In this section, we focus our attention on functions satisfying the pointwise Cauchy–Riemann conditions on a CR singular image with a nonempty CR singu-lar set S. We shall show that for each p∈S there exists a real-analytic function on a neighborhood ofpinM, satisfying all the pointwise Cauchy–Riemann condi-tions, that does not extend to a holomorphic function atp. This result generalizes Lemma4.3.
Theorem 7.1. Let M⊂Cn be a connected real-analytic CR singular subman-ifold such that there are a real-analytic generic submansubman-ifold N⊂Cn and a real-analytic CR map f:N→Cn that is a diffeomorphism ontoM=f(N). Suppose S is the CR singular set ofM andM\S is generic. Then,for anyp∈S,there exist a neighborhoodU ofpand a real-analytic functionuon U∩M such thatLu|q=0for
anyq∈U∩M and anyL∈Tq(0,1)M,butudoes not extend to a holomorphic function on any neighborhood ofpin Cn.
Proof. LetF be the unique holomorphic extension off to a neighborhood V off−1(p) inCn. Letθ=JF and ϕ=θ2◦f−1. Thenϕ is a real-analytic function on f(V∩N). We claim thatϕsatisfies all CR conditions onM. Indeed, for any point q∈M nearp, letL∈Tq(0,1)M. Ifq∈S, thenq:=f−1(q)∈f−1(S)⊂{x∈N:θ=0} and hence θ(q)=0. If q∈M\S, then Xq:=(f−1)∗L∈Tq(0,1) N and thus Xq(θ)(q)=0.
Therefore for allq∈U∩M, we have
(Lϕ)(q) =L(θ2◦f−1)(q) = ((f−1)∗L)θ2(f−1(q))
=Xqθ2(q) = 2θ(q)(Xqθ)(q) = 0.
(56)
Hence, the claim follows.
Ifϕdoes not extend to a holomorphic function in a neighborhood of pinCn, then we are done. Otherwise, suppose that ϕextends to a neighborhood. Notice thatϕ≡0 onS. Without loss of generality we can assume further thatϕis radical.
On the other hand, by Lemma 4.3, we can find a real-analytic function uon a neighborhood U of p in M such that u is CR on U\S and u does not extend holomorphically nearp. By construction,uϕandu2ϕrestricted toU are both CR functions onM∩U. We claim that at least one of the two functions uϕ and u2ϕ does not extend holomorphically pastp. Indeed, assume for a contradiction thatv1
andv2 are holomorphic functions on a neighborhood ofpin Cn whose restrictions toM areuϕandu2ϕ, respectively. Observe that the following equalities hold onU,
(57) v12=u2ϕ2=v2ϕ.
Since M is generic at all points on M\S, and v21 and v2ϕ are holomorphic, we deduce from (57) that v21=v2ϕ in a neighborhood ofp in Cn. In other words, we have the following equality in the ringOp,
(58) v12=v2ϕ.
Note that the ringOp is a unique factorization domain andϕis radical. From (58) we obtain thatv1 dividesv2and hence v2/v1 is holomorphic nearp. Consequently, u extends to the holomorphic function v2/v1 on a neighborhood of p. We have obtained a contradiction.
8. Examples
We start this section by the following proposition, which is helpful in construct-ing examples.
Proposition 8.1. Let w=ρ(z,z)¯ define a connected CR singular manifoldM near the origin in the coordinates (z, w)∈C2×C,where ρis real-analytic and such that ρ=0 anddρ=0at the origin. If ρz¯1≡0,thenM is Levi-flat at CR points and, furthermore,the set S of CR singularities is given by M∩{(z, w):ρz¯2(z)=0}.
Moreover,for each pointp∈M, there exists a neighborhoodU such thatU∩M is the image under a real-analytic CR diffeomorphism of an open subset of R2×C. Note that if w=ρ(z,z) and¯ ρ¯z1≡0, we could also get that M is a complex manifold, but in this caseM is not CR singular.
Proof. If ρz¯2≡0 then ρ is holomorphic and hence M is complex-analytic.
Otherwise,ρ¯z2≡0 and therefore, from Proposition 5.4, we see that (59) S=M∩{(z, w) :ρz¯2(z,z) = 0¯ }.
Hence, S⊂M is a proper real subvariety and M\S is generic. To see that M\S is Levi-flat, observe that in a neighborhood of p /∈S, M\S is foliated by a family of one-dimensional complex submanifolds defined by Lt={(z, w):w=ρ(z1, t,0,¯t)} with complex parametert. Let the local mapf:R2×C→M be given by
(60) f: (x, y, ξ)−→(ξ, x+iy, ρ(ξ, x+iy,0, x−iy)).
The CR structure onR2×Cis given by∂/∂ξ¯and so clearly f is a CR map. Since ρdoes not depend on ¯z1, it follows that f sendsR2×CintoM. The fact thatf is a local diffeomorphism is immediate.
Using the proposition we can easily create many examples showing that the CR singular set of a Levi-flat manifold that is an image of a CR diffeomorphism can have any possible CR structure allowed by Corollary5.2.
Example 8.2. We can obtain a 3-dimensional CR singularity by simply taking a parabolic CR singular Bishop surface in 2 dimensions and considering it in 3 dimensions. For example,
(61) w=|z2|2+¯z22
2 .
The manifold is the image of R2×C by the construction of Proposition 8.1. The CR singular set is the set{(z, w):Rez2=0}∩M, and hence 3 real dimensional.
The submanifold is contained in the nonsingular Levi-flat hypersurface defined by Imw=−Im12z22.
Example 8.3. Next, let us consider
(62) w=z1z¯22.
The manifold is the image of R2×C by the construction of Proposition 8.1. The CR singular set is the set
({(z, w) :z1= 0}∪{(z, w) :z2= 0})∩M,
that is a union of two 2-dimensional sets, both of which are complex-analytic. Note that the set{(z, w):z1=0}∩M is a complex-analytic set that is an image of a totally real submanifold ofR2×Cunder the map of Proposition8.1. We therefore have a complex-analytic set that is a subset of M while not being an image of one of the leaves of the Levi-foliation ofR2×C.
Example 8.4. Consider
(63) w=z1z¯2−¯z22
2 .
Again the manifold is the image ofR2×C. The CR singular setS is the set {(z, w) :z1= ¯z2}∩M,
which is a totally real set; to see this fact simply substitute ¯z2=z1 in the defining equation forM to find thatS is the intersection of {(z, w):z1= ¯z2}with a complex manifold.
Example 8.5. Consider
(64) w=z1z¯2−z2z¯22 2 .
The CR singular set S is the set {(z, w):z1=|z2|2}∩M, which is a CR singular submanifold.
Example 8.6. While we have mostly concerned ourselves with flat manifolds, there is nothing particularly special about flat manifolds. Even a finite-type man-ifold can map to a CR singular manman-ifold. The following example was given as Example 1.6 in [ER]. LetM⊂C3be given by
(65) M= M is taken to be the CR singular manifold
(66) {(z1, z2, w)∈C3:w= (¯z2+i|z1|2+|z1|4)2} via the finite holomorphic map
(67) (z, w1, w2)−→(z, w1+iw2,(w1−iw2)2).
The map is a diffeomorphism onto its image when restricted toM.
References
[AGV] Arnold, V. I., Gusein-Zade, S. M. and Varchenko, A. N., Monodromy and asymptotics of integrals, inSingularities of Differentiable Maps.Vol.II, Mono-graphs in Mathematics83, Birkh¨auser, Boston, MA, 1988.
[BER1] Baouendi, M. S., Ebenfelt, P. and Rothschild, L. P., CR automorphisms of real analytic manifolds in complex space, Comm. Anal. Geom. 6 (1998), 291–315.
[BER2] Baouendi, M. S., Ebenfelt, P. and Rothschild, L. P., Real Submanifolds in Complex Space and Their Mappings, Princeton Mathematical Series 47, Princeton University Press, Princeton, NJ, 1999.
[B] Bishop, E., Differentiable manifolds in complex Euclidean space,Duke Math.J.32 (1965), 1–21.
[BG] Burns, D.andGong, X., Singular Levi-flat real analytic hypersurfaces,Amer.J.
Math.121(1999), 23–53.
[DF] Diederich, K. and Fornæss, J. E., Pseudoconvex domains with real-analytic boundary,Ann.of Math.107(1978), 371–384.
[ER] Ebenfelt, P. and Rothschild, L. P., Images of real submanifolds under finite holomorphic mappings,Comm.Anal.Geom.15(2007), 491–507.
[G] Gong, X., Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics, Indiana Univ.
Math.J.53(2004), 83–95.
[H1] Harris, G. A., The traces of holomorphic functions on real submanifolds,Trans.
Amer.Math.Soc.242(1978), 205–223.
[H2] Harris, G. A., Lowest order invariants for real-analytic surfaces in C2, Trans.
Amer.Math.Soc.288(1985), 413–422.
[Hu] Huang, X., On an n-manifold in Cn near an elliptic complex tangent, J. Amer.
Math.Soc.11(1998), 669–692.
[HK] Huang, X.andKrantz, S. G., On a problem of Moser,Duke Math.J.78(1995), 213–228.
[HY1] Huang, X. and Yin, W., A Bishop surface with a vanishing Bishop invariant, Invent.Math.176(2009), 461–520.
[HY2] Huang, X.andYin, W., Flattening of CR singular points and analyticity of local hull of holomorphy.Preprint, 2012.arXiv:1210.5146.
[KW] Kenig, C. E. andWebster, S. M., On the hull of holomorphy of ann-manifold inCn,Ann.Sc.Norm.Super.Pisa Cl.Sci.11(1984), 261–280.
[L1] Lebl, J., Nowhere minimal CR submanifolds and Levi-flat hypersurfaces,J.Geom.
Anal.17(2007), 321–341.
[L2] Lebl, J., Singular set of a Levi-flat hypersurface is Levi-flat,Math.Ann.355(2013), 1177–1199.
[M] Moser, J., Analytic surfaces inC2 and their local hull of holomorphy,Ann.Acad.
Sci.Fenn.Math.10(1985), 397–410.
[MW] Moser, J. K.andWebster, S. M., Normal forms for real surfaces inC2near com-plex tangents and hyperbolic surface transformations,Acta Math.150(1983), 255–296.
[S] Shabat, B. V., Functions of several variables, inIntroduction to Complex Analysis.
Part II, Translations of Mathematical Monographs 110, Amer. Math. Soc., Providence, RI, 1992.
[W] Whitney, H.,Complex Analytic Varieties, Addison-Wesley, Reading, MA, 1972.
Jiˇr´ı Lebl