Conformal foliations

Transcription

1 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 1/17 Conformal foliations Michael Eastwood [joint work with Paul Baird] Australian National University

2 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 2/17 Disclaimers and references R.P. Kerr, I. Robinson, R. Penrose and W. Rindler, Spinors and Space-time vol. 2, Chapter 7, Cambridge University Press 1986 P. Baird and J.C. Wood, Harmonic morphisms and shear-free ray congruences (2002), P. Baird and J.C. Wood, Harmonic Morphisms between Riemannian Manifolds, Oxford University Press 2003 P. Baird and R. Pantilie, Harmonic morphisms on heaven spaces, Bull. Lond. Math. Soc. 41 (2009) P. Nurowski, Construction of conjugate functions, Ann. Glob. Anal. Geom. 37 (2010) P. Baird and M.G. Eastwood, CR geometry and conformal foliations,

3 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 3/17 Conjugate functions f=f (q, r, s) g=g(q, r, s) s.t. f= q g=r { f, g =0 f 2 = g 2 f= q 2 r 2 s 2 g=2q r 2 + s 2 f= r q2 + r 2 + s 2 g= s q2 + r 2 + s 2 r 2 + s 2 r 2 + s 2 f= (1 q2 r 2 s 2 )r+ 2qs r 2 + s 2 g= (1 q2 r 2 s 2 )s 2qr r 2 + s 2 R 3 S 3 R 2 S 2 Hopf

4 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 4/17 The implicit function theorem If M N is a hypersurface ( non-singular) Then M={ f= 0} locally ( non-degenerate) OK if M and N are smooth ( f smooth) OK if M and N are complex ( f holomorphic) What if M and N are CR? ( f CR)? Yes if M and N are also real-analytic No in general!

5 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 5/17 CR geometry H T M and J : H Hsuch that J 2 = Id ( rank R H is even) [H 0,1, H 0,1 ] H 0,1 where H 0,1 {X JX= ix} Examples H= T M where M is a complex manifold M 2n 1 C n and H T M JT M, a CR manifold of hypersurface type Q {[Z] CP 3 Z Z 2 2 = Z Z 4 2 }, the Levi-indefinite hyperquadric incp 3 f : M Cis a CR function X f= 0 X Γ(H 0,1 )

6 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 6/17 CR functions {[Z] CP 2 Z Z 2 2 = Z 3 2 }= three-sphere CP Theorem (H. Lewy 1956) CR holomorphic extension {[Z] CP 3 Z Z 2 2 = Z Z 4 2 }=Q CP 3 Corollary CR holomorphic extension Hence, a CR function on Q is real-analytic!

7 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 7/17 CR submanifolds of Q Suppose Q open Ω f C is a CR function. Then M { f= 0} Ωis a CR submanifold, i.e. T M H is preserved by J. Conversely, suppose M Ω open Q is a 3-dim l CR submanifold M is real-analytic. Then M= M Q for M CP 3 a complex hypersurface M={ f= 0} for f :Ω C CR and real-analytic. Q: Can we drop real-analyticity? A: NO

8 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 8/17 Conformal foliations U= unit vector field onω open R 3. U ω=0 ω,ω =0 ω dω=0 U is transversally conformal L U preserves the conformal metric orthogonal to its leaves h isothermal coördinates C h= f+ ig d f, dg =0 d f 2 = dg 2 dh, dh = 0 dh=ω ω,ω =0 dω=0 conjugate functions!

9 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 9/17 Integrable Hermitian structures Suppose J : TR 4 TR 4 satisfies J 2 = Id 0 u v w u 0 w v J= v w o u J SO(4) w v u 0 u 2 + v 2 + w 2 = 1 [T 0,1, T 0,1 ] T 0,1 where T 0,1 {X JX= ix} R 3 ={(p, q, r, s) R 4 p=0} R 4 Then U ( J p) R 3= ( u q + v r + w s) R 3 is transversally conformal.

10 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 10/17 Twistor fibration CP 3 \{z 3 = z 4 = 0} [z 1, z 2, z 3, z 4 ] τ [ ] [ p+iq R 4 1 z2 z = 3 + z 4 z 1 r+is z z 4 2 z 1 z 3 z 4 z 2 τ 1 (x) { J : T x R 4 T x R 4 J 2 = Id J SO(T x R 4 ) } ] Theorem A sectionr 4 open Ω J CP 3 ofτdefines an integrable Hermitian structure if and only if M J(Ω) is a complex submanifold.

11 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 11/17 Twistor fibration cont d CP 3 \{z 3 = z 4 = 0} [z 1, z 2, z 3, z 4 ] τ [ ] [ p+iq R 4 1 z2 z = 3 + z 4 z 1 r+is z z 4 2 z 1 z 3 z 4 z 2 R 3 ={p=0} τ 1 (R 3 )={[z] CP 3 R( z 2 z 3 + z 4 z 1 )=0}=Q\I τ 1 (x) {U T x R 3 U 2 =1} Q\I unit sphere bundle Theorem A sectionr 3 open Ω U Q ofτ : Q\I R 3 defines a conformal foliation if and only if M U(Ω) is a CR submanifold. ]

12 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 12/17 The story so far Compactify: CP 3 Q=S (TS 3 ) S 4 S 3 CP 3 Q complex surface = M? M = CR three-fold Hermitian structure? conformal foliation ( ) Proofs: check in local coördinates Real-analytic case: M = M Q et cetera

13 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 13/17 Smooth counterexample ( ) 2 ( ) 2 f f Eikonal equation: + = 1 r s Plenty of non-analytic solutions: s Γ f= signed distance toγ } f (q, r, s) = f (r, s) g(q, r, s) = q d f, dg =0 d f 2 = dg 2 r QED

14 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 14/17 Real-analytic refinements ω= C-valued real-analytic null 1-form onω open R 3 ω dω=0 σ dω=0 σ s.t. σ,ω =0 dω=0 M CP 3 S C 4 \{0} (andπ( S )= M) S C 4 \{0} such that S is Lagrangian

15 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 15/17 Explicit formulæ Q C R 3 (z, q, r, s) CP 3 C 3 (z, z 1, z 2 ) z 1 = (r+is)z iq z 2 = iqz (r is) ω 2z dq+i(1+z 2 ) dr+(1 z 2 ) ds ω,ω =0 ω dω=2 dz dz 1 dz 2 z=z(q, r, s) implicitly by z=φ(z 1, z 2 ) holomorphic dz= Φ z 1 dz 1 + Φ z 2 dz 2 ω dω=0 } {{ }

16 Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 16/17 Explicit formulæ cont d C 2 R 3 (w, z, q, r, s) C 4 (w, z, z 1, z 2 ) z 1 = (r+is)z iqw z 2 = iqz (r is)w ω 2wz dq+i(w 2 + z 2 ) dr+(w 2 z 2 ) ds ω,ω =0 dω = 2i(dz dz 1 dw dz 2 ) = 2id(z dz 1 w dz 2 ) z=z(q, r, s) w=w(q, r, s) by z dz 1 w dz 2 = d(ξ(z 1, z 2 )) NB: Lagrangian w.r.t. dz dz 1 dw dz 2 dω=0

17 THANK YOU Séminaire au Laboratoire J.A. Dieudonné de l Université Nice Sophia Antipolis p. 17/17