topology on Ψ1(∆)) for all z ∈Ψ1(ω);

• Dn(z, ε0) ⊆ Θ(N(Ψ1, r∆)) for all z ∈ Ψ1(ω); and

• π Θ−1(z) ∈ Dn(p, 2ε0) ∩ Ψ1(∆) for all z ∈ Dn(p, ε0) and for all p ∈ Ψ1(ω)

(where we understand that Θ : N(Ψ1, r∆)→ Cn).Now suppose ζ , η ∈ Ψ1(ω) and Dn(ζ, ε0) ∩ Dn(η, ε0) , ∅. Let w ∈ Dn(ζ, ε0) ∩ Dn(η, ε0).

Then, as Θ is an imbedding, there is a unique v ∈ N(Ψ1, r∆) such that Θ(v) = w. By ourconstruction π(v) = π Θ−1(w) ∈ Dn(ζ, 2ε0) ∩ Dn(η, 2ε0) ∩ Ψ1(∆). Combining this with thestatement culminating in (6.11), we have the claim.

Write:

V :=(∪ζ∈D1

Dn(Ψ1(ζ), ε0))∪ (∪z∈KDn(z, ε0)) , W := ∪z∈KDn(z, ε0).

Clearly, W ⊂ V ∩ Ω as ε0 < dΩ(K) and V * Ω as Ψ1(D1) ⊆ V . Let us use the notation (6.3) torepresent Ψ1(ζ) (which is an H(Ω)- germ). We have established in the proof of Proposition 6.4— see Fact 1 stated in Section 6.2 — that there is a representative (U1,ζ , φ

ζf : f ∈ H(Ω)) of Ψ1(ζ)

such that U1,ζ = Dn(Ψ1(ζ), d(K∗)). Given any f ∈ H(Ω), define G f : V → C by

G f (z) :=

φζf (z), if z ∈ Dn(Ψ1(ζ), ε0),f (z), if z ∈ Dn(θ, ε0) for some θ ∈ K \ Ψ1(D1).

Here, for simplicity of notation, we write φζf := φ1,ζf .

We shall prove that the function G f is well-defined on V . To start with, let us assume that forsome z ∈ V , z ∈ Dn(Ψ1(ζ), ε0) ∩ Dn(Ψ1(η), ε0), where ζ , η. By the above claim, there are twopossible outcomes. If Dn(Ψ1(ζ), ε0) ∪ Dn(Ψ1(η), ε0) ⊆ Ω, then φζf = f = φ

ηf by Proposition 6.4.

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Therefore, we must focus on the outcome (6.10) given by the last claim. In this situation, thereexists τ ∈ D1 such that Ψ1(τ) ∈ Dn(Ψ1(ζ), 2ε0) ∩ Dn(Ψ1(η), 2ε0). The same argument that leadsto Fact 2 stated in the proof of Proposition 6.4 can be used to show the following analogue ofFact 2:

Fact 2′: For any t ∈ [0, 1] and ζ ∈ Dt, if Aζ := the path-component of Ψ−1t (U t,ζ) containing ζ,

thenφ

t,ζf |X(ζ,η) ≡ φ

t,ηf |X(ζ,η) for all f ∈ H(Ω) and for all η ∈ Aζ ,

where X(ζ, η) = U t,ζ ∩ U t,η.Since 2ε0 < dΩ(K) ≤ d(K∗), Ψ1(τ) ∈ U1,x, x = ζ, η. Furthermore, under the condition (6.10), wesee that

τ lies in the connectected component of Ψ−11 (U1,x) containing x, x = ζ, η,

since Ψ1 is an imbedding. From Fact 2′, we deduce that

φζf |U1,ζ∩U1,τ ≡ φτf |U1,ζ∩U1,τ , φ

ηf |U1,η∩U1,τ ≡ φτf |U1,η∩U1,τ ,

for all f ∈ H(Ω). Therefore there is a neighbourhood N of Ψ1(τ) such that

φζf |N ≡ φ

ηf |N . (6.12)

As X := Dn(Ψ1(ζ), 2ε0) ∩ Dn(Ψ1(η), 2ε0) is connected, by (6.12) we have φζf |X ≡ φηf |X. Therefore

φζf (z) = φ

ηf (z).

Lastly, we consider the case when z ∈ Dn(Ψ1(ζ), ε0) ∩ Dn(θ, ε0), where ζ ∈ D1 and θ ∈ K. Inthis case, the argument for well-definedness is exactly as presented at the beginning of the lastparagraph. This completes the proof of well-definedness.

By the construction of G f on V , we have, given that z ∈ Dn(θ, ε0) for some θ ∈ K, G f (z) =

f (z). Therefore G f |W ≡ f |W . Finally, by the fact that holomorphicity is a local property, weconclude that G f ∈ H(V).

We now turn to the proof of Theorem 6.3. For this purpose, we will use Theorem 5.4 byJupiter.

The proof of Theorem 6.3. By Theorem 5.4, p : Ω → Cn is an injective map. As p is a localbiholomorphism, p(Ω) is an open connected set in Cn and p−1 is a holomorphic map on it. Letus write

V :=⋃

z∈Ψ1(D1)∪KDn(z, dΩ(K)).

Since p Ψ1 ≡ Ψ1 and Ψ1(D1) ⊂ Ω, we have V * Ω, as Ψ1(D1) * Ω.As dΩ(K) ≤ d(K∗) and Ψ1(D1) ∈ (K∗)Ω, the Cartan–Thullen Theorem gives V ⊂ p(Ω). Thus,

if for any f ∈ H(Ω), we define G f : V → C by

G f := F f p−1,

then G f ∈ H(V) and G f |Ω ≡ f .

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Remark 6.5. Note that we could have taken V = p(Ω) in the above proof. But we want tohighlight the fact that the Kontinuitatssatz is a means of deducing information about p(Ω) whenvery little is known about the geometry of Ω. The spirit of Theorem 6.3 is that, while we knowvery little about the finer geometric properties of Ω, this theorem shows what we can learnabout p(Ω) if we have a little extra information about (Ω, p, j). In using both Theorem 6.3 andTheorem 6.1, it is understood that we have good information about the pair (Ω,K). It is inkeeping with this spirit that we chose V as above.

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