On the smoothness of levi-foliations

Publicacions Matemátiques, Vol 32 (1988), 171-177 . ON THE SMOOTHNESS OF LEVI-FOLIATIONS Abstract 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 . 1 . Introduction Let S be a real hypersurface in C' of class C k (k >_ 2) with vanishing Leviform . The maximal complex subspace TSnJTS is an integrable distribution of class C k-1 and codimension one on S, so the Frobenius theorem [5] guarantees that S admits a (unique) foliation of class C k-1 by complex hypersurfaces . The goal of this paper is to establish the following result . Theorem . The induced foliation of a C k Levi-flat hypersurface is actually of class C k . For k = 1 the Levi-form is not defined, but our proof will nevertheless show that if a real C 1 hypersurface admits a continuous foliation by complex hypersurfaces then that foliation is of class C 1 . In fact, the existence of the Levi-foliation in the C 1 case was proved by Shcherbina for n = 2[6] and by Airepetian for general n [l] ; moreover, Airepetian's paper also establishes the C 1 -smoothness of the foliation . It appears that his technique (which uses both the Bishop disc method and the Frobenius theorem) could also be used to prove our result . Our proof is based on the construction of a family of approximate pluriharmonic defining functions for S . (See the Proposition below) . The existence of such functions is also useful in the study of holonomy of Levi-foliations (see Acknowledgements . The authors wish to thank Franc Forstneric for assistance with translation and Eric Bedford for some helpful remarks . This research was supported in part by an Alfred P . Sloan Foundation Fellowship to the first author and by a NSF Grant to the second author . 172  D .E . B ARRETT AND J .E . F ORNAESS 1 . 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 T O S = {(z', z n ) y n = 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 L t 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 L t 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 h o , h 1 , . . . , 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/az n ~ 0, and (iv)j  Im hj =o(¡ hili) for z E S as z  + L t , uniformly in (t, z) . We may take h o (t, z) = z - O(t, z') . Let T= (a1ax n )+(arIax n )(a1a yn ) ; T is a vector field of class C' -1 tangent to S and transverse to the Levi-foliation near 0 . Let 0 1 (t, z') =  arg (Th o )(T(t, z')) =  arg (1 + i(arlaxn)(XY(t, z')) . Then01 is continuous in (t, z'), and the function Im e-'01(t,z')ho(t, z) vanishes along L t 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 ES as z --> L t , uniformly on a neighborhood of (0, 0) . We have 0 1 (0, 0') = = 0so that working on a smaller neighborhood we may assume that 10,1 < < 7r/4 . Claim . 0 1 (t, z') is pluriharmonic in z' . 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 = 0 1 on áo we have Re f (fió) = 0 1 00), where (ó is the center of A . Consider the two-parameter family of discs We have LEVI-FOLIATIONS  173 (n = xn + 21f . = ho(z) . yn = «(,, xn) = tan 0 11b , ) - í . + o(I-í .I)- rE,a(C , ) _  e{e' f(S') +iñ}), (, EA . yn - r(~ , , gin) = e{A + e1f«') . cos  Re f(~ , ) ' ( tan  Re f - tan 01(C'))} + o(I--I) for ( =  (' E 0 . Suppose that 0100) > Re f « Ó ) . Pick a and b so that and 0 < a < eIf(Só) . cos Re f«ó) - ( tan 0 1«ó) - tan Re f«ó)) b > 2 maX CE Ae - Then for sufficiently small e > 0we have 1 fn < r(S,,xr,) when  = Fe-,a (SÓ), 11 . > r(S ', xn) when  = rE,a(C , )~ ~, E C70, a < A < b, l yn > r« I ,xn) when  = FE,b« i ),  E á . But this violates the dise theorem [4, p . 53], since S is clearly pseudoconvex from both sides . The case 0 1 00) > Re f(CÓ) is similar . Now choose f1 (t, z') continuous in (t, z') and holomorphic in z' with 0 1 = = Re f1 . Let h 1 (t, z) = e =f1 (',z') h o (t, z) . Then h 1 satisfies (i) 1 , (ii) 1 , (üi) 1 , and (iv) 1 on a suitable neighborhood U 1 of (0, 0) . 174  D .E . B ARRETT AND J .E . FORNAESS Let us assume that hj _ 1 has been constructed and proceed . t o construct hj . Let We have and _  (T i Im h j _ 1 )(t, z) C~(t' z)  J!((T Re h j _ 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 zderivatives of h j _ 1 come for free), and the function Im h j _ 1 (t, z) - Oj(t, z') - ( Re hj_ 1 (t, z))j vanishes along L t along with its derivatives of order <_ j with respect to the vector field T . Hence Im h j _ 1 (t, z) - Oj(t, z') - ( Re h j _ 1 (t, z))j = o(1hj_1(t, z)jj) for z E S as z  + L t , uniformly on a neighborhood of (0, 0) . Claim . Oj(t, z') ás pluriharmonic in z' . Proof of Claim : Again we suppress t temporarily and perform a change of coordinates (n = xn + ¡In = h7_1 (z) Thus S is defined by an equation of the form ñn = Oj« l ) - -í - n ' + 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 . Again we consider a two-parameter family of discs rE,a(() = ((', E + ¡Ej {A -1f (( I M, ( 1 E 0 . yn-r((~,xn)=E~{~+ Re f((')-07(C~)Í+o(~E~') for ( = FE,a((~), (' E 0 . Suppose that O ; ((ó) > Re f « Ó ) . Pick a and b so that 0<a<Oi((ó)- Re f((ó) b > 2maAx{Oj((')- Re f((')} C , E Then for sufficiently small e > 0 we have ún < r«', in ) when  = Ff,a«ó), In > í""«', in ) when  = FE,a«'), (' E áo, a < A < b, In > «(', - I n) when ( = FE,b«% (' E 0 . But this violates the disc theorem as before . The case O1(Có) < Re f «ó) is again similar . Again choose f i(t, z') continuous in (t, z') and holomorphic in z' with O ; = = Re f j . Let h ;'(t, z) = hj_ 1 (t, z) - ifi(t, z')(hj-1(t, z))j . Then h j satisfies (i)j, (ii) j , (iii)j, and (iv) j on a suitable neighborhood Uj of (0, 0) . 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 astraightforward 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 h j can actually be chosen to be of class C kj 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 0in S then h(z) _ LEVI-FOLIATIONS  175 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 a j are holomorphic . Thus aj(z') - (h(0, z))j + P(z) - (h(0, z))k+l k E( Im aj(z'))( Re h(0, z))j =  Im h(z) -fo(1h(0, z)I k = o(1 Re h(0, z)jk) j=o on S, forcing Im a j - 0 for 0 _< j < k, so that each a j is a real constant . 6) If S is a real hypersurface of class C k which is pseudoconvex from one side and which contains a complex hypersurface then the functions hj can be 176  D .E . BARRETT AND J.E . FORNAESS constructed for j < some even integer jo ; the corresponding function O ;,, will be sub-or superharmonic . (The pluriharmonicity of 01 has beenused 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 t o , t, z' close enough to zero . But (iv) implies that Re h and h are comparable so it follows that and so Re h(t o , xP(t, z')) = 0(1 Re h(t o , xP(t, 0'))I) h(to, T(t, z')) = 0(I h(to, T(t, 0'»j) Im h(t o , ID(t, z')) = o(I h(to, T(t, 0'))I k ) . Thus from bounds for pluriharmonic conjugates we have h(to, T(t, z')) - h(t o , T(t, 0')) = o(I h(to, p(t, 0'))I k ) = o(I t - to ¡k) after shrinking the domain of z' . By (iii) and the inverse function theorem we may write z, l = -P(t, z', h(t, z)), where ~¿ is continuous in (t, w) and holomorphic in w . Thus «t, z') = d>(to, z', h(to, kP(t, z ' ))) = -¿(to, z', h(to, T(t, z ' ))) + O(I t - to 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)k 0(t, z') by showing that (a1at)kO(t, z ' ) = (alCgt)k~¿(t o , z ' , h(to, 0(t, 0'))) Ito=tLemma . Le¡ f be a Ck -1 function on an interval I C R . Suppose that there is a function g on IxI such that (i)  g(s, t) exists and is continuous on IxI for 0 < j < k, and (ii) f (t) = g ( 3, t) + o(It - S i k) uniformly on IxI . Then f (k, (t) = (alat) k g(t,S) I9=t . Proof of Lemma : It is clear from the hypotheses that f (j)( t ) = (a1at)'g(s,t)19=t for 0 _< j < k - 1 . Let P(s, t) denote the (k - 1)` Taylor polynomial for f at s, and let a(t) = (alat)kg(s,t)19=t . LEVI-FOLIATIONS  17 7 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 V h 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 (V h ) k - I f (t) = f (k-1 )(t) + cka(t)h + o(h), uniformly on compact subsets of I, where ck = h -1 (Vh) k-1 (t - s)k js-t = (k - 1)(kl)/2 . Thus ( 1 )  f (k) (t) = li Ó Vh f(k-1) (t) = li m(Vh)kf(t) - ck{a(t + h) - a(t)} + o = hlm(Vh)kf(t) = hi m a(t) + o(1)  (by (*) again) = a(t) . This completes the proof of the Theorem . References 1 .  R .A . AIRAPETIAN, Extending CR functions from piecewise smooth CR manifolds, Math . Sbornik 134 (1987), 108-118, Russian . 2 .  D . BARRETT, Complex analytic realization of Reeb's foliation of S 3 , PrePrint . 3 .  K . DIEDERICH AND J . FORNAESS, Pseudoconvex domains : an example with non-trivial nebenhülle, Math . Ann . 225 (1977), 275-292 . 4 .  J . FORNAESS AND B . STENSONES, Lectures on Counterexamples in Several Complex Variables, Mathematical Notes 33, Princeton University Press, Princeton (1987) . 5 .  R . NARASIMHAN, Analysis on Real and Complex Manifolds, North-Holland, Amsterdam (1968) . 6 .  N .V . SHCHERBINA, Descomposition of a common boundary of two domains of holomorphy into analytic curves, Iszvestia Akad . Nauk . SSSR Ser . Mat . 46 (1982), 1106-1123 ; Math . USSR Iszvestia 21 (1983), 399-413 . Institut Mittag-Leffer Aurav¿¡gen 17 S 182 62 Djursholm, SWEDEN . Rebut el 19 de Febrer de 1988