ON THE SMOOTHNESS OF LEVI-FOLIATIONS

D.E . BARRETT AND J . E . FORNAESS We study the regularity of the induced foliation of a Levi-flat hypersurface in C'° , showing that the foliation is as many times continuously differentiable as the hypersurface itself. The key step in the proof given here is the construction of a certain family of approximate plurisubharmonic defining functions for the hypersurface in question .


Proof of Theorem
We work locally near some point in S, which we may take to be the origin, and we choose coordinates z = (z', z~), z,l = x, + iy so that TOS = {(z', zn ) yn = 0} .Then we may write y = r(z',xj on S, where r is of class Ck .Let (t, z') by defined for (t, z') near (0, 0') by the condition that ID (t, z')= = (z', O(t, z')) lies on the leaf Lt passing through (0', t + ir(0', t)) .Then 0 is of class Ck -1 in (t, z') and holomorphic in z' .Our goal is to show that 0 is actually of class Ck in (t, z') ; it is easy to see that all partial derivatives involving fewer than k differentiations with respect to t exist and are continuous, so it will suffice to establish the existente and continuity of (a1at)k0 .
The major step in proving the Theorem is the approximation of S along each leaf Lt by the zero set of a pluriharmonic function, as specified in the following Proposition .
Proposition.There is a function h(t, z) defined in a neighborhood U of (0, 0) such that the following conditions hold for (t, z) E U : (i) h is continuous in (t, z) and holomorphic in z, (ii) h(t, z) = 0 for z E Lt, (iii) ah/az 5b 0, and (iv) Im h = o(1hj') for z E S as z ---> Lt, uniformly in (t, z) .
Proof of Proposition: We will construct a sequence of functions ho , h1 , . . ., h,k = h defined in a neighborhood Uj of (0, 0) such that the following conditions hold for (t, z) E Uj : (i)j hj is continuous in (t, z) and holomorphic in z, (ii)j h;(t, z) = 0 for z E Lt, (iii)i ahi/azn ~0, and (iv)j Im hj =o(¡ hili) for z E S as z + Lt , uniformly in (t, z) .We may take ho (t, z) = z -O(t, z').Let T = (a1ax n )+(arIax n)(a1ayn ) ; T is a vector field of class C'-1 tangent to S and transverse to the Levi-foliation near 0 .Let Then 01 is continuous in (t, z'), and the function Im e-'01(t,z')ho(t, z) vanishes along Lt as does its derivative with respect to the vector field T. Thus Im e -'e 1 (t,z') ho(t z) = o(1 ho(t, z) 1) for z E S as z --> Lt , uniformly on a neighborhood of (0, 0) .We have 01 (0, 0') = = 0 so that working on a smaller neighborhood we may assume that 10,1 < < 7r/4 .
Proof of Claim: It will simplify notation to suppress the parameter t temporarily.Also, it will be useful to perform the change of coordinates In the (-coordinates S is defined by an equation of the form To show that 01(x') is pluriharmonic it suffices to show that for every complex-linear dise A near 0' in Cn -1 and for every f continuous on 0 and holomorphic on 0 with Re f = 01 on áo we have Re f (fió) = 0100), where (ó is the center of A.
Consider the two-parameter family of discs We have for ( = (' E 0 .Suppose that 0100) > Re f«Ó ).Pick a and b so that and 0 < a < e-I-f(Só) .cos Re b > 2 maXCEAe-Then for sufficiently small e > 0 we have But this violates the dise theorem [4, p. 53], since S is clearly pseudoconvex from both sides.
Let us assume that hj_1 has been constructed and proceed .to construct hj .Let We have and _ (T i Im hj _ 1 )(t, z) C~(t' z) J!((T Re hj_1 )(t, z))j (Condition (iii) j _ 1 shows that the denominator doesn't vanish at (0, 0')).Then O ; is continuous in (t, z') (recall that z-derivatives of hj_1 come for free), and the function Im hj_1(t, z) -Oj(t, z') -( Re hj_ 1 (t, z))j vanishes along Lt along with its derivatives of order <_ j with respect to the vector field T. Hence for z E S as z + Lt , uniformly on a neighborhood of (0, 0) .
Proof of Claim : Again we suppress t temporarily and perform a change of coordinates Thus S is defined by an equation of the form ñn = Oj«l) --ín ' + o(jxnj') As before, it suffices to show that for every complex-linear disc A near 0' in Cn-1 and for every f continuous on 0 and holomorphic on A with Re f = Oj on áo we have Re f ((ó) = Oj((ó), where (ó is the center of 0 .
But this violates the disc theorem as before.
The proposition is proved, by induction .Remarks . 1) In the case k = oo it need not be the case that S can be approximated to infinite order along a given leaf by the zero set of a pluriharmonic function.For n = 1, for example, a C°°curve need not be approximable to infinite order at a given point by a real-analytic curve.
2) For j < k -2 the claims in the above proof can be proved by a straightforward Levi-form computation .
3) One can avoid explicit mention of pseudoconvexity in the above proof by observing that the winding number of the boundary of a holomorphic disc around a given leaf cannot jump under small perturbations .
4) The functions hj can actually be chosen to be of class Ckj in (t, z). 5) If h(z) is a holomorphic function vanishing on Lo with Im h(z) _ = o(1 h(z)I k ) on some neighborhood of 0 in S then h(z) _ h(z) = P(h(0, z)) + fl(z) -(h(0, z)) k+1 where P is a polynomial of degree k with real coeficients and ,Q is holomorphic .Indeed, we may write where fl and the aj are holomorphic .Thus j=o on S, forcing Im aj -0 for 0 _< j < k, so that each aj is a real constant.
6) If S is a real hypersurface of class Ck which is pseudoconvex from one side and which contains a complex hypersurface then the functions hj can be constructed for j < some even integer jo; the corresponding function O ;,, will be sub-or superharmonic .(The pluriharmonicity of 01 has been used in several papers, for example in [3, p. 290] .) To prove the Theorem we first note that Re h has constant sign on each leaf so that by Harnack's inequality we have for to , t, z' close enough to zero.But (iv) implies that Re h and h are comparable so it follows that and so Re h(to , xP(t, z')) = 0(1 Re h(to , xP(t, 0'))I) h(to, T(t, z')) = 0(I h(to, T(t, 0'»j) Im h(to , ID (t, z')) = o(I h(to, T(t, 0'))I k ) .
Now the main term of this last expression is C k with respect to t, so that the following Lemma will establish the existence and continuity of (alat) k0(t, z') by showing that (a1at)kO(t, z' ) = (alCgt)k~¿(t o , z' , h(to, 0(t, 0'))) Ito=t- Lemma.Le¡ f be a Ck -1 function on an interval I C R .Suppose that there is a function g on I x I such that (i) g(s, t) exists and is continuous on I x I for 0 < j < k, and (ii) f(t) = g( 3, t) + o(It -S i k) uniformly on I x I.
Then applying Taylor's theorem to g in (ii) we have (*) f(t) = P(s, t) + a(t)(t s)k/kl + o(1t -sjk) uniformly on compact subsets of I x I. Let Vh denote the difference operator V h<P(t) = {W(t + h) -4p(t)}/h .Applying (Vh)k -1 to both sides of (*) and taking s = t we get uniformly on compact subsets of I, where This completes the proof of the Theorem .