Some Elliptic and Subelliptic Complexes from Geometry

Transcription

1 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 1/14 Some Elliptic and Subelliptic Complexes from Geometry Michael Eastwood [ based on joint work with Robert Bryant, Rod Gover, and Katharina Neusser Australian National University ]

2 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 2/14 De Rham complex in R 3 f grad i f ω i curl ǫ i jk j ω k φ i div i φ i on a smooth manifold 0 Λ 0 d Λ 1 d Λ 2 d Λ 3 d d Λ n 0 Elliptic and locally exact (mostly) Γ(U,Λ p 1 ) d Γ(U,Λ p ) d Γ(U,Λ p+1 ) is exact ker Γ(U,Λ 0 ) d Γ(U,Λ 1 )=R p 1

3 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 3/14 Coeffective complex M = symplective manifold of dimension 2n. J Γ(M,Λ 2 ) non-degenerate and dj= 0. 0 Λ k Λ k J Λ k+2 0 defines Λ k n k 2n. NB 0 Λ k Λ k J Λ k+2 0 d d 0 Λ k+1 Λ k+1 J Λ k+3 0 Hence (Bouche 1990) Λ n Λ n+1 Λ 2n 2 Elliptic and locally exact Λ 2n 1 d Λ 2n 0

4 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 4/14 Coeffective complex cont d NB J Λ k Λ k+2 is surjective for n k 2n 2 J Λ n 1 Λ n+1 is an isomorphism J Λ k 2 Λ k is injective for 2 k n k 2 J 0 Λ Λ k Λ k 0 defines Λ k 0 k n. Hence 0 Λ 0 d Λ 1 Λ 2 Elliptic Λ n 1 Λ n But Λ k Λ 2n k 0 k n Whence this is just the adjoint of the coeffective complex.

5 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 5/14 Coeffective complex cont d cont d Thus, on a symplectic manifold 0 Λ 0 d Λ 1 0 Λ 0 Λ 1 Λ 2 Λ 2 Λ n 1 Λ n 1 Λ n Λ n d (2) Elliptic local cohomology = R Global cohomology Smith (1976, n=2) Rumin-Seshadri Tseng-Yau H r (M,R) H r J (M) Hr 1 (M,R) J H r+1 (M,R)

6 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 6/14 Calibrated G 2 -manifolds M= smooth manifold of dimension 7 J Γ(M,Λ 3 ) non-degenerate and dj= 0. Λ 0 J Λ 3 and Λ 1 J Λ 4 are injective Λ 2 J Λ 5 is an isomorphism Λ 3 J Λ 6 and Λ 4 J Λ 7 are surjective 0 Λ 0 Λ 1 Λ 2 Λ 3 Λ 4 0 Λ 0 Λ 1 Λ 2 Λ 3 Λ 4 local cohomology = R second order Elliptic

7 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 7/14 Rumin complex in R 3 X,Y vector fields. Suppose X,Y,Z [X,Y] span. NB Xf= 0,Y f= 0 f constant. Let H span{x,y}. on a contact manifold H TM Λ 1 Λ 1 H 0 R Λ 0 Λ 1 Λ 1 H defines d H Λ 0 Λ 1 H s.t. R=ker Λ 0 d H Λ 1 H locally Darboux [X,Z]=0=[Y,Z] wlg Xf = g Y f = h } { XY g X2 h+zg = 0 Y Xh Y 2 g Zh = 0 conversely? yes!

8 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 8/14 Rumin complex cont d Λ 0 Λ 0 d Λ 1 d Λ 2 d Λ 3 d Λ 4 d Λ 5 Λ 0 Λ 1 H Λ 2 H Λ 3 H Λ 4 H L Λ 1 H L Λ2 H L Λ3 H L Λ4 H L inj ve iso sm Diagram chase (spectral sequence) Subelliptic complex d H Λ 1 H d H Λ 2 H d (2) H Λ 2 H L sur ve d H Λ 3 H L d H Λ dim=1 dim=4 dim=5 dim=5 dim=4 dim=1

9 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 9/14 Parabolic geometry Geometries modelled on homogeneous spaces Examples G/P conformal geometry CR geometry projective geometry G simple Lie group { P parabolic subgroup SO(n + 1, 1)/P SU(n+1,1)/P contact projective geometry Sp(2n+2,R)/P SL(n + 1, R)/ 0 0

10 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 10/14 BGG complexes on the 3-sphere Conformal a b a 2 2a+b+2 a b 3 2a+b+2 a b 3 Contact projective a b a 2 a+b+1 a 2b 4 a+b+1 a 2b 4 b b CR a b a 2 a+b+1 a+b+1 b 2 a b 3 a b a b 3 b 2 a 2

11 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 11/14 Applications: linear elasticity displacement X i in R 3 strain i X j + j X i Σ ij Killing Saint-Venant stress ǫ km i ǫ ln j k l Σ mn G ij load i G ij Bianchi BGG (2) on 3-sphere (Arnold-Falk-Winther) new stable finite element schemes

12 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 12/14 Applications: global analysis on a symplectic manifold Rumin-Seshadri complex 0 Λ 0 d Λ 1 Λ 2 Λ 3 Γ(CP n,λ 0 ) d Γ(CP n,λ 1 ) Γ(CP n,λ 2 ) exact Γ(CP n,λ 1 ) Γ(CP n, 2 Λ 1 ) Killing (2) Γ(CP n, Λ 1 ) exact cf. St-Venant on a CR manifold Akahori-Garfield-Lee complex Hodge theory on symplectic manifolds Tseng-Yau

13 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 13/14 Further reading T. Akahori, P.M. Garfield, and J.M. Lee, Deformation theory of 5-dimensional CR-structures and the Rumin complex, Michigan Math. Jour. 50 (2002) D.N. Arnold, R.S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15 (2006) D.N. Arnold, R.S. Falk, and R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. 47 (2010) R.L. Bryant, M.G. Eastwood, A.R. Gover, K. Neusser, Some differential complexes within and beyond parabolic geometry, arxiv: A. Čap and J. Slovák, Parabolic Geometries I, Amer. Math. Soc M.G. Eastwood, Extensions of the coeffective complex, arxiv: M.G. Eastwood and H. Goldschmidt, Zero-energy fields on complex projective space, arxiv: L.-S. Tseng and S.-T. Yau, Cohomology and Hodge theory on symplectic manifolds: I and II, arxiv: and arxiv:

14 Geometric and Nonlinear Partial Differential Equations, Xi An Jiaotong University p. 14/14 THANK YOU HAPPY BIRTHDAY NEIL!