Background material 3

Background material 3

The basic goal

We want an intersection theory for asymp- totically cylindrical holomorphic curves:

u : ˙Σ → W ,^c v : ˙Σ^′ → W^c

u Wc

Σ˙

Desired properties:

1. Homotopy-invariant suÆient onditions

for u and v to be disjoint or transverse

2. Homotopy-invariant suÆient onditions

for simple urves to be embedded

u W_c

Σ˙

Desired properties:

1. Homotopy-invariant sufficient conditions for u and v to be disjoint or transverse

2. Homotopy-invariant suÆient onditions

for simple urves to be embedded

u W_c

Σ˙

Desired properties:

1. Homotopy-invariant sufficient conditions for u and v to be disjoint or transverse 2. Homotopy-invariant sufficient conditions

for simple curves to be embedded

u W_c

Σ˙

It looks promising at first...

Whenever u( ˙Σ) 6= v( ˙Σ^′), we have u · v ≥ {(z, ζ) | u(z) = v(ζ)}

with equality iff u ⋔ v. ^Hene:

u v = 0 , u(

_

) \ v(

_

0

) = ;:

Similarly, if u is simple,

Æ(u)

1

2

f(z;) j u(z) = u(); z 6= g

with equality i u is immersed with all double

points transverse. Therefore:

Æ(u) = 0 , u is embedded:

The basi problem:

Neither u v nor Æ(u) is homotopy invariant!

It looks promising at first...

Whenever u( ˙Σ) 6= v( ˙Σ^′), we have u · v ≥ {(z, ζ) | u(z) = v(ζ)}

with equality iff u ⋔ v. Hence:

u · v = 0 ⇔ u( ˙Σ) ∩ v( ˙Σ^′) = ∅.

Similarly, if u is simple,

Æ(u)

1

2

f(z;) j u(z) = u(); z 6= g

with equality i u is immersed with all double

points transverse. Therefore:

Æ(u) = 0 , u is embedded:

The basi problem:

Neither u v nor Æ(u) is homotopy invariant!

It looks promising at first...

Whenever u( ˙Σ) 6= v( ˙Σ^′), we have u · v ≥ {(z, ζ) | u(z) = v(ζ)}

with equality iff u ⋔ v. Hence:

u · v = 0 ⇔ u( ˙Σ) ∩ v( ˙Σ^′) = ∅.

Similarly, if u is simple, δ(u) ≥ 1

2

{(z, ζ) | u(z) = u(ζ), z 6= ζ}

with equality iff u is immersed with all double points transverse. ^Therefore:

Æ(u) = 0 , u is embedded:

The basi problem:

Neither u v nor Æ(u) is homotopy invariant!

It looks promising at first...

Whenever u( ˙Σ) 6= v( ˙Σ^′), we have u · v ≥ {(z, ζ) | u(z) = v(ζ)}

with equality iff u ⋔ v. Hence:

u · v = 0 ⇔ u( ˙Σ) ∩ v( ˙Σ^′) = ∅.

Similarly, if u is simple, δ(u) ≥ 1

2

{(z, ζ) | u(z) = u(ζ), z 6= ζ}

with equality iff u is immersed with all double points transverse. Therefore:

δ(u) = 0 ⇔ u is embedded.

The basi problem:

Neither u v nor Æ(u) is homotopy invariant!

It looks promising at first...

Whenever u( ˙Σ) 6= v( ˙Σ^′), we have u · v ≥ {(z, ζ) | u(z) = v(ζ)}

with equality iff u ⋔ v. Hence:

u · v = 0 ⇔ u( ˙Σ) ∩ v( ˙Σ^′) = ∅.

Similarly, if u is simple, δ(u) ≥ 1

2

{(z, ζ) | u(z) = u(ζ), z 6= ζ}

with equality iff u is immersed with all double points transverse. Therefore:

δ(u) = 0 ⇔ u is embedded.

The basic problem:

Neither u · v nor δ(u) is homotopy invariant!

Disaster scenario:

Suppose u : ˙Σ → W^c has two ends approach- ing the same Reeb orbit...

Wc

[0, ∞) × M₊

(−∞,0] × M₋ δ(u) > 0

Intersetions an esape to innity!

Solution:

Understand asymptoti behaviour well.

Disaster scenario:

Suppose u : ˙Σ → W^c has two ends approach- ing the same Reeb orbit...

Wc

[0, ∞) × M₊

(−∞,0] × M₋ δ(u) > 0

Intersetions an esape to innity!

Solution:

Understand asymptoti behaviour well.

Disaster scenario:

Suppose u : ˙Σ → W^c has two ends approach- ing the same Reeb orbit...

Wc

[0, ∞) × M₊

(−∞,0] × M₋ δ(u) > 0

Intersetions an esape to innity!

Solution:

Understand asymptoti behaviour well.

Disaster scenario:

Suppose u : ˙Σ → W^c has two ends approach- ing the same Reeb orbit...

Wc

[0, ∞) × M₊

(−∞,0] × M₋ δ(u) > 0

Intersetions an esape to innity!

Solution:

Understand asymptoti behaviour well.

Disaster scenario:

Suppose u : ˙Σ → W^c has two ends approach- ing the same Reeb orbit...

Wc

[0, ∞) × M₊

(−∞,0] × M₋ δ(u) > 0

Intersetions an esape to innity!

Solution:

Understand asymptoti behaviour well.

Disaster scenario:

Suppose u : ˙Σ → W^c has two ends approach- ing the same Reeb orbit...

Wc

[0, ∞) × M₊

(−∞,0] × M₋ δ(u) = 0

Intersections can escape to infinity!

Solution:

Understand asymptoti behaviour well.

Disaster scenario:

Suppose u : ˙Σ → W^c has two ends approach- ing the same Reeb orbit...

Wc

[0, ∞) × M₊

(−∞,0] × M₋ δ(u) = 0

Intersections can escape to infinity! Solution:

Understand asymptotic behaviour well.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0

ind = 1 ind = 1

A

p

is symmetri, so its eigenvalues are real.

Analogy with Morse theory

In Morse homology, one studies gradient-flow lines

x : R → M, x˙ = ∇f(x)

of a function f : M → R on a Riemannian manifold (M, g), where we assume each p ∈ Crit(f) is nondegenerate, i.e. the Hessian

A^{p := ∇(∇f)(p) : TpM → TpM} has trivial kernel.

ind = 2

ind = 0

ind = 1 ind = 1

A^{p is} symmetric, so its eigenvalues are real.

Asymptotic formula for gradient-flow

Theorem

Assume p ∈ Crit(f), h(s) ∈ T_pM is defined for s close to ±∞ and

x(s) = exp_p h(s) ∈ M

is a gradient-flow line approaching p as s →

±∞. ^{Then h(s) satises the deay formula}

h(s) = e s

(v + r(s))

for some eigenvetor v 2 T

p

M of A

p

with

A

p

v = v; < 0;

and a funtion r(s) 2 T

p

M with

r(s) ! 0 as s ! 1.

\Flow lines approah ritial points along

asymptoti eigenvetors."

Asymptotic formula for gradient-flow

Theorem

Assume p ∈ Crit(f), h(s) ∈ T_pM is defined for s close to ±∞ and

x(s) = exp_p h(s) ∈ M

is a gradient-flow line approaching p as s →

±∞. Then h(s) satisfies the decay formula h(s) = e^λs (v + r(s))

for some eigenvector v ∈ T_pM of A^{p with} A^{pv = λv, ±λ < 0,}

and a function r(s) ∈ T_pM with r(s) → 0 as s → ±∞.

\Flow lines approah ritial points along

asymptoti eigenvetors."

Asymptotic formula for gradient-flow

Theorem

Assume p ∈ Crit(f), h(s) ∈ T_pM is defined for s close to ±∞ and

x(s) = exp_p h(s) ∈ M

is a gradient-flow line approaching p as s →

±∞. Then h(s) satisfies the decay formula h(s) = e^λs (v + r(s))

for some eigenvector v ∈ T_pM of A^{p with} A^{pv = λv, ±λ < 0,}

and a function r(s) ∈ T_pM with r(s) → 0 as s → ±∞.

“Flow lines approach critical points along asymptotic eigenvectors.”

“Holomorphic curves are gradient-flow lines”

Choose J ∈ J (α) on a symplectisation (R × M, d(e^sα)). ^{Then a} half-ylinder

u = (f;v) : [0;1) S 1

! R M

is J-holomorphi if and only if

s

f (

t

v) = 0;

t

f + (

s

v) = 0;

s

v + J

t

v = 0;

where

: TM ! is the projetion along R

.

Claim: the third equation an be interpreted

as the L 2

-gradient ow of the ontat a-

tion funtional

: C 1

(S 1

;M) ! R : 7!

Z

S 1

;

whose ritial points are Reeb orbits. Its Hes-

sian at a T-periodi Reeb orbit : S 1

! M is

the L 2

-symmetri operator

r(r

)() : (

) ! (

)

7! J (r

t

Tr

R

)

;

where r is any symmetri onnetion on M.

“Holomorphic curves are gradient-flow lines”

Choose J ∈ J (α) on a symplectisation (R × M, d(e^sα)). Then a half-cylinder

u = (f, v) : [0,∞) × S¹ → R × M is J-holomorphic if and only if

∂_sf − α(∂_tv) = 0,

∂_tf + α(∂_sv) = 0, π_α∂_sv + J π_α∂_tv = 0,

where π_α : T M → ξ is the projection along R_α.

Claim: the third equation an be interpreted

as the L 2

-gradient ow of the ontat a-

tion funtional

: C 1

(S 1

;M) ! R : 7!

Z

S 1

;

whose ritial points are Reeb orbits. Its Hes-

sian at a T-periodi Reeb orbit : S 1

! M is

the L 2

-symmetri operator

r(r

)() : (

) ! (

)

7! J (r

t

Tr

R

)

;

where r is any symmetri onnetion on M.

“Holomorphic curves are gradient-flow lines”

Choose J ∈ J (α) on a symplectisation (R × M, d(e^sα)). Then a half-cylinder

u = (f, v) : [0,∞) × S¹ → R × M is J-holomorphic if and only if

∂_sf − α(∂_tv) = 0,

∂_tf + α(∂_sv) = 0, π_α∂_sv + J π_α∂_tv = 0,

where π_α : T M → ξ is the projection along R_α. Claim: the third equation can be interpreted as the L²-gradient flow of the contact ac- tion functional

Φ_α : C^∞(S¹, M) → R : γ 7→

Z

S¹ γ^∗α,

whose critical points are Reeb orbits. ^{Its Hes-}

sian at a T-periodi Reeb orbit : S 1

! M is

the L 2

-symmetri operator

r(r

)() : (

) ! (

)

7! J (r

t

Tr

R

)

;

where r is any symmetri onnetion on M.

“Holomorphic curves are gradient-flow lines”

Choose J ∈ J (α) on a symplectisation (R × M, d(e^sα)). Then a half-cylinder

u = (f, v) : [0,∞) × S¹ → R × M is J-holomorphic if and only if

∂_sf − α(∂_tv) = 0,

∂_tf + α(∂_sv) = 0, π_α∂_sv + J π_α∂_tv = 0,

where π_α : T M → ξ is the projection along R_α. Claim: the third equation can be interpreted as the L²-gradient flow of the contact ac- tion functional

Φ_α : C^∞(S¹, M) → R : γ 7→

Z

S¹ γ^∗α,

whose critical points are Reeb orbits. Its Hes- sian at a T-periodic Reeb orbit γ : S¹ → M is the L²-symmetric operator

∇(∇Φ_α)(γ) : Γ(γ^∗ξ) → Γ(γ^∗ξ)

η 7→ −J (∇_tη − T∇_ηR_α) , where ∇ is any symmetric connection on M.