REMOVABLE SETS FOR HOLOMORPHIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES Abstract

EDGAR LEE STOUT * We show that every closed subset of Cl that has finite (2N-2)-dimensional measure is a removable set for holomorphic functions, and we obtain a related result on the ball .


Introduction
A colleague has remarked that Everybody knows that a set too small to be a variety is removable.The present paper is devoted to an explication of certain cases of this general philosophy, which are motivated by a result of Shiffman [11], [12], to the effect that a closed subset E of a domain S2 in CN is removable for holomorphic functions in the sense that if f E O(Q\E), then f extends holomorphically to an f E O(9) provided A2N -2 (E) = 0, A 2N-2 denoting (2N -2)-dimensional Hausdorff measure .tBecause of the Hartogs phenomenon, this result is of interest only in the case that the set E is not compact .Our principal result is an extension of this theorem, in the case that S2 is C N itself, that replaces the hypothesis that A2N-2 (E) = 0 by the hypothesis that A2N-2 (E) be finite .
We shall prove the following result .
2 .The main result 1 .Theorem .If E C C N, N >_ 2, is a closed set with A 2N-2 (E) < oo, then E is removable .This is a global theorem in that the conclusion fails for closed sets in bounded domains.For example, if 9 is a bounded domain that contains the origin, and if E = S2 n {ZN = 0}, then A 2N-2 (E) < oo, but E is not removable, as the function f(z) = zñ 1 shows.*Research supported in part by grant DMS-8801032 from the National Science Foundation .tA veision of the result of Shiffman had been found earlier by Caccioppoli [3] .

E . L . STOUT
Proof of the Theorem : We give a direct proof in the case of C2 and then argue by induction .
The proof in C2 depends on a lemma, which is based on work of Alexander Denote by BN the unit ball in CN and by rBN the set {rz : z E BN} when r E (0, oo) .The boundary brBN is the sphere in CN of radius r centered at the origin.
The referee has kindly drawn the author's attention to Théoréme 4, p .309, of Sibony's paper [15], which contains this lemma, with the constant 2 rather than the constant -,/2 -7r, as a special case.It would be of interest to what the best value of the constant is.Proof.. First, let X C bB2 be a compact set with 0 E X, X the polynomially convex hall of X.According to Theorem 1 of [1], if 7rj : C2 --> C is the projection given by ri(zl , z2 ) = zi,i = 1,2 then whence one of the summands, say the first, in (1) is at least ir/2 .
Let Z denote the polynomially convex hull of the set .7r, (X), Le., the union of ir, (X) and the bounded components of C\7r 1 (X).The boundary of Z is the boundary of the unbounded component of the set C\7rl (X), and the set Z does not disconnect the plane.According to the isoperimetric inequality [2, § §14.3, 14.61 Every point of bZ is a peak point for the algebra P(Z),* and so for every point p E bZ, the set 7ri 1 (p) fl X is a peak set for the algebra P(X ), which can be identified with P(X ) .Consequently, the set Sri 1 (p) meets the Silov boundary for P(X ), Le., the set X : We have that r1 (X) D bZ .As rl is a Lipschitz map with Lipschitz constant one, we must have Al (X) > Al (M).As A1 (bZ) >_ V/-27r, we have A1 (X) >_ v~-27r .
The lemma implies that the origin does not lie in the polynomially convex hull of the set E n bt j B2 .If <Pj denotes the restriction to bt jB2\E of the function f, then Pj satisfies the tangential Cauchy-Riemann equations and so ( [6], [7], [8,Appendix]) continues holomorphically into t j B2\(Enbt;B 2 )^, which is a neighborhood of the origin.Denote this extension by (Dj .That o¿j is an extension of f follows from the fact that f and ¿j agree on an open subset of btjB 2 . The theorem is proved now in the two-dimensional case.We next assume it proved in the N-dimensional case and derive the (N + 1)-dimensional case.To this end, it is of some importance to notice that the axgument just given works equally well granted only that A2 (E n {z : Iz1 > 1}) is finite .
We consider in CN + 1 a closed subset E with A2N (E) < oo .Let f E O(CN \E) .Fix a point z E CN+ 1 and denote by CJN+1,N (z) the Grassmannian of all complex af$ne N-planes in CN+ 1 that pass through the point z.There is a natural invariant measure on JCN+1,N (z), which we shall denote by dp(II) .We assume this mea_sure to be normalized so that it has total mass one .We have by [12] that if E = E n {jzj > 1}, then A2N-2 (E n II)du(II) < CNA2N (E) < 00 CJN}1,N for a fixed constant CN .In particular, for almost every II E _ GJN+1,N(z), A 2N-2 (E n II) < oo .Thus, for almost every II, fI(II\E) extends holomorphically through all of II.Denote this extension by fri z .We define This gives a well-defined value for F(z), because fn,z(z) is independent of the choice of II: Two II's, say II, and II2 , in CGN+1,N (z) intersect in an affine subspace of C of positive dimension on which fn l z and frj2 Z agree .Thus they agree at z .The function F defined in this way is defined on all of CN+ 1 , and it agrees with f on CN+1\E .
We have to see that F is holomorphic, and for this, it suffices to show that it is continuous .To do this, let {zn}°°_1 be a sequence in CN that converges to zo ; we shall show that F(zn ) --+ F(zo ).Fix a II o E GJN+1,N(zo) with AIN-2 (no n E) < oo and such that F is holomorphic on IIo .For each n, choose fin E JGN+1,N (Zn) such that A IN-2 (IIn n E) < oo, such that F is holomorphic on II n, and such that II n -+ IIo.
If z E CN and II E ON+1,N (z), denote by P,(II) the projectioe space of all complex lines in II through the point z.We have dimR P., (II) = 2N -2.There are large values of R such that A2N-s (bBN+1 (z, R) n E n IIo) < oo, so if 7r IIo \{z o } -> Pzo(IIo) is the standard projection, then 7r (bBN+1(zo, R)n E n IIo) is a set of measure zero in Pzo .Thus, there is a complex line A o with zo E Ao C IIo and with A o n E n bBN+1 (zo , R) = 0. We may choose An E Pzn (II n ) so that An --+ Ao .For large values of n, An n E n bBN+1 (zo , R) = 01.If we apply the Cauchy integral formula in A n and Ao to represent F(zn ) and F(zo ) as the Cauchy integral of f over the circle A n n bBn+1 (z o , R) and of Ao n bBN+1 (zo, R), respectively, we find that as n ---> oo, F(zn) -> F(zo ) as desired .
Thus, F is continuous and so necessarily holomorphic .This completes the proof of the theorem .

. Variations on the theme
The first variation is to the effect that there is an analogue of the result for submanifolds of CN : Leí M be a k-dimensional complex submanifold of CN , and leí E C M be a closed subset with A 2k-2 (E) < oo .* If f E O(M\E), then f continues holomorphically into all of M.
In the case that M is an algebraic manifold, we can invoke [10, Th. 10, p. 52] to find a projection 7r : CN -> Ck that exhibits M as an analytic cover over Ck .Using symmetric functions and applying the result already established in Ck , we can derive the result on M.
In the case of a general M, there will be no such projection, and, in essence, it is necessary simply to rewrite the proof given above.The case n = 2 proceeds as before : Fix zo E E. For certain large values of t, A1 (E n bBN (zo, t)) will be small and bB N (z o , t) n M will be a smooth (2k -1)-dimensional real hypersurface that bounds the domain 0(t, zo ) = BN(z o , t) n M. By Lemma 2, the polynomially convex hull of E n bBN(zo t) does not contain zo , and by the extension theorem given by Laurent-Thiebeaut [6], flb ,~,(t, zo)\E extends holomorphically into a neighborhood of zo .The rest of the argument in the two-dimensional case is as before .
For the induction step we replace the affine hyperplanes used in the proof of the theorem by intersections M n II, II a codimension one affine hyperplane *Here, as above, we are computing Hausdorff measures with respect to the Euclidean metric on CN = R2N .Below we shall consider the Hausdorff measures associated to certain other metrics, but there we shall be quite ezplicit about the metrics involved.
in CN that is transverse to M. The generic II is transverse to M and so, generically, m n II is a codimension one submanifold of .M .In a bit more detail, if z E Nl, then almost every II E `JN,N-1 (z) is transverse to M and, by [12,Lemma 5] almost every II also satisfies AZk-a (II n E) < oo.Thus, the induction hypothesis applies to extend f¡(M n II\E) to an fri E C(M n II) .We define F(z) = fn(z) ; this is well-defined and gives the desired extension of f throughout .M .
A second variation of the theme is that the hypothesis that A2N-2 (E) be finite can be replaced by the condition that A2N-2 (E n rBN) not grow too rapidly as a function of r,,r --+ oo .In fact if E is a closed subset of CZ that satisfies A2 (En rBN) < ar 2 for all large r, then E is removable provided 2 a < 4f .
That the desired conclusion can be drawn may be seen as follows .Notice first that A2N-2 (E fl rBN) < ar 2 for large r implies that A2N-2 (E n BN(p, r)) < ar 2 for large r, no matter what center p is chosen.Next, we have by [2,4] that for infinitely many arbitraxily large values of t, and this implies that the origin is not in the polynomially convex hull of E n {iz1 = t} for such values of t .Thus, by arguments we have used already, f continues holomorphically into a neighborhood of the origin.Similarly, it continues holomorphically into a neighborhood of every point of C2 , and the result is established .
The example E = {(z1 , 0) : z1 E C} shows that the result just derived cannot be obtained under the hypothesis that AZ (E n rB 2 ) _< 7rr2 .It seemes probable that if A2 (E n rB 2 ) < 7rr2 for all large values of r then E is removable, but no proof has presented itself.The discrepancy between 4~f here arises in part from the integral geometric inequality (2) and in part from Lemma 2.

A result on the ball
We now turn to a result on the ball that is an analogue in the Bergman geometry of the result we obtained above for CN.
The Bergman kernel on the ball in CN is given by t h(z, The corollary follows from the theorem, for codimnension-one subvarieties of the ball are not removable : If V is such a variety, then as we can solve the second Cousin problem on BN, there is f E O(BN) with V as its zero set .The reciprocal of f shows V not to be removable .
As we shall see below, there is a straightforward calculation that shows that if V C BN is a k-dimensional variety, then AB (V) is infinite .
Proof of the Theorem : The proof follows the general lines of the proof in the case of CN, but certain integral-geometric details require attention .We start with the case that N = 2.
Let distB(z, w) denote the Bergman distance between the points z, w E B2 .Fix a point zo E B2 , and define p : B2 --> (0, oo) by p(z) = distB(z, zo ).The triangle inequality in the Bergman distance yields that p is a Lipschitz function : As p satisfies a Lipschitz condition and A2 (E) < oo, we have that oo > AB (E) > const.
This yields a sequence {t l }°_°1 with tj -+ oo and with and this implies that AB (E n {z E B2 : p(z) = t j }) --> 0, (NB .As before, A1 denotes the 1-dimensional Hausdorff measure computed with respect to the Euclidean metric.)Let D(t, zo) = {z E B 2 : p(z, zo ) = t} .This is a ball in the Bergman metric, and its boundary is smooth .
Granted that f E O(B2 \E), we know that fibD(tj, zo )\E continues holomorphically into a neighborhood of zo , at least when j is large, so the result in the two-dimensional case is obtained as before .
To make the induction step work as before, we need two facts.First, we need to know that if E C BN+1 satisfies AB (E) < oo, then for almost every II E J 9N+1,N, A 2 -2 (n n E) < oc where we denote by E the set E n {z distB(z, 0) > 1} .(A2 -2 (II n E) denotes the Hausdorff measure computed with respect to the Bergman metric on BN+1 .)The second point we need is that the finiteness of the quantity AB -2 (II n E) implies the finiteness of the (2N-2)-dimensional Hausdorff measure of the set II nE computed with respect to the Bergman metric on the N-dimensional ball II n BN+1 .
The latter point is straightforward though, for the metric induced on II n BN+1 from the Bergman metric on BN+1 differs only by a constant factor from the Bergman metric on the N-ball II n BN+1 .
That AB (E) < oo implies AB -2 (II n E) < oo for almost all II's is an analogue in the Bergman metric of the result of Shiffman used above.We prove the following integral-geometric fact.5 .Lemma.There is a constant cN such that if S C BN\{z : distB (z,0) < 1}, then The proof of this lemma follows precisely the lines of the proof of Shiffman's Lemma 5 in [12] once we have the following estimate .6. Lemma.There is a constant kN such that for small 6 > 0 if T C BN\{z distB(0, z) < 1} and T has diameter less ¡han 6 in ¡he Bergman distante, then p({II E GN,N-1 : II n T :~0}) < kNO.
For the conveniente of the reader, we recall the argument in [12]  As this is true for all E and as AB-2 (S) = lim AB-2 (S), we have the desired E inequality.Lemma 6 is a consequence of the corresponding Hermitian result.The Bergman diameter of a set is not smaller than the Euclidean diameter .Thus, if T has small Bergman diameter d and is included in BN\{z : distB(0, z) < 1}, then T is contained in a Euclidean ball B of Euclidean diameter 2d.As d is small, B can be choose to lie in {z : ~z1 > 1 -d}.Everything follows from the It is worth noting that our Theorem 3 implies Shiffman's result that for domains in CN , closed sets of vanishing (2N -2)-dimensional measure are removable .Shiffman's result is local, and if A 2N-2 (E) = 0, then for every p E E and every ball BN (p, r) centered at p, the set E n BN (p, r) has zero (2N-2)-dimensional measure with respect to the Bergman metric on BN (p, r).Thus, E is locally removable and so removable.
In the analysis above, we need to know the measure of the set of (N -1)dimensional subspaces of CN that meet a ball.In (4) we stated an estimate that sufilces; in this section, we shall evaluate this volume precisely.We shall, in fact, work in a slightly more general context .(It seems probable that the result obtained here exists somewhere in the published literature, but we know no reference.) We are denoting by GN,k the Grassmannian of all k-dimensional complex subspaces of C N .(Thus, the elements óf GN,k pass.through the origin).The manifold GN,k is a homogeneous space of the unitary group U(N) : If g E U(N) and II E GN,k, then g -II = g(II) E GN,k .There is a unique measure /lk on GN,k with Mk(GN,k) = 1 that is invariant under the action of U(N) .If we denote by II o the element of GN,k and if 7r : U(N) -> GN k is the map given by 7rg = g -IIo, then pk can be calculated by if v denotes the normalized Haar measure on U(N) .
Our problem, precisely formulated, is the following : To determine or, equivalently, to determine Here, zo E C N and R > 0. If IR¡ > zo , then 0 E BN(zo, R), so the measure in question is one .In general, the answer will be a function of zo and R.
The problem is plainly invariant under the action of U(N), so without loss of generality, we may suppose that zo = p = (p,0, . . ., 0) with p = izo¡ .We have that g(II0) f1 BN (p, R) 0 if and only if the distance d(p,g (II o)) is less than R.

j-k+1
We define a map 17 : U(N) -> S IN-1 by rlg = g1 where by 1 we mean the north pole 1= (1, 0. . . .0).The fiber 17 -1 (1) is the subgroup of U(N) isomorphic to U(N -1) that consists of the matrices of the form Thus, wkj = -wjk .In particular, at the identity, da l j = -d&j1 .This implies that at the identity We see then that for a suitable choice of constant EN = fl, at the identity of U(N),77*w and EN( n (wij AW1 j)) A w11 coincide.As each is left invariant, 77 * w -EN( l \ (w1j A W 1j)) Aw11 .1<j<N accordingly, for each z, Thus,
Off the set where zN 7~0, we can solve this for dzN : In the preceding section, we needed the special case of this in which k = N-1 and c is small.we see that in this case, The results on removable singularities we have obtained above are surely not the end of the story.The two results are of the general form: S2 is a domain in an N-dimensional complex manifold, ds 2 is a Hermitian metric on P and A9N -2 denotes the (2N -2)-dimensional measure derived from ds 2 .In two special cases, we have that a closed set E in 2 is removable provided A2N -2 (E) is finite .One may pose the question : What conditions on the metric ds 2 suffice for us to draw this conclusion?In particular, is it sufficient for ds 2 to be a complete Kil,hler metric?Do the metrics of Carathéodory or Kobayashi play a róle here?
Another problem that arises is to stablish a projective version of the result valid for meromorphic functions .Consider the Fubini-Study metric on the complex projective space PN .With respect to this metric, the volumes of the subvaxieties of PN form a countable set ; the volume of a variety in PN is, to within a normalizing constant, its degree.If E is a compact subset of pN that has (2N -2)-dimensional measure (with respect to the Fubini-Study metric) less than the smallest of the volumes of codimension-one hypersurfaces in PN, does it follow that E is removable for meromorphic functions in the evident sense that it F is a function meromorphic on P N \E, then F extends through E to be meromorphic on the whole on pN?
Another question that is suggested by what we have done is the following: If D is a pseudoconvex domain in CN , must bD have dimension at least 2N --2?The removable singularity theorem of Shiffman implies that the Hausdorff dimension or metric dimension is at least 2N -2, and our Theorem 1 implies that bD must have infinite (2N-2)-dimensional measure.The present question understands dimension in the sense of the topological theory of dimension for which one may consult [5] .
with The Bergman area of varieties .
We have reached the result that if V C BN is a k-dimensional variety, then A2 (V n rBN) = (i ra)k k A2k(V n rBN) .
In particular, V has infinite volume in the Bergman metric.
*A discussion of a version of Stokes's theorem sufficient for our present needs is given in [13] .

2 .
Lemma.If Y is a closed subset of brB2 and if the polynomially convex hull of Y contains the origin, then A 1 (Y) > -,/2-wr.
W (Z, (»N+! if ( , ) denotes the Hermitian inner product on CN, and the Bergman metric is given by N ds 2 = Y' Tjkdz j ® dzk j, k-1 with coefficients Tjk given byWe shall denote by AB the a-dimensional Hausdorff measure computed with respect to the distance function on BN derived from the Bergman metric.We shall prove the following analogue of Theorem 1 .3.Theorem .If E Cremovable.4.Corollary .If E C BÑ is a subvariety of codimension one, then E has infinite arca, arca computed with respect to the Bergman metric .