Intersections of totally real and holomorphic disks

Publicacions Matematiques, Vol 37 (1993), 111-126 . A bstract INTERSECTIONS OF TOTALLY REAL AND HOLOMORPHIC DISKS TOM DUCHAMP AND FRANC FORSTNERIC It is shown that a holomorphically embedded open disk in (C2 and a totally real embedded open disk which have a common smooth boundary have nontrivial intersection . 1 . Introduction It is now clear, through the work of Gromov and others, that there is a strong relationship between the theory of Lagrangian immersions into symplectic manifolds and the theory of totally real immersions into complex manifolds . The relation follows from the fact that the Grassmann space of Lagrangian subspaces of R 2 n is homotopy equivalent to the Grassmann space of totally real subspaces of C' . There are surprising differences between the two theories, however . Recall that complex 2-space, (C 2 , is a symplectic manifold with symplectic form w = 2 (dz l A dz l + dz 2 A dz 2 ) . Let 0 : 0 ---> (C 2 be a symplectic embedding of the open unit disk in (C, that is, suppose that the pull-back O*w is a symplectic form on A . It is not difficult to prove that every holomorphic embedding of 0 is symplectic . Assume further that ~b extends to a smooth embedding of the closed disk 0 . There is no Lagrangian embedding of 0 whose boundary coincides with 0(0A) . For suppose that 0 is such an embedding . Then O( 0 ) and 0(,i1) are homologous relative to the boundary ; hence, The second author was partially supported by the Research Council of the Republic of Slovenia . 11 2  T . DUCHAMP, F . FORSTNERIC which is a contraction because the first integral is 0 and the second is positive . On the other hand, it is easy to construct examples of a pair consisting of a holomorphically embedded disk and a totally real disk which have a common smooth boundary (see Section 4 below for one such example) . In this paper the intersection theory of such a pair is investigated . The main result is the following . Intersection Theorem . If a holomorphically embedded disk and a totally real embedded disk in (C Z have a common boundary then they intersect in at least one interior point . More precisely, let 0 : 0 -> (C2 be a smooth embedding of the elosed unit disk, holomorphic on A ; and jet 0 : 0 ~___> C 2 be a smooth totally real embedding such that 0(ao) = 0(ao) . Then the intersection 0(0) f1 O(0) is non-empty . The proof of the theorem is by contradiction : Suppose the theorem is false . Then there are embeddings 0 and 0 of 0 with o(0) f1 o(0) = 0 and 0(8 0 ) = O(aA) . We will show how to deform the images O( 0 ) and 0( 0 ) to produce two totally real embeddings which abut smoothly along the boundary O( 80 ) in such a way that the union of the images of the deformed totally real embeddings defines a totally real embedding of the two sphere into C2 . Bishop [1] (see also Wells [5]) proved that there is no totally real embedding of the two sphere into (C 2 . Acknowledgment . We wish to thank Professor E . L . Stout for bringing this problem to our attention . 2 . Properties of totally real embeddings Before beginning the proof it is necessary to review several conditions which are equivalent to the total reality condition and to define an index for totally real embeddings of annuli . 2 .1 . Conditions for total reality . An embedding ip : U -> (C2, U C cC a region, is said to be totally real if for all ~ E U the intersection 0,T C UnJO * T C U is trivial . (Here TU denotes the real tangent bundle of U and J the complex structure tensor of C 2 .) Let (z, w) denote complex coordinates on C 2 and (u, v) arbitrary real coordinates on U . The next lemma gives several characterizations of the total reality condition . They are more or less well-known and easily verified, so we leave it to the reader to check them . (See Stout-Zame [4] for a discussion of total reality) . TOTALLY REAL AND HOLOMORPHIC DISKS  113 Lemma 1 . Let 0 : U - (C 2 be an embedding, U C (C, defined by the functions z = Z(S), w= W«) . Then the following are equivalent : (i) The embedding 0 is totally real . (ii) For each ( E U the equality (C - 0,,(T S U) = TC 2 holds l . az OZ/a~ (iii) The determinant  Y  never vanishes . aW/a( az az (iv) The determinant  au  av  never vanishes . aW aW au av Remark 1 . Condition (ii) can be restated as follows . Let X = (X 1 , X 2 ) be a basis for the tangent space TAU, ( E U . Then the vectors ~b* (X1) and 0* (X2) form acomplex basis for the complex vector space T, G(C)C 2 . We introduce the notation A~  1 - a < l~l < 1} and Aá - {~ : 1_< l~i ~i <_ 1 + a} for a > 0 . The special case in which ~b is defined on the annular region AQ , a > 0 and 0 is of the form f«)=0 for ~(j=1 is of particular interest to us . First observe that condition (iii) reduces to a~ o . It will prove useful to write equation (1) in polar coordinates, z = reie, (2)  00 - irY  0 . Because f (z) vanishes for all 1 z~ = 1, it follows that áe = 0and total reality implies that -f =,b 0 for all z with Iz1 = 1 . This implies the following lemma . Lemma 2 . Let 0 be a totally real embedding of the annulus A as given above . Then there is an annular region of the form 1 <_ Iz1 < 1 + a', 0 < a' < a < 1, on which f can we written in the form .f (z) = R(z)e¡8(z) 'If V C W is a real subspace of the complex vector space W then (C - V denotes the complex subspace spanned by V . 11 4  T . DUCHAMP, F . FORSTNERIC where R(z) is a non-negative, smooth real-valued function which vanishes for ~z1 =_1 and where O(z) is smoothmodulo 27r . The inequality  8r arí,(z) > 0 is satisfied for all z in a neighborhood of the unit circle Iz1 = 1 . Now suppose that f is any smooth function defined on A of the form .f (z) = R(z)eio(z) Substitution of the formulas _ a  _  aR(z  a0 _ io  af  aR(z)  0" ar - (  a  ) + iR(z) ar ) e  and ó0 = (  00  + iR(z) a0 ) ego into the total reality condition (2) and separating real and imaginary parts of the coefficient of e io yields the condition '9R  + R(z)r ae a0 ) + i (R(z) 50 - r a arz))  0 . In particular, if the imaginary part of the left hand side is negative, the embedding is necessarily totally real . Lemma 3 . Let 0 : Aá --> C2, 0 < a < 1, be an embedding of the form O(z) = (z, R(z)eio(z)), where R and O are real-valued functions with R smooth and O smoothmodulo 27r . If the inequality aR(z) > R(z)a0 r ar  á0 is satisfied then the embedding is totally real . 2 .2 . An index for totally real embeddings of annuli . In this section we define an index for a totally real embedding of an annulus in (C 2 . It is closely related to the Maslov index and is a special case of an index defined by Kamber and Tondeur [3] . A detailed presentation, within the context of totally real embeddings of surfaces in C 2 , is given in [2] . We give a self-contained exposition here . Let 00 : A -> (C 2 be any totally real embedding of an annular region A C C . To define the index of 00 begin by choosing a complex framing 2 2 By a complex framing we mean a pair of complex vector fields which are pointwise independent over (C . f = (f,, f2) for the holomorphic tangent bundle T( 1 , 0 )U, where Lf is any open contractible neighhborhood of A . Next choose a real framing 3 X = (X1, X2) of the real tangent bundle TA which is compatible with the orientation of A as a subset of (C . By virtue of Remark 1, there is a smooth matrix-valued function defined by the formula TOTALLY REAL AND HOLOMORPHIC DISKS  115 Since C is real, so is det(C), hence, MO o : A , GL(2,C) 1 (doo(X1)  dOo(X2)) = (fl  f2) ~m2 1 Definition 1 . The índex of the embedding 0 0 : A -> C 2 , is the degree of the map A -i 80 :  --> det(M) det(M) and is denoted by Ind(0o) E 7G . Remark 2 . (i) It is easily verified that the integer Ind(oo) is independent of the framings X and f . For suppose that X' and f' is another pair of framings, with f' defined on Ll', a contractible neighborhood of Oo(A) . Then there are smooth maps B : un u' , GL(2,C) and C : A E -> GL + (2, R) such that f'=f-B and X'=X .C . If M' : A --> GL(2, C) is the map defined by the formula d0 o (X) = f-M', a straightforward calculation with matrices yields the identity M'=B-1-M .C . l m . 2 m 2 2 det(M')  _  det(B --1 ) det(M) det(C)  _  det(B)j det(M) det(M')¡  det(B -1 ) 11 det(M) 11 det(C) 1  det(B)  Idet(M)1 Since U and U' are contractible, each of the framings f and f' are is homotopic to the framing ( a a l , a a 2 ) . The map B -1 is, therefore, homotopic to the identity . This fact, together with the observation that the 3 By a real framing we mean a pair of real vector fields which are pointwise independent over R . 11 6  T . DUCHAMP, F . FORSTNERIC degree depends only on the homotopy class of the map, completes the argument . (ii) Note also that the above argument shows that if q> : U , (C 2 is a biholomorphism onto an open set in (C 2 then Ind(-P o 0 0 ) = Ind(0o) . (iii) Finally, because the index depends only on the homotopy class of the map, 00, it is determined by the image 00 (A) together with an orientation . Thus, if A C C 2 is a totally real, oriented, embedded annulus, the integer Ind(A) is well-defined . Lemma 4 . If A C C 2 is a totally real embedded annulus which is contained in a totally real embedded disk DC C2 then Ind(A) = 0 . Proof : Let 0 : 0 --> (C 2 a smooth map such that O(0) = D and choose framings X of Tá and f of (C 2 . Then let M(S), E 0, be the GL(2, C)-valued matrix as defined above . The degree of the map ( -> det(M«))/I det(M(~))1,  E  -I (A) is zero because it is homotopy to a constant . 3 . Reduction to the case of real analytic boundary Begin by assuming that there are smooth embeddings 0 and 0 of 0 with 0 holomorphic on 0 and totally real and such that the conditions z/~(0) f1 o(0) = D and O(á0) _ O(á0) are both satisfied . Without loss of generality we may assume that 0 extends holomorphically to a neighborhood of 0 . To see this we observe that, because the condition of total reality is an open condition, any C l -small deformation of the map 0 is also a totally real embedding . In particular, let and in such a manner that 0o6 = {~ : 10 =10, where S > 0 is a small constant to be chosen later . Then 0 can be deformed to a map 0' so that 0'(ao) = 0(aos) o (o) n~b (oó) = 0 . TOTALLY REAL AND HOLOMORPHIC DISKS  117 Oneway to accomplish such a deformation is to let 0' be the composition 0 o vó of  with the flow, v t , of a vector field which is tangent to the image of  and constructed so that v ó (0(OO) =MO5), b > 0 . The map ~b' : 0 --> CZ defined by the equation 00=v)((1b)0 extends holomorphically to a neighborhood of 0 C C . Now replace the pair 0, 0 by the pair 0', 0' . By construction, O(OO) is real analytic . where 4 . Holomorphic disks are relatively isotopic to totally real disks Retum now to the problem of replacing 0 by a totally real embedding . Because 0 extends to a holomorphic embedding of a neighborhood of 0, there is a biholomorphism (D : U -j C 2 , defined on a neighborhood U of ~b (A) such that the composition <D o 0 is the map Consider the family of maps 0 E :0_(C2, ~ H (z~ w) = (S, fE«)), fc .(~) =,E ( 1 - 1(I2) ex(1-I(I2) ~ Note that fE satisfies the conditions : fo = 0, f,(~) = 0 for I(I = 1 . By virtue of equation (1) and the computation OfE(~) = E {(1 _ 21(1 2 ) - ( 1 - I(I 2 )I(I 2 i} e'(' _1C12) 0 0 the embedding defined by 0 .is totally real for all E > 0 . Because for E sufficiently small the image of 0 . lies in the set <D(u), the map 0E =P -1 o 0E is well-defined . Lemma 5 . For e sufficiently small, the family 0E has the following properties : ( 1 ) O,Iao = Ojao for all E . (ii) 0E is a totally real embedding of 0 for E > 0 . (üi) z/~ E (0) CU . (iv)  E(A) n o(o) _ 11 8  T . DUCHAMP, F . FORSTNERIC Proof . Properties (i) through (iii) are immediate from the definition of f, The verification of (iv) is based on the implicit function theorem . First note that since O( 0 ) is totally real and V)0(0) is holomorphic, Tp(oo( 0 )) n Tp(o(0)) = Tp(o(a 0 )) for all p E 0(á 0 ) . Because ~b E depends smoothly on e, the condition Tp(,pE( 0 )) n Tp(O(0)) = Tp(O(a0)) holds for all e suficiently small . By the implicit function theorem and compactness of 80, it follows that there is a number S > 0 such that 0E(A) U O(A6) _ 0 for all sufficiently small c . Moroeover, since 0o(A) n ~(0) = 0 and PE depends smoothly on e, it follows from the compactness of 0(0 \ Ab ) that 0, (A) n O(0 \ A6) _ 0 for all sufficiently small e . Hence, for e sufficiently small, 0, (A) n 0(0) = 0 . 5 . Modifying two totally real disks to abut smoothly Consider the small annular neighborhood Ab C 0, 0 < S < 1, of the boundary 80 . We will modify 0(0) on O(A6) so that O( 0 ) and 0F( 0 ) abut in a C l manner along <b(á0) and thus define a totally real embedding of the 2-sphere into (C 2 . By virtue of the equality O(ó0) = 0( 80 ), for S sufficiently small the imago O(A, )is contained in the neighborhood U of the previous section . For this reason the map 4) o0 : A6__+ C 2 is well-defined and, since we will modify 0 only along A6 , the deformation of 0 can be reinterpreted as a deformation of 0' . The modification will be done in two stages : (i) we first deform 0' so that O'(A, )is the graph of a function g ; (ii) then we deform g so that O(0) and 0( 0 ) abut smoothly . 5 .1 . Replacing O'(A . ) by the graph of a function . Consider the neighborhoods of O'(0) of the form N E , Q = {(z, w)  :  Iz1 < 1 + e, IWI < Q} with e > 0 and a > 0 chosen so small that there is an inclusion N3E,, C -D(U) and such that the condition ~ _1 (N3e,a)n0( 0 \A6) = is satisfied . By choosing S' < S suficiently small we can insure that inclusion 0'(A, - ,) CV,, is satisfied . TOTALLY REAL AND HOLOMORPIIIC DISKS  119 We are going to replace 0' by another totally real embedding ~" A 6 -~ NzE,Q which satisfies the two conditions : and "(Ab ) f1  {(z, g (z)) : z E Aó,}  for o,' < v sufficiently small, where g is a complex-valued function defined on Aá, . By construction, the map 4> -1 o A b _ (C 2 agrees with 0 on the interior boundary component of the annulus and so defines another totally real embedding, 01 : 0 --> (C 2 which intersects O(0) along the circle O(a0) . To begin the construction of 0" observe that the image O'(A6 ) is of the form where the functions Z and W satisfy the conditions (4)  Z(~) = ( and W«) = 0for 1(1 = 1 . Hence, in polar coordinates ( = pe e ", z = re te we can write with R(1, a) = 1 and O(1, a) = a . Applying condition (iv) of Lemma 1 yields the inequality 0 :~ aZ aZ áp 8a aw aw ap aa on the annular regiorl Aó \ Aó, z = Z(0,  w = W(O ,  1 - S < j(j< 1, Z= R(p, a)e2o(P,a) aZ aRe io áw ap aa 0 Thus áP is non-zero for (p, a) = (1, a) .  Continuity implies that for S' > 0 sufficiently small OP does not vanish anywhere on the annulus Ab . This and equation (4) show that, after possibly decreasing S' still further, the map ( H (e z© , W) is an embedding of A,, ; and, therefore, that for any smooth function R(p, a) > 0 the map 0' , : ( - (Z (0, w (S)) = (Re 20 , w) _ -i eZa áPW when p = 1 . defines an embedding of A, - , . Of course, we wish to choose R so that the map 0" satisfies the conditions stated above . That we can do so is implied by the following lemma . 12 6  T . DUCHAMP, F . FORSTNERIC Choose u" > 0 so that m/M >C . We claim that there is a constantQ"' < v" and a function h such that h(r) = 0 for 1 < r < 1 + o,"' , h(r) = 1 for r > 1+o,"/2, and h'(r) < H(r) where H(r) = (Mm)-1 ((r/M -C) . That such a function exists is clear because the right hand side of the last inequality is positive for 1 < r < 1 + u" and because the integral fl+," H(r) dr diverges to +oo . Hence we may set h(r) = fi k(t)dt where k(r) is any function satisfying the conditions, (i) 0 <_ k(r) < H(r), (ii) fi +alt k(r)dr = 1, and (iii) k(r) = 0 for 1 <_ r < o,"' < u"/2 and for r>o,"/2 . 1 .  E . BISHOP, Differentiable manifolds in complex Euclidean space, Duke Math . J . 32 (1965), 1-21 . 2 .  F . FORSTNERIC, Analytic disks with boundaries in a maximal real submanifold of (C 2 , Ann . Inst . Fourier 37 (1987), 1-44 . 3 .  F . W . KAMBER AND PH . TONDEUR, Characteristic invariants of foliated bundles, Manuscripta Mathematica 11 (1974), 51-89 . 4 .  E . L . STOUT AND W . ZAME, Totally real imbeddings and the universal covering spaces of domains of holomorphy : some examples, Manuscripta Mathematica 50 (1985), 29-48 . 5 .  R . O . WEI,I,S, Compact real submanifolds of a complex manifold with nondegenerate holomorphic tangent bundles, Math . Ann . 179 (1969),123-129 . 6 . M . GROMOV, "Partial differential relations," Ergebnisse 3 folge Bd .9, Springer-Verlag, Berlin-Heidelberg-New York, 1986 . Tom Duchamp : Department of Mathematics University of Washington GN-50 SeattIe, WA 98195 U .S .A . References F~anc Forstneric : Department of Mathematics University of Wisconsin Madison, WI 53706 U .S .A . Primera versió rebuda el 12 de Mari de 1992, darrera versió rebuda el 10 de Juny de 1992