Geometry and Analysis, Flinders University, Adelaide p. 1/16 Conformal foliations and CR geometry Michael Eastwood [joint work with Paul Baird] University of Adelaide
Geometry and Analysis, Flinders University, Adelaide p. 2/16 Disclaimers and references R.P. Kerr, I. Robinson,... 1961 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) 275 285 P. Baird and J.C. Wood, Harmonic morphisms and shear-free ray congruences (2002), http://www.maths.leeds.ac.uk/pure/staff/wood/bwbook/bwbook.html 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) 198 204 P. Nurowski, Construction of conjugate functions, Ann. Glob. Anal. Geom. 37 (2010) 321 326 P. Baird and M.G. Eastwood, CR geometry and conformal foliations, Ann. Glob. Anal. Geom. 44 (2013) 73 90
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
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
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 { }
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
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
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.
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 3 2 + z 4 2 z 2 z 3 + z 4 z 1 z 1 z 3 z 4 z 2
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 1 2 + Z 2 2 = Z 3 2 + Z 4 2 } Levi-indefinite hyperquadric CP 3 (cf. saddle)
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.
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.
Geometry and Analysis, Flinders University, Adelaide p. 13/16 CR functions {[Z] CP 2 Z 1 2 + Z 2 2 = Z 3 2 }= three-sphere CP 2......... Theorem (H. Lewy 1956) CR holomorphic extension {[Z] CP 3 Z 1 2 + Z 2 2 = Z 3 2 + Z 4 2 }=Q......... CP 3 Corollary CR holomorphic extension Hence, a CR function on Q is real-analytic!
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
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
THANK YOU Geometry and Analysis, Flinders University, Adelaide p. 16/16