# Background material 3

## Full text

(1)

### Background material 3

The basic goal

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

u : ˙Σ → W ,c v : ˙Σ → Wc

u Wc

Σ˙

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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 Wc

Σ˙

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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 Wc

Σ˙

(4)

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 Wc

Σ˙

(5)

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!

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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!

(7)

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!

(8)

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!

(9)

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!

(10)

Disaster scenario:

Suppose u : ˙Σ → Wc 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.

(11)

Disaster scenario:

Suppose u : ˙Σ → Wc 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.

(12)

Disaster scenario:

Suppose u : ˙Σ → Wc 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.

(13)

Disaster scenario:

Suppose u : ˙Σ → Wc 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.

(14)

Disaster scenario:

Suppose u : ˙Σ → Wc 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.

(15)

Disaster scenario:

Suppose u : ˙Σ → Wc 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.

(16)

Disaster scenario:

Suppose u : ˙Σ → Wc 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.

(17)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

(18)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

(19)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

(20)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

(21)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

(22)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

(23)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0 ind = 1

A

p

is symmetri, so its eigenvalues are real.

(24)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0

ind = 1 ind = 1

A

p

is symmetri, so its eigenvalues are real.

(25)

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

Ap := ∇(∇f)(p) : TpM TpM has trivial kernel.

ind = 2

ind = 0

ind = 1 ind = 1

Ap is symmetric, so its eigenvalues are real.

(26)

Asymptotic formula for gradient-flow

Theorem

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

x(s) = expp 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."

(27)

Asymptotic formula for gradient-flow

Theorem

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

x(s) = expp h(s) ∈ M

is a gradient-flow line approaching p as s →

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

for some eigenvector v ∈ TpM of Ap with Apv = λv, ±λ < 0,

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

\Flow lines approah ritial points along

asymptoti eigenvetors."

(28)

Asymptotic formula for gradient-flow

Theorem

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

x(s) = expp h(s) ∈ M

is a gradient-flow line approaching p as s →

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

for some eigenvector v ∈ TpM of Ap with Apv = λv, ±λ < 0,

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

“Flow lines approach critical points along asymptotic eigenvectors.”

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“Holomorphic curves are gradient-flow lines”

Choose J ∈ J (α) on a symplectisation (R × M, d(esα)). 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.

(30)

“Holomorphic curves are gradient-flow lines”

Choose J ∈ J (α) on a symplectisation (R × M, d(esα)). 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, παsv + J παtv = 0,

where πα : T M → ξ is the projection along Rα.

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.

(31)

“Holomorphic curves are gradient-flow lines”

Choose J ∈ J (α) on a symplectisation (R × M, d(esα)). 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, παsv + J παtv = 0,

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

Φα : C(S1, M) → R : γ 7→

Z

S1 γα,

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.

(32)

“Holomorphic curves are gradient-flow lines”

Choose J ∈ J (α) on a symplectisation (R × M, d(esα)). 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, παsv + J παtv = 0,

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

Φα : C(S1, M) → R : γ 7→

Z

S1 γα,

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

∇(∇Φα)(γ) : Γ(γξ) → Γ(γξ)

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

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