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The hyperkhler geometry of CP ( S ) Brice Loustau Complex projective structures The hyperkhler geometry of the deformation space of The character variety complex projective structures on a surface The Schwarzian parametrization The

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The hyperkähler geometry of the deformation space of complex projective structures on a surface

Brice Loustau August 3, 2012

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Outline

1 Complex projective structures 2 The character variety 3 The Schwarzian parametrization 4 The minimal surface parametrization

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

1 Complex projective structures 2 The character variety 3 The Schwarzian parametrization 4 The minimal surface parametrization

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

What is a complex projective structure?

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

What is a complex projective structure?

Let S be a closed oriented surface of genus g 2.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

What is a complex projective structure?

Let S be a closed oriented surface of genus g 2.

Definition

A complex projective structure on S is a (G, X)-structure on S where the model space is X = CP1 and the Lie group of transformations of X is G = PSL2(C).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

What is a complex projective structure?

Let S be a closed oriented surface of genus g 2.

Definition

A complex projective structure on S is a (G, X)-structure on S where the model space is X = CP1 and the Lie group of transformations of X is G = PSL2(C).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

CP(S) and Teichmüller space T (S)

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

CP(S) and Teichmüller space T (S)

Definition

CP(S) is the deformation space of all complex projective structures

n S:

CP(S) = {all CP1-structures on S}/Diff +

0 (S) .

A point Z ∈ CP(S) is called a marked complex projective surface.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

CP(S) and Teichmüller space T (S)

Definition

CP(S) is the deformation space of all complex projective structures

n S:

CP(S) = {all CP1-structures on S}/Diff +

0 (S) .

A point Z ∈ CP(S) is called a marked complex projective surface. CP(S) is a complex manifold of dimension dimC CP(S) = 6g − 6.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

CP(S) and Teichmüller space T (S)

Definition

CP(S) is the deformation space of all complex projective structures

n S:

CP(S) = {all CP1-structures on S}/Diff +

0 (S) .

A point Z ∈ CP(S) is called a marked complex projective surface. CP(S) is a complex manifold of dimension dimC CP(S) = 6g − 6. Note: A complex projective atlas is in particular a complex atlas on S (transition functions are holomorphic).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

CP(S) and Teichmüller space T (S)

Definition

CP(S) is the deformation space of all complex projective structures

n S:

CP(S) = {all CP1-structures on S}/Diff +

0 (S) .

A point Z ∈ CP(S) is called a marked complex projective surface. CP(S) is a complex manifold of dimension dimC CP(S) = 6g − 6. Note: A complex projective atlas is in particular a complex atlas on S (transition functions are holomorphic).

Definition

There is a forgetful map p : CP(S) → T (S) where T (S) = {all complex structures on S}/Diff +

0 (S)

is the Teichmüller space of S.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Fuchsian and quasifuchsian structures

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Fuchsian and quasifuchsian structures

If any Kleinian group Γ (i.e. discrete subgroup of PSL2(C)) acts freely and properly on some open subset U of CP1, the quotient inherits a complex projective structure.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Fuchsian and quasifuchsian structures

If any Kleinian group Γ (i.e. discrete subgroup of PSL2(C)) acts freely and properly on some open subset U of CP1, the quotient inherits a complex projective structure.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Fuchsian structures

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Fuchsian structures

In particular, any Riemann surface X can be equipped with a compatible CP1-structure by the uniformization theorem: X = H2 where Γ ⊂ PSL2(R) is a Fuchsian group.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Fuchsian structures

In particular, any Riemann surface X can be equipped with a compatible CP1-structure by the uniformization theorem: X = H2 where Γ ⊂ PSL2(R) is a Fuchsian group. Note: This defines a Fuchsian section σF : T (S) → CP(S).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Quasifuchsian structures

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Quasifuchsian structures

By Bers’ simultaneous uniformization theorem, given two complex structures (X +, X −) ∈ T (S) × T (S), there exists a unique Kleinian group Γ such that:

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Quasifuchsian structures

By Bers’ simultaneous uniformization theorem, given two complex structures (X +, X −) ∈ T (S) × T (S), there exists a unique Kleinian group Γ such that:

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

1 Complex projective structures 2 The character variety 3 The Schwarzian parametrization 4 The minimal surface parametrization

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Holonomy

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Holonomy

Any complex projective structure Z ∈ CP(S) defines a holonomy representation ρ : π1(S) → G = PSL2(C).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Holonomy

Any complex projective structure Z ∈ CP(S) defines a holonomy representation ρ : π1(S) → G = PSL2(C).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The character variety

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The character variety

Holonomy defines a map hol : CP(S) → X (S, G) ; where X (S, G) = Hom(π1(S), G)//G is the character variety of S.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The character variety

Holonomy defines a map hol : CP(S) → X (S, G) ; where X (S, G) = Hom(π1(S), G)//G is the character variety of S. hol is a local biholomorphism.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The character variety

Holonomy defines a map hol : CP(S) → X (S, G) ; where X (S, G) = Hom(π1(S), G)//G is the character variety of S. hol is a local biholomorphism. By a general construction of Goldman, the character variety X (S, G) enjoys a natural complex symplectic structure ωG.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The character variety

Holonomy defines a map hol : CP(S) → X (S, G) ; where X (S, G) = Hom(π1(S), G)//G is the character variety of S. hol is a local biholomorphism. By a general construction of Goldman, the character variety X (S, G) enjoys a natural complex symplectic structure ωG. Abusing notations, we also let ωG denote the complex symplectic structure on CP(S) obtained by pulling back ωG by the holonomy map hol : CP(S) → X (S, G).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The character variety (continued)

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The character variety (continued)

Theorem (Goldman)

The restriction of the complex symplectic structure on the Fuchsian slice F(S) is the Weil-Petersson Kähler form: σ∗

F(ωG) = ωWP .

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The character variety (continued)

Theorem (Goldman)

The restriction of the complex symplectic structure on the Fuchsian slice F(S) is the Weil-Petersson Kähler form: σ∗

F(ωG) = ωWP .

Theorem (Platis, L)

Complex Fenchel-Nielsen coordinates (li, τi) associated to any pants decomposition are canonical coordinates for the symplectic structure: ωG =

dli ∧ dτi .

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Hitchin-Kobayashi correspondence

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Hitchin-Kobayashi correspondence

Theorem (Hitchin, Simpson, Corlette, Donaldson)

Fix a complex structure X on S. There is a real-analytic bijection HX : X 0(S, G)

∼

→ M0

Dol(X, G)

where M0

Dol(X, G) is the moduli space of topologically trivial

polystable Higgs bundles on X.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Hitchin-Kobayashi correspondence

Theorem (Hitchin, Simpson, Corlette, Donaldson)

Fix a complex structure X on S. There is a real-analytic bijection HX : X 0(S, G)

∼

→ M0

Dol(X, G)

where M0

Dol(X, G) is the moduli space of topologically trivial

polystable Higgs bundles on X. Note: HX is not holomorphic, in fact:

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

Hitchin-Kobayashi correspondence

Theorem (Hitchin, Simpson, Corlette, Donaldson)

Fix a complex structure X on S. There is a real-analytic bijection HX : X 0(S, G)

∼

→ M0

Dol(X, G)

where M0

Dol(X, G) is the moduli space of topologically trivial

polystable Higgs bundles on X. Note: HX is not holomorphic, in fact:

Theorem (Hitchin)

There is a natural hyperkähler structure (g, I, J, K) on M0

Dol(X, G).

The map HX is holomorphic with respect to J. It is also a symplectomorphism for the appropriate symplectic structures.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

1 Complex projective structures 2 The character variety 3 The Schwarzian parametrization 4 The minimal surface parametrization

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The cotangent hyperkähler structure

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The cotangent hyperkähler structure

Recall that if M is any complex manifold, its holomorphic cotangent bundle T ∗M is equipped with a canonical complex symplectic structure ωcan.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The cotangent hyperkähler structure

Recall that if M is any complex manifold, its holomorphic cotangent bundle T ∗M is equipped with a canonical complex symplectic structure ωcan.

Theorem (Feix, Kaledin)

If M is a real-analytic Kähler manifold, then there exists a unique hyperkähler structure in a neighborhood of the zero section in T ∗M such that:

it refines the complex symplectic structure
it extends the Kähler structure off the zero section
the U(1)-action in the fibers is isometric.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization

Recall that there is a canonical holomorphic projection p : CP(S) → T (S).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization

Recall that there is a canonical holomorphic projection p : CP(S) → T (S). The Schwarzian derivative is an operator on maps between projective surfaces such that:

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization

Recall that there is a canonical holomorphic projection p : CP(S) → T (S). The Schwarzian derivative is an operator on maps between projective surfaces such that:

It turns a fiber p−1(X) into a complex affine space modeled
n the vector space H0(X, K 2) = T ∗

XT (S).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization

Recall that there is a canonical holomorphic projection p : CP(S) → T (S). The Schwarzian derivative is an operator on maps between projective surfaces such that:

It turns a fiber p−1(X) into a complex affine space modeled
n the vector space H0(X, K 2) = T ∗

XT (S).

Globally, CP(S) ≈σ T ∗T (S) but this identification depends on

the choice of a “zero section” σ : T (S) → CP(S).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization

Recall that there is a canonical holomorphic projection p : CP(S) → T (S). The Schwarzian derivative is an operator on maps between projective surfaces such that:

It turns a fiber p−1(X) into a complex affine space modeled
n the vector space H0(X, K 2) = T ∗

XT (S).

Globally, CP(S) ≈σ T ∗T (S) but this identification depends on

the choice of a “zero section” σ : T (S) → CP(S). For each choice of σ, we thus get a symplectic structure ωσ on the whole space CP(S) (pulling back ωcan) and a hyperkähler structure

n some neighborhood of the Fuchsian slice.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Theorem (L)

CP(S) ≈σ T ∗T (S) is a complex symplectomorphism iff d(σ − σF) = ωWP (on T (S)).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Theorem (L)

CP(S) ≈σ T ∗T (S) is a complex symplectomorphism iff d(σ − σF) = ωWP (on T (S)). Using results of McMullen (also Takhtajan-Teo, Krasnov-Schlenker):

Theorem (Kawai, L)

If σ is a (generalized) Bers section, CP(S) ≈σ T ∗T (S) is a complex symplectomorphism.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Theorem (L)

CP(S) ≈σ T ∗T (S) is a complex symplectomorphism iff d(σ − σF) = ωWP (on T (S)). Using results of McMullen (also Takhtajan-Teo, Krasnov-Schlenker):

Theorem (Kawai, L)

If σ is a (generalized) Bers section, CP(S) ≈σ T ∗T (S) is a complex symplectomorphism.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Consequences:

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Consequences:

Fibers of p and Bers slices are Lagrangian complex

submanifolds.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Consequences:

Fibers of p and Bers slices are Lagrangian complex

submanifolds.

(generalized) Quasifuchsian reciprocity.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Consequences:

Fibers of p and Bers slices are Lagrangian complex

submanifolds.

(generalized) Quasifuchsian reciprocity.
If σ is elected among Bers sections,

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Consequences:

Fibers of p and Bers slices are Lagrangian complex

submanifolds.

(generalized) Quasifuchsian reciprocity.
If σ is elected among Bers sections,
The hyperkähler stucture we get on CP(S) refines the complex

symplectic structure,

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Consequences:

Fibers of p and Bers slices are Lagrangian complex

submanifolds.

(generalized) Quasifuchsian reciprocity.
If σ is elected among Bers sections,
The hyperkähler stucture we get on CP(S) refines the complex

symplectic structure,

but the new complex structure J depends on the choice of the

Bers section. In other words the bunch of hyperkähler structures we get is parametrized by T (S).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Consequences:

Fibers of p and Bers slices are Lagrangian complex

submanifolds.

(generalized) Quasifuchsian reciprocity.
If σ is elected among Bers sections,
The hyperkähler stucture we get on CP(S) refines the complex

symplectic structure,

but the new complex structure J depends on the choice of the

Bers section. In other words the bunch of hyperkähler structures we get is parametrized by T (S).

This is similar to the situation we saw with the

Hitchin-Kobayashi correspondence.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The Schwarzian parametrization (continued)

Consequences:

Fibers of p and Bers slices are Lagrangian complex

submanifolds.

(generalized) Quasifuchsian reciprocity.
If σ is elected among Bers sections,
The hyperkähler stucture we get on CP(S) refines the complex

symplectic structure,

but the new complex structure J depends on the choice of the

Bers section. In other words the bunch of hyperkähler structures we get is parametrized by T (S).

This is similar to the situation we saw with the

Hitchin-Kobayashi correspondence. Quiz : what is a significant difference though?

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

1 Complex projective structures 2 The character variety 3 The Schwarzian parametrization 4 The minimal surface parametrization

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization

The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is a neighborhood of the Fuchsian slice such that if Z ∈ AF(S), the hyperbolic 3-manifold associated to Z contains a unique minimal surface Σ.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization

The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is a neighborhood of the Fuchsian slice such that if Z ∈ AF(S), the hyperbolic 3-manifold associated to Z contains a unique minimal surface Σ. The Gauss-Codazzi equations satisfied by the second fundamental form II Σ are equivalent to the fact that II Σ is the real part of a unique holomorphic quadratic ϕ.

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization

The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is a neighborhood of the Fuchsian slice such that if Z ∈ AF(S), the hyperbolic 3-manifold associated to Z contains a unique minimal surface Σ. The Gauss-Codazzi equations satisfied by the second fundamental form II Σ are equivalent to the fact that II Σ is the real part of a unique holomorphic quadratic ϕ. This defines a map AF(S) → T ∗T (S) Z → ([I Σ], ϕ) .

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization

The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is a neighborhood of the Fuchsian slice such that if Z ∈ AF(S), the hyperbolic 3-manifold associated to Z contains a unique minimal surface Σ. The Gauss-Codazzi equations satisfied by the second fundamental form II Σ are equivalent to the fact that II Σ is the real part of a unique holomorphic quadratic ϕ. This defines a map AF(S) → T ∗T (S) Z → ([I Σ], ϕ) . It is a diffeomorphism of AF(S) onto some neighborhood of the zero section of T ∗T (S).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization

The space of almost-Fuchsian structures AF(S) ⊂ QF(S) is a neighborhood of the Fuchsian slice such that if Z ∈ AF(S), the hyperbolic 3-manifold associated to Z contains a unique minimal surface Σ. The Gauss-Codazzi equations satisfied by the second fundamental form II Σ are equivalent to the fact that II Σ is the real part of a unique holomorphic quadratic ϕ. This defines a map AF(S) → T ∗T (S) Z → ([I Σ], ϕ) . It is a diffeomorphism of AF(S) onto some neighborhood of the zero section of T ∗T (S). Again, one can use this “minimal surface parametrization” to pull back the hyperkähler structure of T ∗T (S) on CP(S).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization (continued)

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization (continued)

The notion of renormalized volume of almost-Fuchsian manifolds defines a function W on AF(S).

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization (continued)

The notion of renormalized volume of almost-Fuchsian manifolds defines a function W on AF(S). Using arguments of Krasnov-Schlenker to compute the variation of W under an infinitesimal deformation of the metric, one shows:

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The hyperkähler geometry of CP(S) Brice Loustau Complex projective structures The character variety The Schwarzian parametrization The minimal surface parametrization

The minimal surface parametrization (continued)

The notion of renormalized volume of almost-Fuchsian manifolds defines a function W on AF(S). Using arguments of Krasnov-Schlenker to compute the variation of W under an infinitesimal deformation of the metric, one shows: