Sobolev spaces

In each marked point there should be a chosen directionv⁽ⁱ⁾ v⁽ⁱ⁾∈P(T_p(i)Σ) := (T_p(i)Σ\{0})/R⁺.

Figure 6.3: Surface with marked point and small embedded discs around each puncture, along which we compute the holonomy around each puncture in the direction induced from the orientation of the surface.

Recall that the punctured surface is denoted byΣ. As before˜ Σσ denotesΣequipped with a complex stucture. For eachiwe letz⁽ⁱ⁾:U⁽ⁱ⁾→Ddenote a complex analytic isomorphism with D⊂Cthe unit disk, such thatp⁽ⁱ⁾= (z⁽ⁱ⁾)⁻¹(0)andDz⁽ⁱ⁾(v⁽ⁱ⁾)∈R+(_dx^d)⊆P(T0D). We may define new coordinates on(U⁽ⁱ⁾)^∗= (U⁽ⁱ⁾)\{p⁽ⁱ⁾}. Setw⁽ⁱ⁾:=−logz⁽ⁱ⁾, thenw⁽ⁱ⁾maps(U⁽ⁱ⁾)^∗ analytically to the semi-infinite cylinder C⁽ⁱ⁾ ={(τ, θ)|τ≥0,0≤θ≤2π}/(τ,0)∼ (τ,2π).

Around each puncture, we find a neighborhood, conformally equivalent to the standard semi- infinite straight cylinder S¹×[0,∞), see Figure 6.4, and the directionv⁽ⁱ⁾corresponds to the line (0,0)×[0,∞)on the cylinder in polar coordinates.

Figure 6.4: Surface with semi-infinite cylinders

Fix a metrichonΣ˜ compatible with the complex structureΣσ, such that it restricts to the standard flat metric on the semi-infinite ends ofΣ,h|_U(i)\{p⁽ⁱ⁾}=d(τ⁽ⁱ⁾)²+d(θ⁽ⁱ⁾)².

We are going to construct a further moduli space denoted byM( ˜Σ, λ, )σ. To do this we need the notion of Sobolev spaces, which we will introduce in the following section. We will follow the construction of Andersen in [1] closely.

Recall that we have our surface Σ with punctures P = {p⁽¹⁾, . . . , p^(b)}. Around each puncture we have an open neighborhood U⁽ⁱ⁾ that we can map analytically to a semi-infinite cylinderC⁽ⁱ⁾'S¹×[1,∞).

First we will take a look at the semi-infinite ends of the surface,C⁽ⁱ⁾=S¹×[0,∞). Over C⁽ⁱ⁾consider the trivial principal G-bundleQ⁽ⁱ⁾ with connection∇0=d+A⁽ⁱ⁾, whereA⁽ⁱ⁾ is a constant 1-form with values ing. inQ⁽ⁱ⁾ overC⁽ⁱ⁾. Letd_A(i) denote the covariant derivative in the associated adjoint bundle ofQ⁽ⁱ⁾, AdQ⁽ⁱ⁾.

Define the∗ operator such that

α∧ ∗β =hα, βiVol

If we let(θ⁽ⁱ⁾, r⁽ⁱ⁾)be the coordinates on the cylinder, then we haveVol =dθ⁽ⁱ⁾∧dr⁽ⁱ⁾. By defining

∗dθ⁽ⁱ⁾=dr⁽ⁱ⁾,∗dθ⁽ⁱ⁾=−dθ⁽ⁱ⁾ we see that

dθ⁽ⁱ⁾∧dr⁽ⁱ⁾=dθ⁽ⁱ⁾∧ ∗dθ⁽ⁱ⁾=hdθ⁽ⁱ⁾, dθ⁽ⁱ⁾iVol = Vol.

We can calculate∗(dθ⁽ⁱ⁾∧dr⁽ⁱ⁾)by looking atα=β=dθ⁽ⁱ⁾∧dr⁽ⁱ⁾.

(dθ⁽ⁱ⁾∧dr⁽ⁱ⁾)∧ ∗(dθ⁽ⁱ⁾∧dr⁽ⁱ⁾) =|dθ⁽ⁱ⁾∧dr⁽ⁱ⁾|²dθ⁽ⁱ⁾∧dr⁽ⁱ⁾

∗(dθ⁽ⁱ⁾∧dr⁽ⁱ⁾) = 1.

We can now defined^∗_A(i) :=− ∗d_A(i)∗and we get a commutative diagram Ω²(C⁽ⁱ⁾,AdQ⁽ⁱ⁾) ^∗ //

d^∗

A(i)

Ω⁰(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)

d_A(i)

Ω¹(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)oo ^−∗ Ω¹(C⁽ⁱ⁾,AdQ⁽ⁱ⁾) .

We can combine the operatorsd_A(i) andd^∗_A(i) to get an operator

˜δ_A(i): Ω⁰(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)⊕Ω²(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)→Ω¹(C⁽ⁱ⁾,AdQ⁽ⁱ⁾).

Weighted Sobolev spaces on C⁽ⁱ⁾

We are now going to construct some weighted Sobolev spaces on the cylinder C⁽ⁱ⁾. The ideas used here will be the same, when we construct Sobolev spaces on the whole surface.

Let∈Randψ∈Ω^j(C⁽ⁱ⁾,AdQ⁽ⁱ⁾). We define a norm

|ψ|²_,k:=

Z

C⁽ⁱ⁾

X

0≤l≤k

|∇^l(e^uψ(u, θ))|²dudθ.

LetΩ^j_,k(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)denote the completion ofΩ^j(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)in the norm | · |,k. LetN_A⁽ⁱ⁾_(i) denote the kernel of _∂θ^∂_(i) + [A⁽ⁱ⁾,·]

N_A⁽ⁱ⁾_(i)=

ϕ∈Ω⁰(S¹,AdQ⁽ⁱ⁾)

∂ϕ

∂θ⁽ⁱ⁾+ [A⁽ⁱ⁾, ϕ] = 0

.

Fix a functionρ⁽ⁱ⁾ with support in (0,∞)and constant 1near∞. For j = 0,2define the subspacesΩ^j

N⁽ⁱ⁾

A(i)

(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)⊆Ω^j(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)given by Ω^j

N⁽ⁱ⁾

A(i)

(C⁽ⁱ⁾,AdQ⁽ⁱ⁾) :=n

ϕ∈Ω^j(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)

∃ϕ_∞∈N_A⁽ⁱ⁾_(i) s.t. ϕ−ρ⁽ⁱ⁾ϕ_∞∈Ω^j_C(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)o . In the same way we defineΩ¹

N⁽ⁱ⁾

A(i)

(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)⊆Ω¹(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)as Ω¹

N⁽ⁱ⁾

A(i)

(C⁽ⁱ⁾,AdQ⁽ⁱ⁾) :={ϕ∈Ω¹(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)| ∃ϕ_∞∈N_A⁽ⁱ⁾_(i)⊕N_A⁽ⁱ⁾_(i)

s.t. ϕ−ρ⁽ⁱ⁾ϕ_∞∈Ω^j_C(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)}.

On these spaces we can consider the norm

|ϕ|²_,k,∞:=

Z

C⁽ⁱ⁾

X

0≤l≤k

|∇^l(e^u(ϕ(u, θ)−ρϕ_∞(θ)))|²dudθ+ Z

S¹

|ϕ_∞(θ)|²dθ.

LetΩ^j_,k,∞(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)denote the completion of Ω^j

N⁽ⁱ⁾

A(i)

(C⁽ⁱ⁾,AdQ⁽ⁱ⁾)in the norm| · |_,k,∞.

Weighted Sobolev spaces on Σ˜

We will now define Sobolev spaces onΣ, which locally on the ends of˜ Σ˜ are equivalent to the ones we have just defined on the semi-infinite cylindersC⁽ⁱ⁾.

Let Q = ˜Σ×G be the trivial bundle. As in the beginning of Section 6 we have disk neighborhoodsU⁽ⁱ⁾of each puncturep⁽ⁱ⁾,i= 1, . . . , b. LetA⁰ be a flat connection inQoverΣ,˜ [A⁰]∈ M( ˜Σ, λ). By Lemma 2.7 in [22] for each puncturep⁽ⁱ⁾there exists a gauge-equivalent connection A⁽ⁱ⁾ via a gauge-transformationg⁽ⁱ⁾, such that in the chosen trivialization A⁽ⁱ⁾ is given by ξ⁽ⁱ⁾dθ⁽ⁱ⁾, where ξ⁽ⁱ⁾ ∈g. Note that ξ⁽ⁱ⁾ does not depend on r⁽ⁱ⁾. Let S˜⁽ⁱ⁾ be a circle around p⁽ⁱ⁾ insideU⁽ⁱ⁾. LetD⁽ⁱ⁾denote the disk with boundaryS˜⁽ⁱ⁾. OnS˜⁽ⁱ⁾ we have g⁽ⁱ⁾:S¹→G. We have assumed thatGis simply connected, so we can use a homotopy from g⁽ⁱ⁾to the identity, to expandg⁽ⁱ⁾to a gauge transformation on the cylinder, that isg⁽ⁱ⁾in one end and the identity in the other. Now for each iwe have a gauge transformation˜g⁽ⁱ⁾, that is the identity on all of Σ\U⁽ⁱ⁾ andg⁽ⁱ⁾ onD⁽ⁱ⁾. Since theU⁽ⁱ⁾’s are disjoint, we can compose all the ˜g⁽ⁱ⁾’s to get a gauge transformationg. LetAbe the connection gauge equivalent toA⁰ via g. LetS⁽ⁱ⁾be a circle aroundp⁽ⁱ⁾inside D⁽ⁱ⁾.

LetA⁽ⁱ⁾:=ξ⁽ⁱ⁾dθand N_A(i) :=

ϕ∈Ω⁰(S¹,AdQ⁽ⁱ⁾)

∂ϕ

∂θ⁽ⁱ⁾ + [A⁽ⁱ⁾, ϕ] = 0

.

Definition of the Sobolev spaces

We want to construct the Sobolev spaces as before, but since our surface is now more complicated than a simple cylinder, we have to do it in a little more complicated way. Earlier we haduthat ran from 0to∞along the cylinder. Pick a Riemannian metricg˜onΣ, such˜ that it is equal to the standard metric on the cylinders. Now we pick a point q∈Σ, and let˜ d be the function that measures the distance from any pointp∈Σ˜ toq. This dwill now play the role that udid before. Fix a functionρonΣ˜ with support on the cylinders and such that ρis1 near the ends of the cylinders. Consider the spaces

Ω^j_,k,∞( ˜Σ,AdQ) :=n

f ∈Ω^j_L2 loc

( ˜Σ,AdQ)|

∃f∞∈N_A(1)× · · · ×N_A(b): X

0≤l≤k

Z

Σ˜

|∇^l(e^d(f−ρf∞))|²+ Z

S

iS⁽ⁱ⁾

|f∞|²<∞o

forj= 0,2. Similarly let Ω¹_,k,∞( ˜Σ,AdQ) :=n

f ∈Ω¹_L2 loc

( ˜Σ,AdQ)|∃f_∞¹, f_∞² ∈NA₁× · · · ×NA_n: X

0≤l≤k

Z

Σ˜

∇^l(e^d(f−ρ(f_∞¹ ⊕f_∞² )))

2

+ Z

S

iS⁽ⁱ⁾

|f_∞¹ ⊕f_∞²|²<∞o ,

and

Ω^j_,k( ˜Σ,AdQ) :=n

ϕ∈Ω^j_L2 loc

( ˜Σ,AdQ)

X

0≤l≤k

Z

Σ˜

|∇^l(e^dϕ)|²<∞o .

Note that Ω^j_,k,∞( ˜Σ,AdQ)⊆Ω^j_,k( ˜Σ,AdQ), since we can choosef∞= 0.

The operator δA

The flat connection A in Q gives us a covariant derivative inAdQ over Σ, and we get a˜ complex

0→Ω⁰_,k,∞( ˜Σ,AdQ)→Ω¹_,k( ˜Σ,AdQ)→Ω²_,k,∞( ˜Σ,AdQ)→0

withd_A the boundary map. We will denote the first cohomology group of this complex by H_,k¹ ( ˜Σ, d_A) = kerd_A

ImdA

.

By using the Hodge-star operator associated to the metricg˜onΣwe can considerd^∗_A=

− ∗d_A∗ onΩⁱ( ˜Σ,AdQ). As before we consider

δ˜_A: Ω⁰_,k+1,∞( ˜Σ,AdQ)⊕Ω²_,k+1,∞( ˜Σ,AdQ)→Ω¹_,k( ˜Σ,AdQ).

Similarly we can look atd_A: Ω¹_,k,∞( ˜Σ,AdQ)→Ω²_,k−1( ˜Σ,AdQ), and d^∗_A as an operator d^∗_A: Ω¹_,k,∞( ˜Σ,AdQ)→Ω⁰_,k−1( ˜Σ,AdQ). Denote the sum of these two byδAas in [1],

δ_A: Ω¹_,k,∞( ˜Σ,AdQ)→Ω⁰_,k−1( ˜Σ,AdQ)⊕Ω²_,k−1( ˜Σ,AdQ).

It is shown in [1] thatδAand˜δA are both Fredholm forpositive and sufficiently small. For sufficiently small and positive, theΩ¹_,k-kernel and theL2-kernel ofδA are the same, see [1] .

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