• Nebyly nalezeny žádné výsledky


detFn−1,detHn−1 − ||rn1||2 ≤0.

Locally ΛF¯

detFk,detHk = ¯∂∂log(detHk). We can either apply Lemma6.3or directly argue and obtain that for eachk, detHk/hk(n−k)2 <1.

This is the main step for the proof. The interested reader may refer to [37] for the rest of the


Part III

Selected topics on harmonic maps and minimal surfaces

7 Labourie’s conjecture

For a fixed Riemann surface Σ, the Hitchin component Hitn is parametrized by Ln


H0 Σ, Ki . Denote byV the vector bundle over the Teichm¨uller space whose fiber at Σ is the vector space

Ln i=3

H0 Σ, Ki

and Labourie in [35] considered the map V −→Hitn⊂Rep(π1S,PSL(n,R)), (Σ,(q3, . . . , qn))7−→NAHΣ(0, q3, . . . , qn).

The left hand side has the same dimension as the right hand side. The map is equivariant with respect to the mapping class group action. In the same paper, Labourie asked the following question.

Question 7.1. Is this map a bijection?

As noted in Remark3.1, the vanishing of the first term in the image h(E, φ) in the Hitchin fibration is equivalent to the vanishing of the Hopf differential of the associated harmonic map.

Also, if the Hopf differential vanishes, the harmonic map is conformal and thus minimal. There-fore, Question7.1 can be generalized and rephrased as follows:

Conjecture 7.2 (Labourie’s conjecture). For ρ a Hitchin representation into a split real Lie group G or a maximal representation into a Hermitian Lie group G, does there exist a unique ρ-equivariant minimal surface in G/K?

• Existence (shown by Labourie [35].) Labourie in [35] showed that for a fixed Anosov representation ρ, the function fρ:T(S) → R sending each Σ to E NAHΣ1(ρ)

is proper. So the function fρ has a critical point. By the classical results of Sacks–Uhlenbeck [46, 47] and Schoen–Yau [50], the critical Riemann surface Σ is such that the corresponding harmonic map is conformal. That is, tr φ2

= 0. Since both Hitchin representations and maximal representations are Anosov, the existence follows.

•Uniqueness. This is the main part of Labourie’s conjecture.

Labourie’s conjecture is proven for Hitchin representations into SL(2,R)×SL(2,R), by Schoen in [51]; SL(3,R), independently by Labourie [34], Loftin [40] and all the remaning rank 2 split real Lie groups SL(3,R), Sp(4,R), G2 by Labourie [36]. The property for cyclic Higgs bundles in Lemma 3.5is essential in Labourie’s proof [36].

Labourie’s conjecture is proven for maximal representations into Sp(4,R) by Collier [8];

PSp(4,R) by Alessandrini–Collier [1] and all the remaining rank 2 Hermitian Lie groups by Collier–Tholozan–Toulisse [10].

8 Asymptotics

Before we introduce the asymptotic question, let us first recall Thurston’s compactification [19,54] of the Teichm¨uller space with measured foliations. Let S denote the space of isotopy classes of simple closed curves and denote the projectivization of the space of nonnegative func-tions onS byPRS+. The map which assigns the projectivized length spectrum of each hyperbolic metric is an embedding of the Teichm¨uller space insidePRS+. The boundary corresponds to the image of the intersection length spectrum of the space of projective measured foliations. In terms of the representation variety, there are also algebraic techniques on the compactification by Morgan–Shalen [43] and generalized by Parreau in [44] for the representation variety for higher rank Lie groups.

Wolf in [57] recovers Thurston’s compactification using harmonic maps. Fix a Riemann surface structure Σ, the Teichm¨uller space is homeomorphic to the vector spaceH0 Σ, K2

, see Section6.1. Roughly, the harmonic map compactification of the Teichm¨uller space is by adding the space of rays in the vector space. Let q2 be a holomorphic quadratic differential, consider the raytq2 and letht be the corresponding family of hyperbolic metrics such that the associated

harmonic maps ft: Σ → (S, ht) have Hopf differential tq2. Away from the zeros of q2 choose a coordinatezsuch thatq2 = dz2. In such coordinates we have local measured foliations (F, µ) = ({Re(z) = const},|d Re(z)|) which piece together to form the vertical measured foliationF(q2) of q2. The key step in showing the harmonic map compactification agrees with Thurston’s compactification is to show that the length spectrum of ht is asymptotically the same as the length spectrum of the vertical measured foliation oftq2. That is, for any closed curveγ on Σ, ast→ ∞,

l(ft(γ)) =lγ(ht)∼i(F(tq2), γ). (8.1)

Here, i(F(tq2), γ) is the intersection number of γ with the vertical measured foliation F(tq2).

This compactification was further extended to the character variety for SL(2,C) (see [4] and [14]).

As a generalization of the harmonic map compactification, we discuss the asymptotic behavior of the equivariant harmonic mapsft:Σe →N along theC-familyt·[(E, φ)]∈ MHiggs(SL(n,C)) ast→ ∞ and aim to generalize the asymptotic formula (8.1).

For the left hand side of formula (8.1), we generalize the notion of length of a curve to a vector distance between two points in the target space N. For two Hermitian metrics h1, h2 on an n-dimensional complex vector spaceV, we can take a basee1, e2, . . . , enofV which is orthogonal with respect to both h1 and h2. We have the real numbers kj (j = 1,2, . . . , n) determined by kj = log|ej|h2−log|ej|h1. We imposek1≥k2 ≥ · · · ≥kn and setd(h~ 1, h2) := (k1, . . . , kn)∈Rn. For the right hand side of formula (8.1), we use Higgs field to generalize holomorphic quadratic differential. Unfortunately we only have a local geometric object as a natural generalization of measured foliations. Therefore, instead of working with any closed curve on Σ, we restrict to consider “nice” paths on the universal cover Σ. Denote bye D(E, φ) the set of points where the Higgs fieldφfails to have ndistinct eigen 1-forms, called thediscriminant of the Higgs bundle.

Take a universal covering π:Y →Σ\D(E, φ), we have a decomposition of the Higgs bundle π E,∂¯E, φ

= Mn


Ei,∂¯Ei, φi·idEi ,

where φi are holomorphic 1-forms, rankEi = 1, and φi−φj (i6=j) have no zeros. Let γ: [0,1]

→ Y be a C-path, we have the expression γi) = ai(s)ds, where ai are C-functions on [0,1]. A path γ is called non-critical if Re(ai(s)) 6= Re(aj(s)) (i6= j) for any s∈ [0,1]. Let’s reorder the ai(s) such that Re(ai(s))>Re(aj(s)) fori < j and setαi :=−R1

0 Re(ai(s))ds. The vector (α1, . . . , αn) generalizes the intersection number of the measured foliation.

With the above preparation, we finally state the following conjecture as a local generalization of the asymptotic formula (8.1) to higher rank Higgs bundles.

Conjecture 8.1 (Hitchin WKB problem, Katzarkov–Noll–Pandit–Simpson [31]). As t → ∞, the harmonic map ft satisfies for a non-critical path γ: [0,1]→Y,


td(f~ t(γ(0), ft(γ(1)))∼2(α1, . . . , αn).

To answer the conjecture, we introduce the following notion.

Definition 8.2. We call a Higgs bundle (E, φ)generically regular semisimpleif the discriminant set D(E, φ) is finite.

Remark 8.3. A SL(2,C)-Higgs bundle is either in the nilpotent cone or is generically regular semisimple. However, for n ≥ 3, there are many SL(n,C)-Higgs bundles which are neither generically regular semisimple nor nilpotent.

Theorem 8.4 (Mochizuki [42]). Let (E, φ) be a stable Higgs bundle of degree 0 on Σ. Suppose it is generically regular semisimple. If γ: [0,1] → Y is non-critical, then there exist positive constants C0 and 0 such that the following holds:


td(f~ t(γ(0), ft(γ(1)))−2(α1, . . . , αn)


The constants C0 and 0 may depend only on Σ, φ1, . . . , φn and γ.

The proof of Theorem8.4is based on the following key estimate as “decoupling the Hitchin equation”.

Theorem 8.5 (Mochizuki [42]). Under the same assumptions in Theorem 8.4. Then take any neighborhood N0 of D(E, φ), there exists a constantC0 >0 and 0 >0 such that the following holds on Σ\N0,


∂E ,Ht||=|t|2||[φ, φHt]|| ≤C0exp(−0|t|).

The constants C0, 0 only depend on Σ, g0, N0 and (E, φ).

Remark 8.6. There is also another approach to obtain the decoupling phenomenon in Theo-rem 8.5 for the Hitchin equation in [20,41] for generic Higgs bundles, which will be addressed in the survey paper [21] of L. Fredrickson.

Remark 8.7.

(1) For cyclic Higgs bundles in the Hitchin component, Theorems8.4and8.5were first proven in Loftin [38] forn= 3 and in Collier–Li [9] forn >3.

(2) The full picture of Conjecture 8.1 remains open for Higgs bundles which are neither ge-nerically regular semisimple nor nilpotent. In fact, for those Higgs bundles which are not generically regular semisimple, Conjecture 8.1is not necessarily true. So we need a more refined description of the asymptotics for such families of Higgs bundles.