Contact and Symplectic Geometry of Monge-Ampère Equations: Introduction and Examples

Contact and Symplectic Geometry of Monge-Ampère Equations: Introduction and Examples Vladimir Rubtsov, ITEP,Moscow and LAREMA, Université d Angers Workshop "Geometry and Fluids" Clay Mathematical Institute, Oxford University UK, April 07, 2014

Plan Introduction Effective forms and Monge-Ampère operators Symplectic Transformations of MAO Solutions of symplectic MAE Monge-Ampère and Geometric Structures Classification of SMAE on R 2 SMAE in 3D

Bibliography: Figure: Cambridge University Press, 2007

Basic object Figure: Monge and Ampère A 2 f q 2 1 + 2B 2 f q 1 q 2 + C 2 f q 2 2 + D ( 2 f q 2 1 2 f q 2 2 f ( ) 2) + E = 0 q 1 q 2

Global Solutions: Monge Figure: sphere and pseudosphere An example: curvature of a surface in R 3 u q1 q 1 u q2 q 2 u 2 q 1 q 2 (1+u 2 q 1 + u 2 q 2 ) 2 = K(u)

Main idea Let F : R n (i)r n be a vector-function and its graph is a subspace in T (R n ) = R n (i)r n.

Main idea Let F : R n (i)r n be a vector-function and its graph is a subspace in T (R n ) = R n (i)r n. The tangent space to the graph at the point (x,f(x)) is the graph of (df) x - the differential of F at the point x. This graph is a Lagrangian subspace in T (R n ) iff (df) x is a symmetric endomorphism. The matrix F i x j is symmetric x iff the differential form i F idx i Λ 1 (R n ) is closed or, equivalently, exact: F i = f x i = F = f.

Symplectic Linear Algebra Digression Let (V,Ω) be a symplectic 2n-dimensional vector space over R and Λ (V ) the space of exteriors forms on V. Let Γ : V V be the isomorphism determined by Ω and let X Ω Λ 2 (V) be the unique bivector such that Γ (X Ω ) = Ω. We introduce the operators : Λ k (V ) Λ k+2 (V ), ω ω Ω and : Λ k (V ) Λ k 2 (V ), ω i XΩ (ω). They have the followings properties: [, ](ω) = (n k)ω, ω Λ k (V ); : Λ k (V ) Λ k 2 (V ) is into for k n+1; : Λ k (V ) Λ k+2 (V ) is into for k n 1.

Hodge-Lepage-Lychagin theorem Figure: Hodge and Lychagin The next theorem plays the fundamental role played by the effective forms in the theory of Monge-Ampère operators : Theorem (Hodge-Lepage-Lychagin) Every form ω Λ k (V ) can be uniquely decomposed into the finite sum ω = ω 0 + ω 1 + 2 ω 2 +..., where all ω i are effective forms.

MAO definition Let M be an n-dimensional smooth manifold. Denote by J 1 M the space of 1-jets of smooth functions on M and by j 1 (f) : M J 1 M, x [f] 1 x the natural section associated with a smooth function f on M. The Monge-Ampère operator ω : C (M) Ω n (M) associated with a differential n-form ω Ω n (J 1 M) is the differential operator ω (f) = j 1 (f) (ω).

Contact Structure A contact structures is some analogue of symplectic structure on odd-dimensional manifold. A differential 1-form U on a smooth manifold M is called nondegenerate if the following conditions hold: The map P : a M ker U a T a M is a 1-codimensional distribution; A restriction of du to the hyperplane P(a) is a symplectic structure on P(a) for any a M. The last condition means that if a vector X a P(a) and X a ( du P(a) ) = 0, then Xa = 0. In other words, a differential 1-form U is nondegenerate if the distribution P has no characteristic symmetries.

Example: Cartan form The Cartan form U = du pdx on R 3 is a nondegenerate 1-form.

Example: Cartan form The Cartan form U = du pdx on R 3 is a nondegenerate 1-form. Figure: Contact structure in R 3

Generalized Solutions Let U be the contact 1-form on J 1 M and X 1 be the Reeb s vector field. Denote by C(x) the kernel of U x for x J 1 M.

Generalized Solutions Let U be the contact 1-form on J 1 M and X 1 be the Reeb s vector field. Denote by C(x) the kernel of U x for x J 1 M. (C(x),dU x ) is a 2n-dimensional symplectic vector space and T x J 1 M = C(x) RX 1x. A generalized solution of the equation ω = 0 is a legendrian submanifold L n of (J 1 M,U) such that ω L = 0. Note that T x L is a lagrangian subspace of (C(x),dU x ) in each point x L, and that L is locally the graph of a section j 1 (f), where f is a regular solution of the equation ω (f) = 0, if and only if the projection π : J 1 M M is a local diffeomorphism on L.

Effective Forms - MAO Correspondence Denote by Ω (C ) the space of differential forms vanishing on X 1. In each point x, (Ω k (C )) x can be naturally identified with Λ k (C(x) ).

Effective Forms - MAO Correspondence Denote by Ω (C ) the space of differential forms vanishing on X 1. In each point x, (Ω k (C )) x can be naturally identified with Λ k (C(x) ). Let Ω ε (C ) be the space of forms which are effective on (C(x),dU x ) in each point x J 1 M. The first part of the Hodge-Lepage-Lychagin theorem means that Ω ε(c ) = Ω (J 1 M)/I C, where I C is the Cartan ideal generated by U and du.

Contact Groupe Action Ct(M), the pseudo-group of contact diffeomorphisms on J 1 M, naturally acts on the set of Monge-Ampère operators in the following way

Contact Groupe Action Ct(M), the pseudo-group of contact diffeomorphisms on J 1 M, naturally acts on the set of Monge-Ampère operators in the following way F( ω ) = F (ω), and the corresponding infinitesimal action is X( ω ) = LX (ω).

Symplectic MAO-1 We are interested in particular in a more restrictive class of operators, the class of symplectic operators. These operators satisfy X 1 ( ω ) = LX1 (ω) = 0.

Symplectic MAO-1 We are interested in particular in a more restrictive class of operators, the class of symplectic operators. These operators satisfy X 1 ( ω ) = LX1 (ω) = 0. Let T M be the cotangent space and Ω be the canonical symplectic form on it. Let us consider the projection β : J 1 M T M, defined by the following commutative diagram:

Symplectic MAO-2 We can naturally identify the space {ω Ω ε(c ) : L X1 ω = 0} with the space of effective forms on (T M,Ω) using this projection β. Then, the group acting on these forms is the group of symplectomorphisms of T M. Definition A Monge-Ampère structure on a 2n-dimensional manifold X is a pair of differential form (Ω,ω) Ω 2 (X) Ω n (X) such that Ω is symplectic and ω is Ω-effective i.e. Ω ω = 0. When we locally identify the symplectic manifold (X,Ω) with (T R n,ω 0 ), we can associate to the pair (Ω,ω) a symplectic Monge-Ampère equation ω = 0.

Correspondence: Forms -Symplectic MAO Let M be a smooth n dimensional manifold and ω is a differential n-form on T M. A (symplectic) Monge-Ampère operator ω : C (M) Ω n (M) is the differential operator defined by ω (f) = (df) (ω), where df : M T M is the natural section associated to f.

Examples ω ω = 0 dq 1 dp 2 dq 2 dp 1 f = 0 dq 1 dp 2 + dq 2 dp 1 f = 0 dp 1 dp 2 dp 3 dq 1 dq 2 dq 3 Hess(f) = 1 dp 1 dq 2 dq 3 dp 2 dq 1 dq 3 f Hess(f) = 0 +dp 3 dq 1 dq 2 dp 1 dp 2 dp 3

Generic types of singularities for Generalized solutions of MAE Figure: Lagrangian singularities (Wave fronts, foldings etc.)

Symplectic Monge-Ampère Equations: Solutions A generalized solution of a MAE ω = 0 is a lagrangian submanifold of (T M,Ω) which is an integral manifold for the MA differential form ω: ω L = 0.

Symplectic Monge-Ampère Equations: Solutions A generalized solution of a MAE ω = 0 is a lagrangian submanifold of (T M,Ω) which is an integral manifold for the MA differential form ω: ω L = 0. A generalized solution (generically) locally is the graph of an 1-forme df for a regular solution f.

Generalized solution T* R n df L dg π dh R n Figure: Generalized solution of a MAE

Symplectic Equivalence-1 Two SMAE ω1 = 0 and ω2 = 0 are locally equivalent iff there is exist a local symplectomorphism F : (T M,Ω) (T M,Ω) such that F ω 1 = ω 2.

Symplectic Equivalence-1 Two SMAE ω1 = 0 and ω2 = 0 are locally equivalent iff there is exist a local symplectomorphism F : (T M,Ω) (T M,Ω) such that F ω 1 = ω 2. L is a generalized solution of F ω 1 = 0 iff F(L) is a generalized solution of ω = 0.

Legendre partial transformation Figure: Legendre u q1 q 1 + u q2 q 2 = 0 v q1 q 1 v q2 q 2 v 2 q 1 q 2 = 1 Φ ω = dq 1 dp 2 dq 2 dp 1 ω = dp 1 dp 2 dq 1 dq 2

Legendre partial transformation-2 L u = ( q 1,q 2,u q1,u q2 ) Φ L v = ( q 1, q 2,v q1,v q2 ) = ( q 1, u q2,u q1,q 2 ) e q 1 cos(q 2 ) q 2 arcsin(q 2 e q 1 ) + e 2q 1 q 2 2 with Φ : T R 2 T R 2, (q 1,q 2,p 1,p 2 ) (q 1, p 2,p 1,q 2 ).

Table 1. ω = 0 ω pf(ω) f = 0 dq 1 dp 2 dq 2 dp 1 1 f = 0 dq 1 dp 2 + dq 2 dp 1 1 2 f q 2 1 = 0 dq 1 dp 2 0

Geometric Structures on T R 2. Let (Ω,ω) be a Monge-Ampère structure on X = R 4. The field of endomorphisms A ω : X TX T X is defined by REMARK The tensor ω(, ) = Ω(A ω, ). J ω = A ω pf(ω) gives an almost-complex structure on X if pf(ω) > 0.

THEOREM (Lychagin-R.) Let ω Ω 2 ε (R4 ) be an effective non-degenerate 2-form on (R 4,Ω). The following assertions are equivalent:

THEOREM (Lychagin-R.) Let ω Ω 2 ε (R4 ) be an effective non-degenerate 2-form on (R 4,Ω). The following assertions are equivalent: The equation ω = 0 is locally equivalent to one of two linear equations: f = 0 ou f = 0;

Courant Bracket T tangent bundle of M and T cotangent bundle. (X +ξ,y +η) = 1 2 (ξ(y)+η(x)), -natural indefinite interior product on T T. The Courant bracket on sections of T T is [X +ξ,y +η] = [X,Y]+L X η L Y ξ 1 2 d(ι Xη ι Y ξ).

Generalized Complex Geometry Figure: Hitchin DEFINITION [Hitchin]: An almost generalized complex structure is a bundle map J : T T T T with and J 2 = 1, (J, ) = (,J ). An almost generalized complex structure is integrable if the spaces of sections of its two eigenspaces are closed under the Courant bracket.

2D SMAE and Generalized Complex Geometry DEFINITION A Monge-Ampère equation ω = 0 has a divergent type if the corresponded form can be chosen closed : ω = ω +µω.

2D SMAE and Generalized Complex Geometry DEFINITION A Monge-Ampère equation ω = 0 has a divergent type if the corresponded form can be chosen closed : ω = ω +µω. PROPOSITION (B.Banos) Let ω = 0 be a Monge-Ampère divergent type equation on R 2 with closed ω (which might be non-effective). The generalized almost-complex structure defined by ( A J ω = ω Ω 1 ) Ω(1+A 2 ω, ) A ω is integrable.

Hitchin pairs (after M.Crainic) A Hitchin pair is a pair of bivectors π and Π, Π non-degenerate, satisfying { [Π,Π] = [π,π] (1) [Π,π] = 0. PROPOSITION There is a 1-1 correspondence between Generalized complex structure ( ) A πa J = σ A with σ non degenerate and Hitchin pairs of bivector (π,π). In this correspondence σ = Π 1 A = π Π 1 π A = (1+A 2 )Π

Hitchin pair of bivectors in 4D Π is non-degenerate two 2-forms ω and Ω, not necessarily closed and ω(, ) = Ω(A, ). A generalized lagrangian surface: closed under A, or equivalently, bilagrangian: ω L = Ω L = 0. Locally, L is defined by two functions u and v satisfying a first order system:

Jacobi systems { a+b u x + c u y + d v x + e v y + f det J u,v = 0 A+B u x + C u y + D v x + E v y + E det J u,v = 0 ) J u,v = ( u x v x u y v y Such a system generalizes both MAE and Cauchy-Riemann systems and is called a Jacobi system.

Invariants for effective 3-forms To each effective 3-form ω Ω 3 ε (R6 ), we assign the following geometric invariants:

Invariants for effective 3-forms To each effective 3-form ω Ω 3 ε (R6 ), we assign the following geometric invariants: the Lychagin-R. metric defined by g ω (X,Y) = (ι Xω) (ι Y ω) Ω Ω 3,

Invariants for effective 3-forms To each effective 3-form ω Ω 3 ε (R6 ), we assign the following geometric invariants: the Lychagin-R. metric defined by the Hitchin tensor defined by g ω (X,Y) = (ι Xω) (ι Y ω) Ω Ω 3, g ω = Ω(A ω, ),

Invariants for effective 3-forms To each effective 3-form ω Ω 3 ε (R6 ), we assign the following geometric invariants: the Lychagin-R. metric defined by the Hitchin tensor defined by The Hitchin pfaffian defined by g ω (X,Y) = (ι Xω) (ι Y ω) Ω Ω 3, g ω = Ω(A ω, ), pf(ω) = 1 6 tra2 ω.

(0,3) 0 (2,1) 0 ω = 0 ω ε(q ω ) pf(ω) 1 hess(f) = 1 dq 1 dq 2 dq 3 +ν dp 1 dp 2 dp 3 (3,3) ν 4 2 f hess(f) = 0 dp 1 dq 2 dq 3 dp 2 dq 1 dq 3 (0,6) ν 4 +dp 3 dq 1 dq 2 ν 2 dp 1 dp 2 dp 3 3 f + hess(f) = 0 dp 1 dq 2 dq 3 + dp 2 dq 1 dq 3 (4,2) ν 4 +dp 3 dq 1 dq 2 +ν 2 dp 1 dp 2 dp 3 4 f = 0 dp 1 dq 2 dq 3 dp 2 dq 1 dq 3 + dp 3 dq 1 dq 2 5 f = 0 dp 1 dq 2 dq 3 +dp 2 dq 1 dq 3 + dp 3 dq 1 dq 2 6 q2,q 3 f = 0 dp 3 dq 1 dq 2 dp 2 dq 1 dq 3 (0,1) 0 7 q2,q 3 f = 0 dp 3 dq 1 dq 2 + dp 2 dq 1 dq 3 (1,0) 0 8 2 f q 2 1 = 0 dp 1 dq 2 dq 3 (0,0) 0 9 0 (0, 0) 0 Table: Classification of effective 3-formes in dimension 6

3D Generalized Calabi-Yau structures A generalized almost Calabi-Yau structure on a 6D-manifold X is a 5-uple (g,ω,a,α,β) where g is a (pseudo) metric on X,

3D Generalized Calabi-Yau structures A generalized almost Calabi-Yau structure on a 6D-manifold X is a 5-uple (g,ω,a,α,β) where g is a (pseudo) metric on X, Ω is a symplectic on X, A is a smooth section X TX T X such that A 2 = ±Id and such that g(u, V) = Ω(AU, V) for all tangent vectors U, V, A generalized Calabi-Yau structure (g, Ω, K, α, β) is integrable if α and β are closed.

3D Generalized Calabi-Yau structures A generalized almost Calabi-Yau structure on a 6D-manifold X is a 5-uple (g,ω,a,α,β) where g is a (pseudo) metric on X, Ω is a symplectic on X, A is a smooth section X TX T X such that A 2 = ±Id and such that g(u, V) = Ω(AU, V) for all tangent vectors U, V, α and β are (eventually complex) decomposable 3-forms whose associated distributions are the distributions of A eigenvectors and such that α β Ω 3 is constant. A generalized Calabi-Yau structure (g, Ω, K, α, β) is integrable if α and β are closed.

Generalized CY and MA Each nondegenerate Monge-Ampère structure (Ω,ω 0 ) defines a generalized almost Calabi-Yau structure (q ω,ω,a ω,α,β) with ω 0 ω = λ(ω0 ). 4

The generalized Calabi-Yau structure associated with the equation (f) hess(f) = 0 is the canonical Calabi-Yau structure of C 3 g = 3 dx j.dx j + dy j.dy j j=1 A = 3 y j dx j x j dy j j=1 Ω = 3 dx j dy j j=1 α = dz 1 dz 2 dz 3 β = α

The generalized Calabi-Yau associated with the equation (f)+hess(f) = 0 is the pseudo Calabi-Yau structure g = dx 1.dx 1 dx 2.dx 2 + dx 3.dx 3 + dy 1.dy 1 dy 2.dy 2 + dx 3.dx 3 A = x 1 dy 1 y 1 dx 1 + y 2 dx 2 x 2 dy 2 y 3 dx 3 + x 3 dy 3 Ω = 3 dx j dy j j=1 α = dz 1 dz 2 dz 3 β = α

The generalized Calabi-Yau structure associated with the equation is the real Calabi-Yau structure g = 3 dx j.dy j j=1 hess(f) = 1 A = 3 x j dx j y j dy j j=1 Ω = 3 dx j dy j j=1 α = dx 1 dx 2 dx 3 β = dy 1 dy 2 dy 3

THEOREM A SMAE ω = 0 on R 3 associated to an effective non-degenerated form ω is locally equivalent to on of three following equations:

THEOREM A SMAE ω = 0 on R 3 associated to an effective non-degenerated form ω is locally equivalent to on of three following equations: hess(f) = 1 f hess(f) = 0 f + hess(f) = 0

THEOREM A SMAE ω = 0 on R 3 associated to an effective non-degenerated form ω is locally equivalent to on of three following equations: hess(f) = 1 f hess(f) = 0 f + hess(f) = 0 iff the correspondingly defined generalized Calabi-Yau structure is integrable and locally flat.