A Counterexample to C k -regularity for the Newlander-Nirenberg Theorem
aa r X i v : . [ m a t h . C V ] J u l A Counterexample to C k -regularity for theNewlander-Nirenberg Theorem Yao, Liding
University of Wisconsin-Madison
Abstract
We give an example of C k -integrable almost complex structure that does not admita corresponding C k +1 -complex coordinate system. The celebrated Newlander-Nirenberg theorem [5] states that given an integrable almostcomplex structure, it is locally induced by some complex coordinate system.Malgrange [4] proved the existence of such complex coordinate chart when the almostcomplex structure is not smooth, and he obtained the following sharp H¨older regularity forthis chart:
Theorem 1 (Sharp Newlander-Nirenberg) . Let k ∈ Z + and < α < , let M n be a C k +1 ,α -manifold endowed with a C k,α -almost complex structure J : T M → T M . If J is integrable,then for any p ∈ M there is a C k +1 ,α -complex coordinate chart ( w , . . . , w n ) : U → C n near p such that J ∂∂w j = i ∂∂w j for j = 1 , . . . , n . There are several equivalent characterizations for integrability. One of which is the van-ishing of the
Nijenhuis tensor N J ( X, Y ) := [
J X, J Y ] − J [ J X, Y ] − J [ X, J Y ] − [ X, Y ].Our main theorem is to show that this is not true when α ∈ { , } : Theorem 2.
Let k, n ∈ Z + . There is a C k -integrable almost complex structure J on R n ,such that there is no C k, -complex coordinate chart w : U ⊂ R n → C n near satisfying J ∂∂w j = i ∂∂w j for j = 1 , . . . , n . For convenience we use the viewpoint of eigenbundle of J : Set V n = ` p { v ∈ C T p R n : J p v = iv } . So J ∂∂w j = i ∂∂w j for all j iff d ¯ w , . . . , d ¯ w n spans V ⊥ n | U ≤ C T ∗ R n | U . And J isintegrable if and only if X, Y ∈ Γ( V n ) ⇒ [ X, Y ] ∈ Γ( V n ) for all complex vector fields X, Y .See [1] Chapter 1 for details.First we can restrict our focus to the 1-dimensional case:
Proof of 1-dim ⇒ n-dim. Suppose J is a C k -almost complex structure on C z (not com-patible with the standard complex structure), such that near 0, there is no C k, -complexcoordinate ϕ satisfying V ⊥ = Span d ¯ ϕ in the domain. Here V ⊥ is the dual eigenbundle of J .Denote θ = θ ( z ) as a C k C z that spans V ⊥ .Consider R n ≃ C n ( z ,...,z n ) . We identify θ as the C k R n . Take an n -dim almostcomplex structure on R n such that the dual of eigenbundle V ⊥ n is spanned by θ, d ¯ z , . . . , d ¯ z n .In other words, V n is the “tensor” of V with the standard complex structure of C n − z ,...,z n ) .If w = ( w , . . . , w n ) is a corresponding C k, -complex chart for V n near 0, then there is a1 ≤ j ≤ n such that d ¯ w j d ¯ z , . . . , d ¯ z n ) near 0. In other words, we have linear1ombinations d ¯ w j = λ θ + λ d ¯ z + · · · + λ n d ¯ z n for some non-vanishing function λ ( z , . . . , z n )near z = 0.Therefore w j ( · , n − ) is a complex coordinate chart defined near z = 0 ∈ R whosedifferential spans V ⊥ n | R z ∼ = V ⊥ near 0. By our assumption on V , we have w j ( · , / ∈ C k, .So w j / ∈ C k, , which means w / ∈ C k, .Now we focus on the one-dimensional case. Note that a 1-dim structure is automaticallyintegrable.Fix k ≥
1. Define an almost complex structure by setting its eigenbundle V ≤ C T R equals to the span of ∂∂z + a ( z ) ∂∂ ¯ z , where a ∈ C k ( R ; C ) has compact support that satisfiesthe following:(i) a ∈ C ∞ loc ( R \{ } ; C );(ii) ∂ − z ∂ ¯ z a / ∈ C k − , near 0;(iii) za ∈ C k +1 ( R ; C ) and z − a ∈ C k − ( R ; C ) (which implies a ( z ) = o ( | z | ) as z → a ⊂ B ;(v) k a k C < δ for some small enough δ > δ = 10 − will be ok).Here we take ∂ − z to be the conjugated Cauchy-Green operator on the unit disk : ∂ − z φ ( z ) = ∂ − z, B φ ( z ) := 1 π Z B φ ( ξ + iη ) dξdη ¯ z − ξ + iη . We use notation ∂ − z because it is an right inverse of ∂ z . And ∂ − z : C m,β ( B ; C ) → C m +1 ,β ( B ; C ) is bounded linear for all m ∈ Z ≥ , 0 < β <
1. See [8] theorem 1.32 in section8.1 (page 56), or [2] lemma 2.3.4 for example.We can take supp a ⊂ B such that when | z | < , a ( z ) := ¯ z k +1 ∂ z (cid:0) ( − log | z | ) (cid:1) = ∂ z (cid:0) ¯ z k +1 ( − log | z | ) (cid:1) . (1)Note that for this a we have a ( z ) = O (cid:0) | z | k ( − log | z | ) − (cid:1) = o ( | z | k ). Remark 3.
Roughly speaking, Property (i) Singsupp a = { } says that the regularity of a ( z ) corresponds to the vanishing order of a at 0. To some degree, by multiplying with a ( z ),a function gains some regularity at the origin.We check Property (ii) that ∂ − z ∂ ¯ z a / ∈ C k − , here. Lemma 4.
Let a ( z ) be given by (1) , and let χ ∈ C ∞ c ( B ) satisfies χ ≡ in a neighborhoodof 0. Then ∂ − z ( χa ¯ z ) / ∈ C k − , near z = 0 .Proof. Denote b ( z ) := ¯ z k +1 ( − log | z | ) , so b ∈ C ∞ ( B \{ } ; C ), χa = χb z and χa ¯ z = χb z ¯ z .First we show that ∂ − z ( χb z ¯ z ) − b ¯ z ∈ C ∞ ( B ; C ). We write ∂ − z ( χb z ¯ z ) − b ¯ z = ∂ − z ∂ z ( χb ¯ z ) − χb ¯ z − (1 − χ ) b ¯ z − ∂ − z ( χ z b ¯ z ) . Since ∂ z ∂ − z ∂ z ( χb ¯ z ) = ∂ z ( χb ¯ z ), we know ∂ − z ∂ z ( χb ¯ z ) − χb ¯ z is anti-holomorphic. By Cauchyintegral formula we get ∂ − z ∂ z ( χb ¯ z ) − χb ¯ z ∈ C ∞ .By assumption 0 / ∈ supp χ z , 0 / ∈ supp(1 − χ ) and b ∈ C ∞ ( B \{ } ; C ), we know χ z b ¯ z ∈ C ∞ c and (1 − χ ) b ¯ z ∈ C ∞ ( B ; C ). Throughout the paper, B = B (0 ,
1) refers to the unit disk in C . ∂ − z = π ¯ z ∗ ( · ) is aconvolution operator with kernel π ¯ z ∈ L , so ∂ − z ( χ z b ¯ z ) = π ¯ z ∗ ( χ z b ¯ z ) ∈ C ∞ .It remains to show b ¯ z / ∈ C k − , near 0. Indeed one has ∂ k +1¯ z (¯ z k +1 ( − log | z | ) ) = ( k + 1)!( − log | z | ) + O (1) , as z → , because by Leibniz rule ∂ k ¯ z ∂ ¯ z (cid:0) ¯ z k +1 ( − log | z | ) (cid:1) = P k +1 j =0 (cid:0) k +1 j (cid:1) ∂ k +1 − j ¯ z ( z k +1 ) · ∂ j ¯ z ( − log | z | ) =( k + 1)!( − log | z | ) + P k +1 j =1 O ( z j ) O (cid:0) z − j ( − log | z | ) − (cid:1) = ( k + 1)!( − log | z | ) + O (cid:0) ( − log | z | ) − (cid:1) . Now assume w : ˜ U ⊂ R → C is a 1-dim C -complex coordinate chart defined near 0that represents V , then Span d ¯ w = Span( d ¯ z − adz ) | ˜ U = V ⊥ | ˜ U . So d ¯ w = ¯ w ¯ z d ¯ z + ¯ w z dz =¯ w ¯ z ( d ¯ z − adz ), that is, ∂w∂ ¯ z ( z ) + ¯ a ( z ) ∂w∂z ( z ) = 0 , z ∈ ˜ U .
Remark 5.
It is worth noticing that ∂ w = ∂ z + a∂ ¯ z . Indeed ∂ w is only a scalar multiple of ∂ z + a∂ ¯ z .Note that w z (0) = 0 because ( d ¯ z − adz ) | = d ¯ z | ∈ Span d ¯ w | . So by multiplying w z (0) − ,we can assume w z (0) = 1 without loss of generality. Then f := log ∂ z w is a well-definedfunction in a smaller neighborhood U ⊂ ˜ U of 0, which solves ∂f∂ ¯ z ( z ) + a ( z ) ∂f∂z ( z ) = − ∂ ¯ a∂z ( z ) (cid:16) = − ∂a∂ ¯ z ( z ) (cid:17) , z ∈ U. (2)Property (v) indicates that the operator ∂ ¯ z +¯ a∂ z is a first order elliptic operator. Thereforewe can consider a second order divergence form elliptic operator L := ∂ z ( ∂ ¯ z + ¯ a∂ z )whose coefficients are C k globally and are C ∞ outside the origin.By the classical Schauder’s estimate (see [7] Theorem 4.2, or [3] Chapter 6 & 8), we havethe following: Lemma 6 (Schauder’s interior estimate) . Assume u, ψ ∈ C , ( B ; C ) satisfy Lu = ψ z . Let U ⊂ R be a neighborhood of . The following hold:(a) If ψ ∈ C ∞ loc ( U \{ } ; C ) , then u ∈ C ∞ loc ( U \{ } ; C ) .(b) If ψ ∈ C k − , ( R ; C ) , then u ∈ C k, − ε loc ( R ; C ) for all < ε < . Our Theorem 2 for 1-dim case is done by the following proposition:
Proposition 7.
For any neighborhood U ⊂ R z of , there is no f ∈ C k − , ( U ; C ) solving (2) .Proof. Suppose there is a neighborhood U ⊂ B of the origin, and a solution f ∈ C k − , ( U ; C )to (2).Applying Lemma 6 (a) on (2) with u = f and ψ = − ¯ a z , we know that f ∈ C ∞ ( U \{ } ; C ).Take χ ∈ C ∞ c ( U, [0 , χ ≡ g ( z ) := χ ( z ) f ( z ) , h ( z ) := ¯ zg ( z ) . g, h are C k − , -functions defined in R that are also smooth away from 0 , and satisfythe following: g ¯ z + ¯ ag z = χ ¯ z f + χ z ¯ af − χ ¯ a z , (3) h ¯ z + ¯ ah z = g + ¯ z ( χ ¯ z f + χ z ¯ af ) − χ ¯ z ¯ a z . (4)By construction ∇ χ ≡ χ ¯ z f + χ z ¯ af ∈ C ∞ c ( U ; C ).Under our assumption that f ∈ C k − , ( U ; C ), then the key is to show that(I) h ∈ C k, − εc ( R ; C ), ∀ ε ∈ (0 , ag z ∈ C k − , − εc ( R ; C ), ∀ ε ∈ (0 , (I) By assumption ¯ z ( χ ¯ z f + χ z ¯ af ) ∈ C ∞ c , g ∈ C k − , , and by Property (iii), χ ¯ z ¯ a z ∈ C k ( R ; C ).Applying Lemma 6 (b) to (4), with u = h and ψ = g + ¯ z ( χ ¯ z f + χ z ¯ af ) − χ ¯ z ¯ a z ∈ C kc ( R ; C ),we get h ∈ C k, − ε ( B ; C ), for all 0 < ε < (II) When k ≥
2, we know z − a ∈ C k − for Property (iii). So for any ε ∈ (0 , ag z ∈ C k − , − ε because ∇ z, ¯ z (¯ ag z ) = ¯ a · ( ∂ z g z , ∂ ¯ z g z ) + g z ∇ z, ¯ z ¯ a = ¯ z − ¯ a · ( h zz , h z ¯ z − g z ) + O ( C k − , ) ∈ C k − , − ε . When k = 1, for any z , z ∈ R \{ } , note that g ¯ z is smooth outside the origin, so¯ ag z ∈ C , − ε : | ¯ ag z ( z ) − ¯ ag z ( z ) | ≤ | ¯ a ( z ) || g z ( z ) − g z ( z ) | + | ¯ a ( z ) − ¯ a ( z ) || g z ( z ) |≤| ¯ z − ¯ a ( z ) || ¯ z g z ( z ) − ¯ z g z ( z ) | + | ¯ z − ¯ a ( z ) || ¯ z − ¯ z || g z ( z ) | + | ¯ a ( z ) − ¯ a ( z ) || g z ( z ) |≤k z − a k C k∇ h k C − ε | z − z | − ε + k z − a k C k g k C , | z − z | + k a k C k g k C , | z − z | . Here as a remark, | ¯ ag z ( z ) − ¯ ag z ( z ) | . a,g,h | z − z | − ε still makes sense when z or z = 0 , though ¯ g z (0) may not be defined. Indeed lim z → ¯ ag z ( z ) exists because ¯ ag z itself hasbounded C , − ε -oscillation on B \{ } , and then the limit defines the value of ¯ ag z at z = 0 . So for either case of k , we have ¯ ag z ∈ C k − , − ε ( R ; C ) for all ε ∈ (0 , g = ( g − ∂ − z g ¯ z ) + ∂ − z ( χ ¯ z f + χ z ¯ af ) − ∂ − z (¯ ag z ) − ∂ − z ( χ ¯ a z ) , in B . (5)The right hand side of (5) consists of four terms, the first to the third are all C k , whilethe last one is not C k − , . We explain these as follows: • Since ∂ z ( g − ∂ − z g ¯ z ) = 0 in B , we know g − ∂ − z g ¯ z is anti-holomorphic, which is smoothin B . • By assumption χ ¯ z f + χ z ¯ af ∈ C ∞ c ( R ; C ), so ∂ − z ( χ ¯ z f + χ z ¯ af ) ∈ C ∞ ( B ; C ) as well. • By consequence (II) ¯ ag z ∈ C k − , − ε ( R ; C ), for all ε ∈ (0 , ∂ − z (¯ ag z ) ∈ C k, − ε ⊂ C k ( B ; C ). • However by Lemma 4, ∂ − z ( χ ¯ a z ) / ∈ C k − , near 0.Combining each term to the right hand side of (5), we know g / ∈ C k − , near 0. Contra-diction! Remark 8.
The key to the proof is the non-surjectivity of ∂ z : C k → C k − , which we use toconstruct a function a ( z ) such that a (0) = 0, Singsupp a = { } , and ∂ − z ∂ ¯ z a / ∈ C k .4 emark 9. For positive integer k , Malgrange’s sharp estimate of Theorem 1 still holds forZygmund spaces C k = B k ∞∞ , that is, given J ∈ C k , there exists a C k +1 -coordinate chart( w , . . . , w n ), such that J ∂∂w j = i ∂∂w j . One can also see [6] for details. A reason why ourproof does not give a counterexample for Zygmund spaces, is that there does exist a f ∈ C k defined in a neighborhood of 0 that solves (2). Acknowledgement
The author would like to express his appreciation to his advisor Prof. Brian T. Street forhis help.
References [1] Shiferaw Berhanu, Paulo D. Cordaro, and Jorge Hounie.
An introduction to involutivestructures , volume 6 of
New Mathematical Monographs . Cambridge University Press,Cambridge, 2008.[2] So-Chin Chen and Mei-Chi Shaw.
Partial differential equations in several complex vari-ables , volume 19 of
AMS/IP Studies in Advanced Mathematics . American MathematicalSociety, Providence, RI; International Press, Boston, MA, 2001.[3] David Gilbarg and Neil S. Trudinger.
Elliptic partial differential equations of second order .Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.[4] B. Malgrange. Sur l’int´egrabilit´e des structures presque-complexes. In
Symposia Mathe-matica, Vol. II (INDAM, Rome, 1968) , pages 289–296. Academic Press, London, 1969.[5] A. Newlander and L. Nirenberg. Complex analytic coordinates in almost complex mani-folds.
Ann. of Math. (2) , 65:391–404, 1957.[6] Brian Street. Sharp regularity for the integrability of elliptic structures.
J. Funct. Anal. ,278(1):108290, 2020.[7] Michael E. Taylor.
Partial differential equations III. Nonlinear equations , volume 117 of
Applied Mathematical Sciences . Springer, New York, second edition, 2011.[8] I. N. Vekua.
Generalized analytic functions . Pergamon Press, London-Paris-Frankfurt;Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962.