Conformal foliations and CR geometry

Transcription

1 Geometry and Analysis, Flinders University, Adelaide p. 1/16 Conformal foliations and CR geometry Michael Eastwood [joint work with Paul Baird] University of Adelaide

2 Geometry and Analysis, Flinders University, Adelaide p. 2/16 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 L.P. Hughston and L.J. Mason, A generalised Kerr-Robinson theorem, Class. Quant. Grav. 5 (1988) 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, Ann. Glob. Anal. Geom. 44 (2013) 73 90

3 Geometry and Analysis, Flinders University, Adelaide p. 3/16 Conformal foliations U= unit vector field onω open R 3. h C U is (transversally) conformal L U preserves the conformal metric orthogonal to its leaves isothermal coördinates h= f+ ig f, g =0 f = g conjugate functions

4 Geometry and Analysis, Flinders University, Adelaide p. 4/16 Conjugate functions on R 2 f=f(r, s) g=g(r, s) s.t. { f= r g= s f, g =0 f = g f= r 2 s 2 f= g=2rs r r 2 + s 2 g= f= e r cos s s r 2 + s 2 g=e r sin s h f+ ig is (anti-)holomorphic in z r+ is

5 Geometry and Analysis, Flinders University, Adelaide p. 5/16 Conjugate functions on R 3 f=f(q, r, s) g=g(q, r, s) s.t. { f= r g= s f, g =0 f = g 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 Hopf R 2 S 2 { }

6 Geometry and Analysis, Flinders University, Adelaide p. 6/16 Almost Hermitian structures NB: J(p, q, r, s) R 4 R 4 satisfies J 2 = Id 0 u v w u 0 w v J= v w 0 u J SO(4) w v u 0 u 2 + v 2 + w 2 = 1, two-sphere ConsiderR 3 ={(p, q, r, s) R 4 p=0} R 4 NB: U (J p ) R 3 =(u q + v r + w s ) R 3 unit vector field also two-sphere

7 Geometry and Analysis, Flinders University, Adelaide p. 7/16 Sphere bundles bundle of unit vectors bundle of almost Hermitian structues Q Z τ R 3 R 4 section unit vector field section almost Hermitian structure

8 Geometry and Analysis, Flinders University, Adelaide p. 8/16 Hermitian structures Lemma J is integrable U (J p ) R 3 is conformal Conversely?? NB: J integrable J real-analytic Question: U conformal U real-analytic?? Answer: NO! WHY? However: U real-analytic and conformal U extends uniquely to an integrable J.

9 Geometry and Analysis, Flinders University, Adelaide p. 9/16 Twistor geometry Q Z τ R 3 R 4 compactify bundle of almost Hermitian structues CP3 {z 3 = z 4 = 0} [z 1, z 2, z 3, z 4 ] τ C 2 p+iq r+ is CP 3 τ twistor fibration (cf. Hopf) S 4 = 1 z z 4 2 z 2 z 3 + z 4 z 1 z 1 z 3 z 4 z 2

10 Geometry and Analysis, Flinders University, Adelaide p. 10/16 Twistor geometry cont d Q Z R 3 R 4 compactify Q CP 3 τ S 3 S 4 Q Q = {[z] CP 3 R( z 2 z 3 + z 4 z 1 )=0} {[Z] CP 3 Z Z 2 2 = Z Z 4 2 } Levi-indefinite hyperquadric CP 3 (cf. saddle)

11 Geometry and Analysis, Flinders University, Adelaide p. 11/16 Twistor results Q CP 3 τ S 3 S 4 Q CP 3 τ Theorem A section S 4 open Ω J CP 3 ofτdefines an integrable Hermitian structure if and only if M J(Ω) is a complex submanifold. Theorem A section S 3 open Ω U Q ofτ Q S 3 defines a conformal foliation if and only if M U(Ω) is a CR submanifold.

12 Geometry and Analysis, Flinders University, Adelaide p. 12/16 CR submanifolds and functions M Q CP 3 is a CR submanifold? It means: T M JT Q is preserved by J. It does not mean: M={ f= 0} where f is a CR function: (X+ ijx) f= 0 X Γ(T Q JT Q). Implicit function theorem is false in the CR category CR functions on Q are real-analytic. conformal foliations on S 3 need not be.

13 Geometry and Analysis, Flinders University, Adelaide p. 13/16 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!

14 Geometry and Analysis, Flinders University, Adelaide p. 14/16 Smooth conjugate functions Eikonal equation:( f 2 r ) +( f 2 s ) Plenty of non-analytic solutions: s Γ = 1 f= signed distance toγ r f(q, r, s) = f(r, s) g(q, r, s) = q } f, g =0 f = g QED

15 Geometry and Analysis, Flinders University, Adelaide p. 15/16 Real-analytic refinements U= unit vector field onω open R 3. ChooseωaC-valued null 1-form onωs.t. U Lemma U conformal ω dω=0. ω=0. Consider h= f+ ig Ω Cand letω=dh. Remark dω=0 f and g conjugate ωis null. Theorem Suppose ω Ω C is real-analytic null. ω dω=0 M CP 3 (s.t. M= M Q) ω dω=0 S C 4 {0} s.t.π( S)= M dω=0 S C 4 {0} s.t. S is Lagrangian Holomorphic function of two complex variables

16 THANK YOU Geometry and Analysis, Flinders University, Adelaide p. 16/16