contact and symplectic geometry of monge-ampère … · contact and symplectic geometry of...

Contact and Symplectic Geometry of

Monge-Ampere Equations: Introduction

and Examples

Vladimir Rubtsov,ITEP,Moscow and LAREMA, Universite d’Angers

Workshop "Geometry and Fluids"Clay Mathematical Institute,

Oxford University UK, April 07, 2014

Plan

Introduction

Effective forms and Monge-Ampere operators

Symplectic Transformations of MAOSolutions of symplectic MAE

Monge-Ampere and Geometric StructuresClassification of SMAE on R2

SMAE in 3D

Bibliography:

Figure: Cambridge University Press, 2007

Basic object

Figure: Monge and Ampere

A∂2f

∂q21

+ 2B∂2f

∂q1∂q2

+ C∂2f

∂q22

+ D(∂2f

∂q21

· ∂2f

∂q22

− (∂f

∂q1∂q2

)2)

+ E = 0

Global Solutions: Monge

Figure: sphere and pseudosphere

An example: curvature of a surface in R3

uq1q1· uq2q2

− u2q1q2

(1 + u2q1

+ u2q2)2

= K(u)

Main idea

Let F : Rn → (i)Rn be a vector-function and its graph is asubspace in T ∗(Rn) = Rn ⊕ (i)Rn.

Main idea


The tangent space to the graph at the point (x ,F (x)) is thegraph of (dF )x - the differential of F at the point x .

Main idea



This graph is a Lagrangian subspace in T ∗(Rn) iff (dF )x is asymmetric endomorphism . The matrix || ∂Fi

∂xj|| is symmetric

∀x iff the differential form∑

i Fidxi ∈ Λ1(Rn) is closed or,equivalently, exact:

Fi =∂f

∂xi=⇒ F = ∇f .

Main idea



This graph is a Lagrangian subspace in T ∗(Rn) iff (dF )x is asymmetric endomorphism . The matrix || ∂Fi

∂xj|| is symmetric

∀x iff the differential form∑

i Fidxi ∈ Λ1(Rn) is closed or,equivalently, exact:

Fi =∂f

∂xi=⇒ F = ∇f .

The projection of the graph of ∇f on (Rn)x is given in

coordinates by ∇2(f ) = det || ∂2fi∂x2

j

|| .

Symplectic Linear Algebra Digression

Let (V ,Ω) be a symplectic 2n-dimensional vector space over Rand Λ∗(V ∗) the space of exteriors forms on V . LetΓ : V → V ∗ be the isomorphism determined by Ω and letXΩ ∈ Λ2(V ) be the unique bivector such that Γ∗(XΩ) = Ω.



We introduce the operators ⊤ : Λk(V ∗) → Λk+2(V ∗),ω 7→ ω ∧ Ω and ⊥ : Λk(V ∗) → Λk−2(V ∗), ω 7→ iXΩ

(ω). Theyhave the followings properties:





[⊥,⊤](ω) = (n − k)ω , ∀ω ∈ Λk(V ∗);

⊥ : Λk(V ∗) → Λk−2(V ∗) is into for k ≥ n + 1;

⊤ : Λk(V ∗) → Λk+2(V ∗) is into for k ≤ n − 1.





[⊥,⊤](ω) = (n − k)ω , ∀ω ∈ Λk(V ∗);

⊥ : Λk(V ∗) → Λk−2(V ∗) is into for k ≥ n + 1;

⊤ : Λk(V ∗) → Λk+2(V ∗) is into for k ≤ n − 1.

A k-form ω is effective if ⊥ω = 0 and we will denote byΛkε (V

∗) the vector space of effective k-forms on V . Whenk = n, ω is effective if and only if ω ∧ Ω = 0.

Hodge-Lepage-Lychagin theorem

Figure: Hodge and Lychagin

The next theorem plays the fundamental role played by the effectiveforms in the theory of Monge-Ampere operators :

Theorem (Hodge-Lepage-Lychagin)

Every form ω ∈ Λk(V ∗) can be uniquely decomposed into the

finite sum

ω = ω0 +⊤ω1 +⊤2ω2 + . . . ,

where all ωi are effective forms.

Hodge-Lepage-Lychagin theorem

Figure: Hodge and Lychagin

The next theorem plays the fundamental role played by the effectiveforms in the theory of Monge-Ampere operators :

Theorem (Hodge-Lepage-Lychagin)

Every form ω ∈ Λk(V ∗) can be uniquely decomposed into the

finite sum

ω = ω0 +⊤ω1 +⊤2ω2 + . . . ,

where all ωi are effective forms.

If two effective k-forms vanish on the same k-dimensional

isotropic vector subspaces in (V ,Ω), they are proportional.

MAO definition

Let M be an n-dimensional smooth manifold. Denote by J1M

the space of 1-jets of smooth functions on M and byj1(f ) : M → J1M, x 7→ [f ]1x the natural section associatedwith a smooth function f on M.

MAO definition

Let M be an n-dimensional smooth manifold. Denote by J1M

the space of 1-jets of smooth functions on M and byj1(f ) : M → J1M, x 7→ [f ]1x the natural section associatedwith a smooth function f on M.

The Monge-Ampere operator

∆ω : C∞(M) → Ωn(M)

associated with a differential n-form ω ∈ Ωn(J1M) is thedifferential operator

∆ω(f ) = j1(f )∗(ω).

Contact Structure

A contact structures is some analogue of symplectic structure onodd-dimensional manifold. A differential 1-form U on a smoothmanifold M is called nondegenerate if the following conditionshold:

The map P : a ∋ M 7→ ker Ua ⊂ TaM is a 1-codimensionaldistribution;

Contact Structure



A restriction of dU to the hyperplane P(a) is a symplecticstructure on P(a) for any a ∈ M .

Contact Structure



A restriction of dU to the hyperplane P(a) is a symplecticstructure on P(a) for any a ∈ M .

The last condition means that if a vector Xa ∈ P(a) andXa⌋

(

dU|P(a)

)

= 0, then Xa = 0. In other words, a differential1-form U is nondegenerate if the distribution P has nocharacteristic symmetries.

Example: Cartan form

The Cartan form U = du − pdx on R3 is a nondegenerate1-form.

Example: Cartan form

The Cartan form U = du − pdx on R3 is a nondegenerate1-form.

Figure: Contact structure in R3

Generalized Solutions

Let U be the contact 1-form on J1M and X1 be the Reeb’svector field. Denote by C (x) the kernel of Ux for x ∈ J1M.



(C (x), dUx ) is a 2n-dimensional symplectic vector space and

TxJ1M = C (x)⊕ RX1x .



(C (x), dUx ) is a 2n-dimensional symplectic vector space and

TxJ1M = C (x)⊕ RX1x .

A generalized solution of the equation ∆ω = 0 is a legendriansubmanifold Ln of (J1M,U) such that ω|L = 0. Note that TxL

is a lagrangian subspace of (C (x), dUx ) in each point x ∈ L,and that L is locally the graph of a section j1(f ), where f is aregular solution of the equation ∆ω(f ) = 0, if and only if theprojection π : J1M → M is a local diffeomorphism on L.

Effective Forms - MAO Correspondence

Denote by Ω∗(C ∗) the space of differential forms vanishing onX1. In each point x , (Ωk(C ∗))x can be naturally identifiedwith Λk(C (x)∗).



Let Ω∗ε(C

∗) be the space of forms which are effective on(C (x), dUx ) in each point x ∈ J1M.



Let Ω∗ε(C


The first part of the Hodge-Lepage-Lychagin theorem meansthat

Ω∗ε(C

∗) = Ω∗(J1M)/IC ,

where IC is the Cartan ideal generated by U and dU.



Let Ω∗ε(C


The first part of the Hodge-Lepage-Lychagin theorem meansthat

Ω∗ε(C

∗) = Ω∗(J1M)/IC ,

where IC is the Cartan ideal generated by U and dU.

The second part means that two differential n-forms ω and θon J1M determine the same Monge-Ampere operator if andonly if ω − θ ∈ IC .

Contact Groupe Action

Ct(M), the pseudo-group of contact diffeomorphisms on J1M,naturally acts on the set of Monge-Ampere operators in thefollowing way

Contact Groupe Action

Ct(M), the pseudo-group of contact diffeomorphisms on J1M,naturally acts on the set of Monge-Ampere operators in thefollowing 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 ofoperators, the class of symplectic operators. These operatorssatisfy

X1(∆ω) = ∆LX1(ω) = 0.

Symplectic MAO-1


X1(∆ω) = ∆LX1(ω) = 0.

Let T ∗M be the cotangent space and Ω be the canonicalsymplectic form on it. Let us consider the projectionβ : J1M → T ∗M, defined by the following commutativediagram:

Symplectic MAO-1


X1(∆ω) = ∆LX1(ω) = 0.

Let T ∗M be the cotangent space and Ω be the canonicalsymplectic form on it. Let us consider the projectionβ : J1M → T ∗M, defined by the following commutativediagram:

R J1Mαoo β // T ∗M

M

f

``BBBBBBBBBBBBBBBBB

j1(f )

OO

df

<<zzzzzzzzzzzzzzzzzz

Symplectic MAO-2

We can naturally identify the space ω ∈ Ω∗ε(C

∗) : LX1ω = 0

with the space of effective forms on (T ∗M,Ω) using thisprojection β. Then, the group acting on these forms is thegroup of symplectomorphisms of T ∗M.

Symplectic MAO-2


∗) : LX1ω = 0


DefinitionA Monge-Ampere structure on a 2n-dimensional manifold X is a

pair of differential form (Ω, ω) ∈ Ω2(X )× Ωn(X ) such that Ω is

symplectic and ω is Ω-effective i.e. Ω ∧ ω = 0.

Symplectic MAO-2


∗) : LX1ω = 0





When we locally identify the symplectic manifold (X ,Ω) with(T ∗Rn,Ω0), we can associate to the pair (Ω, ω) a symplecticMonge-Ampere equation ∆ω = 0.

Symplectic MAO-2


∗) : LX1ω = 0





When we locally identify the symplectic manifold (X ,Ω) with(T ∗Rn,Ω0), we can associate to the pair (Ω, ω) a symplecticMonge-Ampere equation ∆ω = 0.

Conversely, any symplectic Monge-Ampere equation ∆ω = 0on a manifold M is associated with Monge-Ampere structure(Ω, ω) on T ∗M.

Correspondence: Forms -Symplectic MAO

Let M be a smooth n−dimensional manifold and ω is a differentialn-form on T ∗M. A (symplectic) Monge-Ampere operator∆ω : C∞(M) → Ωn(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

dq1 ∧ dp2 − dq2 ∧ dp1 ∆f = 0

dq1 ∧ dp2 + dq2 ∧ dp1 f = 0

dp1 ∧ dp2 ∧ dp3 − dq1 ∧ dq2 ∧ dq3 Hess(f ) = 1

dp1 ∧ dq2 ∧ dq3 − dp2 ∧ dq1 ∧ dq3 ∆f − Hess(f ) = 0+dp3 ∧ dq1 ∧ dq2 − dp1 ∧ dp2 ∧ dp3

Generic types of singularities for Generalized solutions ofMAE

Figure: Lagrangian singularities (Wave fronts, foldings etc.)

Symplectic Monge-Ampere Equations: Solutions

A generalized solution of a MAE ∆ω = 0 is a lagrangiansubmanifold of (T ∗M,Ω) which is an integral manifold for theMA differential form ω:

ω|L = 0.

Symplectic Monge-Ampere Equations: Solutions

A generalized solution of a MAE ∆ω = 0 is a lagrangiansubmanifold of (T ∗M,Ω) which is an integral manifold for theMA differential form ω:

ω|L = 0.

A generalized solution (generically) locally is the graph of an1-forme df for a regular solution f .

Generalized solution

Rn

T Rn

π

L

*

df

dg

dh

Figure: Generalized solution of a MAE

Symplectic Equivalence-1

Two SMAE ∆ω1= 0 and ∆ω2

= 0 are locally equivalent iffthere is exist a local symplectomorphismF : (T ∗M,Ω) → (T ∗M,Ω) such that

F ∗ω1 = ω2.

Symplectic Equivalence-1

Two SMAE ∆ω1= 0 and ∆ω2

= 0 are locally equivalent iffthere is exist a local symplectomorphismF : (T ∗M,Ω) → (T ∗M,Ω) such that

F ∗ω1 = ω2.

L is a generalized solution of ∆F∗ω1= 0 iff F (L) is a

generalized solution of ∆ω = 0.

Legendre partial transformation

Figure: Legendre

uq1q1+ uq2q2

= 0 oo //_________

vq1q1vq2q2

− v2q1q2

= 1

ω = dq1 ∧ dp2 − dq2 ∧ dp1 ω = dp1 ∧ dp2 − dq1 ∧ dq2

Φ∗oo

Legendre partial transformation-2

Lu =(

q1, q2, uq1, uq2

) Φ //

Lv =(

q1, q2, vq1, vq2

)

=(

q1,−uq2, uq1

, q2

)

eq1 cos(q2) oo //_________q2 arcsin(q2e

−q1)

+√

e2q1 − q22

with Φ : T ∗R2 → T ∗R2, (q1, q2, p1, p2) 7→ (q1,−p2, p1, q2).

Table 1.

∆ω = 0 ω pf (ω)

∆f = 0 dq1 ∧ dp2 − dq2 ∧ dp1 1

f = 0 dq1 ∧ dp2 + dq2 ∧ dp1 −1∂2f∂q2

1

= 0 dq1 ∧ dp2 0

Geometric Structures on T∗R2.

Let (Ω, ω) be a Monge-Ampere structure on X = R4. The field ofendomorphisms Aω : X → TX ⊗ T ∗X is defined by

ω(·, ·) = Ω(Aω·, ·).

REMARK The tensor

Jω =Aω

√

|pf (ω)|

gives

an almost-complex structure on X if pf (ω) > 0.

Geometric Structures on T∗R2.

Let (Ω, ω) be a Monge-Ampere structure on X = R4. The field ofendomorphisms Aω : X → TX ⊗ T ∗X is defined by

ω(·, ·) = Ω(Aω·, ·).

REMARK The tensor

Jω =Aω

√

|pf (ω)|

gives

an almost-complex structure on X if pf (ω) > 0.

an almost-product structure on X if pf (ω) < 0.

THEOREM (Lychagin-R.)

Let ω ∈ Ω2ε(R

4) be an effective non-degenerate 2-form on (R4,Ω).

The following assertions are equivalent:


Let ω ∈ Ω2ε(R



The equation ∆ω = 0 is locally equivalent to one of two linearequations: ∆f = 0 ou f = 0;


Let ω ∈ Ω2ε(R




The tensor Jω is integrable;


Let ω ∈ Ω2ε(R




The tensor Jω is integrable;

the normalized form ω√|pf (ω)|

is closed.

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 ] + LXη − LY ξ − 1

2d(ιXη − ιY ξ).

Generalized Complex Geometry

Figure: Hitchin

DEFINITION [Hitchin]: An almost generalized complex structureis a bundle map J : T ⊕ T ∗ → T ⊕ T ∗ with

J2 = −1,

and(J·, ·) = −(·, J·).

An almost generalized complex structure is integrable if the spacesof sections of its two eigenspaces are closed under the Courantbracket.

2D SMAE and Generalized Complex Geometry

DEFINITION A Monge-Ampere equation ∆ω = 0 has adivergent type if the corresponded form can be chosen closed :ω′ = ω + µΩ.

2D SMAE and Generalized Complex Geometry

DEFINITION A Monge-Ampere equation ∆ω = 0 has adivergent type if the corresponded form can be chosen closed :ω′ = ω + µΩ.

PROPOSITION (B.Banos)Let ∆ω = 0 be a Monge-Ampere divergent type equation onR2 with closed ω (which might be non-effective). Thegeneralized almost-complex structure defined by

Jω =

(

Aω Ω−1

−Ω(1 + A2ω·, ·) −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]

[Π, π] = 0.(1)

PROPOSITION There is a 1-1 correspondence betweenGeneralized complex structure

J =

(

A πA

σ −A∗

)

with σ non degenerate and Hitchin pairs of bivector (π,Π). In thiscorrespondence

σ = Π−1

A = π Π−1

πA = −(1 + A2)Π

Hitchin pair of bivectors in 4D

Π is non-degenerate ⇒ two 2-forms ω and Ω, not necessarily closedand 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 ordersystem:

Jacobi systems

a + b ∂u∂x

+ c ∂u∂y

+ d ∂v∂x

+ e ∂v∂y

+ f det Ju,v = 0

A + B ∂u∂x

+ C ∂u∂y

+ D ∂v∂x

+ E ∂v∂y

+ E det Ju,v = 0

Ju,v =

(

∂u∂x

∂u∂y

∂v∂x

∂v∂y

)

Such a system generalizes both MAE and Cauchy-Riemann systemsand is called a Jacobi system.

Invariants for effective 3-forms

To each effective 3-form ω ∈ Ω3ε(R

6), we assign the followinggeometric invariants:




the Lychagin-R. metric defined by

gω(X ,Y ) =(ιXω) ∧ (ιYω) ∧ Ω

Ω3,





gω(X ,Y ) =(ιXω) ∧ (ιYω) ∧ Ω

Ω3,

the Hitchin tensor defined by

gω = Ω(Aω·, ·),





gω(X ,Y ) =(ιXω) ∧ (ιYω) ∧ Ω

Ω3,

the Hitchin tensor defined by

gω = Ω(Aω·, ·),

The Hitchin pfaffian defined by

pf (ω) =1

6trA2

ω.

∆ω = 0 ω ε(qω) pf (ω)

1 hess(f ) = 1 dq1∧dq2∧dq3+ν ·dp1∧dp2∧dp3 (3, 3) ν4

2 ∆f − hess(f ) = 0 dp1 ∧ dq2 ∧ dq3 − dp2 ∧ dq1 ∧ dq3 (0, 6) −ν4

+dp3∧dq1∧dq2−ν2·dp1∧dp2∧dp3

3 f + hess(f ) = 0 dp1 ∧ dq2 ∧ dq3 + dp2 ∧ dq1 ∧ dq3 (4, 2) −ν4

+dp3∧dq1∧dq2+ν2·dp1∧dp2∧dp3

4 ∆f = 0 dp1∧dq2∧dq3−dp2∧dq1∧dq3+dp3 ∧ dq1 ∧ dq2

(0, 3) 0

5 f = 0 dp1∧dq2∧dq3+dp2∧dq1∧dq3+dp3 ∧ dq1 ∧ dq2

(2, 1) 0

6 ∆q2,q3f = 0 dp3 ∧ dq1 ∧ dq2 − dp2 ∧ dq1 ∧ dq3 (0, 1) 0

7 q2,q3f = 0 dp3 ∧ dq1 ∧ dq2 + dp2 ∧ dq1 ∧ dq3 (1, 0) 0

8 ∂2f∂q2

1

= 0 dp1 ∧ dq2 ∧ dq3 (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 a6D-manifold X is a 5-uple (g ,Ω,A, α, β) where

g is a (pseudo) metric on X ,



g is a (pseudo) metric on X , Ω is a symplectic on X ,

A generalized Calabi-Yau structure (g ,Ω,K , α, β) is integrableif α and β are closed.



g is a (pseudo) metric on X , Ω is a symplectic on X , A is a smooth section X → TX ⊗ T ∗X such that A2 = ±Id

and such thatg(U ,V ) = Ω(AU ,V )

for all tangent vectors U ,V ,




g is a (pseudo) metric on X , Ω is a symplectic on X , A is a smooth section X → TX ⊗ T ∗X such that A2 = ±Id

and such thatg(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 eigenvectorsand such that

α ∧ β

Ω3is constant.


Generalized CY and MA

Each nondegenerate Monge-Ampere structure (Ω, ω0) defines ageneralized almost Calabi-Yau structure (qω ,Ω,Aω, α, β) with

ω =ω0

4

√

|λ(ω0)|.

The generalized Calabi-Yau structure associated with the equation

∆(f )− hess(f ) = 0

is the canonical Calabi-Yau structure of C3

g = −3∑

j=1

dxj .dxj + dyj .dyj

A =3∑

j=1

∂∂yj

⊗ dxj − ∂∂xj

⊗ dyj

Ω =3∑

j=1

dxj ∧ dyj

α = dz1 ∧ dz2 ∧ dz3

β = α

The generalized Calabi-Yau associated with the equation

(f ) + hess(f ) = 0

is the pseudo Calabi-Yau structure

g = dx1.dx1 − dx2.dx2 + dx3.dx3 + dy1.dy1 − dy2.dy2 + dx3.dx3

A = ∂∂x1

⊗ dy1 − ∂∂y1

⊗ dx1 +∂∂y2

⊗ dx2 − ∂∂x2

⊗ dy2 − ∂∂y3

⊗ dx3

+ ∂∂x3

⊗ dy3

Ω =3∑

j=1

dxj ∧ dyj

α = dz1 ∧ dz2 ∧ dz3

β = α

The generalized Calabi-Yau structure associated with the equation

hess(f ) = 1

is the “real” Calabi-Yau structure

g =3∑

j=1

dxj .dyj

A =3∑

j=1

∂∂xj

⊗ dxj − ∂∂yj

⊗ dyj

Ω =3∑

j=1

dxj ∧ dyj

α = dx1 ∧ dx2 ∧ dx3

β = dy1 ∧ dy2 ∧ dy3

THEOREM A SMAE ∆ω = 0 on R3 associated to aneffective non-degenerated form ω is locally equivalent to on ofthree following equations:


hess(f ) = 1

∆f − hess(f ) = 0

f + hess(f ) = 0


hess(f ) = 1

∆f − hess(f ) = 0

f + hess(f ) = 0

iff the correspondingly defined generalized Calabi-Yau structureis integrable and locally flat.

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