Asymptotics of Hitchin's metric on the Hitchin section
AASYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION
DAVID DUMAS AND ANDREW NEITZKE
Abstract.
We consider Hitchin’s hyperk¨ahler metric g on the moduli space M of degreezero SL(2)-Higgs bundles over a compact Riemann surface. It has been conjectured that,when one goes to infinity along a generic ray in M , g converges to an explicit “semiflat”metric g sf , with an exponential rate of convergence. We show that this is indeed the casefor the restriction of g to the tangent bundle of the Hitchin section B ⊂ M . Introduction
Summary.
Fix a compact Riemann surface C . In [9] Hitchin studied the moduli space M of degree zero SL(2) -Higgs bundles on C , and showed in particular that M admits acanonically defined hyperk¨ahler metric g .In [6, 8] a new conjectural construction of g was given. The full conjecture is complicatedto state (see [13] for a review), but one of its consequences is a concrete picture of the genericasymptotics of g , as follows.The non-compact space M is fibered over the space B of holomorphic quadratic differen-tials on C . We consider a path to infinity in M , lying over a generic ray { tφ } t ∈ R + ⊂ B ,where φ has only simple zeroes. Along such a path, the prediction is that g = g sf + O (cid:16) e − αt (cid:17) , (1.1)where g sf is the semiflat metric, given by a simple explicit formula (see § α is anyconstant with α < M ( φ ), where M ( φ ) is the length of the shortest saddle connection inthe metric | φ | (see § g − g sf does decay at least polynomially in t . This work motivated us to wonderwhether one could show directly that the decay is actually exponential. In this paper weshow that this is indeed the case for the restriction of g to the tangent bundle of a certainembedded copy of B inside M , the Hitchin section : (1.1) holds there for any α < M ( φ ).(Unfortunately, we miss the conjectured sharp constant by a factor of 2.) The precisestatement is given in Theorem 1 below.1.2. The strategy.
Points of B correspond to holomorphic quadratic differentials φ on C .Since these form a linear space, tangent vectors to B likewise correspond to holomorphicquadratic differentials ˙ φ . Given ( φ, ˙ φ ) ∈ T B , both g φ ( ˙ φ, ˙ φ ) and g sf φ ( ˙ φ, ˙ φ ) arise as integralsover C (which can be found in (4.23) and (4.24) below). The integrand in g sf φ is completelyexplicit, while the integrand in g φ depends on the solutions of two elliptic scalar PDEs onthe surface C . To prove (1.1) for some given α , we need to show that these two integralsagree up to O (e − αt ). Date : March 27, 2018. (v1: February 20, 2018). a r X i v : . [ m a t h . DG ] M a r D. DUMAS AND A. NEITZKE
Figure 1.
A genus 2 surface C equipped with a holomorphic quadratic dif-ferential φ which has 4 simple zeroes (orange crosses). The shortest saddleconnection is shown in green; its length is M ( φ ). We have chosen α slightlysmaller than M ( φ ). C near is the union of 4 disks D i centered on the zeroes,all with the same radius α . The complementary region C far is shaded.To do this, we let r ( z ) denote the | φ | -distance from z to the closest zero of φ , and dividethe surface C into two regions, as illustrated in Figure 1: • The “far” region C far = { z : r ( z ) > α } . In this region we can show that the integrands agree to order O (e − αt ): indeed, we show that the difference δ of the integrands decaysas δ = O (e − γt r ( z ) ) for any γ <
4. This part of our analysis contains no big surprises,and is closely parallel to the analysis carried out by Mazzeo-Swoboda-Weiss-Witt inthe more general setup of arbitrary SL(2)-Higgs bundles in [11]. (However, because werestrict to the Hitchin section
B ⊂ M , our job is somewhat simpler: we only have todeal with scalar PDEs, and use more-or-less standard techniques. The specific estimateswe use in this part are built on the work of Minsky in [12].) • The “near” region C near = { z : r ( z ) ≤ α } . This region looks more difficult becauseour estimates do not show that δ is close to zero here. The happy surprise—which wasreally the reason for writing this paper—is that when α < M ( φ ), δ turns out to beclose to an exact form that we can control, as follows. For any α < M ( φ ), C near is a disjoint union of disks D i centered on the zeros of φ . On each D i we show that δ = d β i + O (e − αt ), for a 1-form β i which has the same decay property as δ , namely β i = O (e − γt r ( z ) ). Thus β i is exponentially small on the boundary of D i , and Stokes’stheorem gives (cid:82) D i δ = (cid:82) ∂D i β i + O (e − αt ) = O (e − αt ).Combining these contributions we obtain the desired estimate (cid:82) C δ = O (e − αt ).1.3. Outline.
We carry out the strategy described above as follows. In §§ § g and g sf to T B . In §§ §§ β i which we use in the “near” region. In §
10 we put all this together to complete the proof ofthe main theorem.1.4.
Origin in experiment.
This work was initially inspired by computer experiments(using programs developed by the authors, and building on work of the first author and Wolfin [4]) that seemed to show exponential decay of g − g sf in certain cases, despite the lack of anexponentially decaying bound on the integrand near the zeros of φ . While these experimentswere conducted in a slightly different setting—namely, meromorphic Higgs bundles on CP with a single pole—all of the essential features and challenges are present in both cases.The experimental results therefore suggested that some “cancellation” would occur in C near . SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 3
Further investigation of the integrand in this region led to the results of §§ Outlook.
It would be very desirable to understand how to extend Theorem 1 to Higgsbundles of higher rank, say SL( N )-Higgs bundles. There is a conjecture very similar to (1.1)in that case, but instead of the shortest saddle connection, it involves the lightest finite web as defined in [7]. While the analysis of C far should extend to this case using methods similarto those of [11], it is not clear how our approach to C near should be generalized.Similarly, one would like to extend Theorem 1 to work on the full M instead of only B ⊂ M . The analysis of C far has already been done on the full M in [11], so again the issueis whether the analysis of C near can be extended.In another direction, it would be desirable to improve Theorem 1 to show that the expo-nential estimate holds for all α < M ( φ ) instead of just α < M ( φ ). However, this mightrequire a new method; in our computation we meet several different corrections which arenaively of the same order e − M ( φ ) t ; one would need to find some mechanism by which thesedifferent corrections can cancel one another.1.6. Acknowledgements.
The authors thank Rafe Mazzeo, Jan Swoboda, Hartmut Weiss,and Michael Wolf for helpful discussions related to this work, and also thank the anonymousreferee for a careful reading and helpful comments and corrections. The authors also grate-fully acknowledge support from the U.S. National Science Foundation through individualgrants DMS 1709877 (DD), DMS 1711692 (AN), and through the GEAR Network (DMS1107452, 1107263, 1107367, “RNMS: GEometric structures And Representation varieties”)which supported a conference where some of this work was conducted.2.
Background
Higgs bundles.
Recall that a stable
SL(2) -Higgs bundle over C of degree zero is a pair( E , ϕ ) where • E is a rank 2 holomorphic vector bundle over C , equipped with a trivialization of det E , • ϕ is a traceless holomorphic section of End E ⊗ K C , • all ϕ -invariant subbundles of E have negative degree.There is a (coarse) moduli space M parameterizing stable SL(2)-Higgs bundles over C ofdegree zero [9, 10].2.2. Harmonic metrics.
For each stable SL(2)-Higgs bundle ( E , ϕ ) of degree zero, it isshown in [9] that there is a distinguished unit-determinant Hermitian metric h on E , the harmonic metric . The metric h is determined by solving an elliptic PDE: letting D denotethe Chern connection in E , with curvature F D ∈ Ω ( su ( E , h )), and letting Φ = ϕ − ϕ † ∈ Ω ( su ( E , h )) with ϕ † the h -adjoint of ϕ , we require F D −
12 [Φ , Φ] = 0 . (2.1)In this equation both F D and Φ depend on h . D. DUMAS AND A. NEITZKE
The hyperk¨ahler metric.
Now we recall Hitchin’s hyperk¨ahler metric g on the modulispace M . A beautiful description of this metric was given by Hitchin in [9] in terms of aninfinite-dimensional hyperk¨ahler quotient. In this paper we will not use the hyperk¨ahlerstructure; all we need is a practical recipe for computing the metric. In this section wereview that recipe.Let v be tangent to an arc in M , and lift this arc to a family of Higgs bundles ( E t , ϕ t ),equipped with harmonic metrics h t . Identify all the ( E t , h t ) with a fixed C ∞ SU(2)-bundle E . Then we have a family of unitary connections D t on E and 1-forms Φ t ∈ Ω ( su ( E ))which for all t satisfy (2.1). For brevity, let D := D and Φ := Φ denote these objects at t = 0. Differentiating at t = 0 we obtain a pair of 1-forms( ˙ A, ˙Φ) = ( ∂ t D t | t =0 , ∂ t Φ t | t =0 ) ∈ Ω ( su ( E )) . (2.2)Given α ∈ Ω ( su ( E )) we define a nonnegative density | α | on C by | α x d x + α y d y | = − Tr( α x + α y ) d x d y. (2.3)Here z = x + i y is a local conformal coordinate on C . In coordinate-independent terms, thedensity | α | corresponds (using the orientation of C ) to the 2-form − Tr( α ∧ (cid:63)α ), where (cid:63) denotes the Hodge star operator on 1-forms. Now we equip Ω ( su ( E )) with the L metric (cid:107) ( ˙ A, ˙Φ) (cid:107) = (cid:90) C (cid:16) | ˙ A | + | ˙Φ | (cid:17) . (2.4)Let ρ : Ω ( su ( E )) → Ω ( su ( E )) be the linearized gauge map, defined by ρ ( X ) = ( − d D X, [ X, Φ]) . (2.5)We consider the orthogonal decomposition of ( ˙ A, ˙Φ) relative to the image of ρ ,( ˙ A, ˙Φ) = ( ˙ A, ˙Φ) (cid:107) + ( ˙ A, ˙Φ) ⊥ (2.6)with ( ˙ A, ˙Φ) (cid:107) ∈ ρ (Ω ( su ( E )) and ( ˙ A, ˙Φ) ⊥ ∈ ρ (Ω ( su ( E )) ⊥ . Hitchin’s hyperk¨ahler metric g is g ( v, v ) = (cid:107) ( ˙ A, ˙Φ) ⊥ (cid:107) . (2.7)2.4. The Hitchin section.
Fix a spin structure on the compact Riemann surface C . Thespin structure determines a holomorphic line bundle L equipped with an isomorphism L (cid:39) K C , and thus a rank 2 holomorphic vector bundle E = L ⊕ L − . (2.8)This bundle has det E = L ⊗ L − which is canonically trivial. Let B be the space of holo-morphic quadratic differentials on C , B = H ( C, K C ) . (2.9)For each φ ∈ B there is a corresponding Higgs field, ϕ = (cid:18) − φ (cid:19) ∈ H ( C, End
E ⊗ K C ) . (2.10)The Higgs bundles ( E , ϕ ) are all stable, and thus determine a map ι : B → M . The image ι ( B ) ⊂ M is an embedded submanifold, the Hitchin section . Moreover, ι is a holomorphicmap, with respect to the complex structure on M induced from its realization as moduli More precisely there are 2 × genus( C ) Hitchin sections, corresponding to the equivalence classes of spinstructures on C . All of our discussion applies to any of them. SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 5 space of Higgs bundles (which is the complex structure denoted I in [9]). Thus ι ( B ) is acomplex submanifold of M . From now on, by abuse of notation, we identify B with ι ( B ).Our interest in this paper is in the restriction of the hyperk¨ahler metric g from the full T M to T B . This restriction is a K¨ahler metric on B , which we will also denote g .3. Metric estimate
The semiflat metric.
Let B (cid:48) ⊂ B be the locus of quadratic differentials with onlysimple zeros, which is an open and dense set. On B (cid:48) we define an explicit K¨ahler metric g sf as follows. A tangent vector to B can be represented by a quadratic differential ˙ φ . We define g sf φ ( ˙ φ, ˙ φ ) = 2 (cid:90) C | ˙ φ | | φ | . (3.1)Note that the integrand on the right hand side is a smooth density on C \ φ − (0). Thecondition that φ ∈ B (cid:48) implies that this integral is convergent.We remark that g sf is a “(rigid) special K¨ahler” metric on B (cid:48) in the sense of [5]. It doesnot extend to a Riemannian metric on the full B .3.2. Threshold and radius.
Any nonzero quadratic differential φ ∈ B induces a flat metric | φ | on C , which is smooth except for conical singularities at the zeros of φ . From now onwe always use this metric to define geodesics and lengths on C , unless a different metric isexplicitly referenced. A saddle connection of φ is a geodesic segment on C which begins andends on zeros of φ (not necessarily two distinct zeros), and which has no zeros of φ in itsinterior.We define the threshold M : B → R (cid:62) by M ( φ ) = (cid:40) the minimum length of a saddle connection of φ for φ ∈ B (cid:48) , φ ∈ B \ B (cid:48) . (3.2)Then M is continuous and has the homogeneity property M ( tφ ) = t M ( φ ) , t ∈ R + . (3.3)The threshold measures the distance “between zeros” of φ (including the possibility of asegment between a zero and itself). In what follows it will also be important to consider thedistance from an arbitrary point to the zeros of φ . We define the radius function r : C → R of φ by r ( z ) = d ( z, φ − (0)) . (3.4)The main technical estimates that are used in the proof of Theorem 1 are all phrased interms of bounds on various functions on C in terms of the radius.3.3. The estimate.
Now we can state the main result of this paper:
Theorem 1. If φ ∈ B (cid:48) , and ˙ φ is any holomorphic quadratic differential on C , then for any α < M ( φ ) we have | ( g tφ − g sf tφ )( ˙ φ, ˙ φ ) | = O (e − α t (cid:107) ˙ φ (cid:107) ) (3.5)as t → ∞ , where (cid:107) · (cid:107) denotes any norm on the vector space B . Having fixed such a norm,the implicit multiplicative constant in (3.5) can be taken to depend only on α , M ( φ ), andthe genus of C . D. DUMAS AND A. NEITZKE Coordinate computations
Self-duality equation and variation in coordinates.
To set the stage for the proofof Theorem 1 we start by deriving local coordinate expressions for the self-duality equation(2.1) at a point φ ∈ B , and for its first variation in the direction of ( ˙ A, ˙Φ) representing˙ φ ∈ T φ B .In a local conformal coordinate z = x + i y on C we write φ = P ( z ) d z for a holomorphicfunction P . Let d z denote a local section of L satisfying d z ⊗ d z = d z ; there are two suchlocal sections, the choice of which will not matter in the sequel. Using the local trivializationof E = L ⊕ L − given by the frame (d z , d z − ), which we call the holomorphic gauge , wecan write ϕ = (cid:18) − P (cid:19) d z, h = (cid:18) e − u
00 e u (cid:19) , ϕ † = (cid:18) u − e − u P (cid:19) d z, (4.1) D = d + A, A = (cid:18) − ∂u ∂u (cid:19) . (4.2)This diagonal form for h reflects that the splitting L ⊕ L − is orthogonal for the harmonicmetric in this case [9, Theorem 11.2].Then (2.1) reduces to a scalar equation for u ,∆ u − u − e − u | P | ) = 0 , (4.3)where ∆ = 4 ∂ ¯ ∂ is the flat Laplacian.In more invariant terms, (4.3) is an equation for the globally defined metric e u | d z | on C .For Higgs bundles of this type, the Hermitian metric h , the K¨ahler metric e u | d z | , and the(local) scalar function u all contain equivalent information. In most of what follows we workwith u , which unlike h and e u | d z | is a coordinate-dependent quantity: Under a conformalchange of coordinates z (cid:55)→ w it transforms as u (cid:55)→ u − log (cid:12)(cid:12) d w d z (cid:12)(cid:12) . We refer to objects withthis transformation property as log densities . Note that the difference of two log densities isa function. Also, if φ = P d z is a quadratic differential, then log | P | is a log density.When considering the density u on C which corresponds to the unique harmonic metric onthe Higgs bundle associated to φ ∈ B , we sometimes write u ( φ ) to emphasize its dependenceon φ , and to distinguish it from other local solutions to (4.3) on domains in C or in the planethat we consider.Next, we consider a variation ˙ φ ∈ T φ B expressed locally as ˙ φ = ˙ P ( z ) d z . Differentiating(4.3) we find that the corresponding first order variation ˙ u , describing the infinitesimal changein h , satisfies the inhomogeneous linear equation∆ ˙ u − u (e u + e − u | P | ) + 8e − u Re( P ˙ P ) = 0 . (4.4)Unlike u , ˙ u is a well-defined global function on C (independent of the coordinate z ). Sincethe operator ∆ − u + e − u | P | ) is negative definite, (4.4) uniquely determines ˙ u .4.2. Unitary gauge.
In preparation for calculating the L inner product of variations it ismore convenient to work in unitary gauge , expressing the Higgs field and connection relativeto the frame (e u d z , e − u d z − ); then (4.1)-(4.2) become ϕ = (cid:18) − e − u P e u (cid:19) d z, ϕ † = (cid:18) u − e − u P (cid:19) d z, A = i2 (cid:18) (cid:63) d u − (cid:63) d u (cid:19) , (4.5) SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 7 with infinitesimal variations given by˙ ϕ = (cid:18) − u P ˙ u − e − u ˙ P e u ˙ u (cid:19) d z, ˙ ϕ † = (cid:18) u ˙ u e − u P ˙ u − e − u ˙ P (cid:19) d z, (4.6a)˙ A = i2 (cid:18) (cid:63) d ˙ u − (cid:63) d ˙ u (cid:19) , (4.6b)which of course gives a corresponding expression for ˙Φ = ˙ ϕ − ˙ ϕ † .4.3. Orthogonal decomposition.
Let ( A, Φ) be obtained from a solution of the self-dualityequation (2.1). Define the linear map µ = µ ( A, Φ) : Ω ( su ( E , h )) → Ω ( su ( E , h )) by µ ( ˙ A, ˙Φ) = d D (cid:63) ˙ A − [ ˙Φ , (cid:63) Φ] . (4.7)A variation ( ˙ A, ˙Φ) is L -orthogonal to the image of the linearized gauge map ρ if and onlyif it satisfies µ ( ˙ A, ˙Φ) = 0. We say that such a variation is in gauge .For a general variation ( ˙ A, ˙Φ), the orthogonal decomposition of (2.6) is given by( ˙ A, ˙Φ) ⊥ = ( ˙ A, ˙Φ) − ρ ( X ) (4.8)where X ∈ Ω ( su ( E )) satisfies µ ( ρ ( X )) = µ ( ˙ A, ˙Φ).For the specific variation obtained in (4.6) we find that d D (cid:63) ˙ A = 0, and a straightforwardcalculation yields Q := µ ( ˙ A, ˙Φ) = − [ ˙Φ , (cid:63) Φ]= − ϕ z , ˙ ϕ † ¯ z ] + [ ϕ † ¯ z , ˙ ϕ z ])d x d y = − − u ( P ˙ P − P ˙ P ) (cid:18) − (cid:19) d x d y. (4.9)The computation of ( ˙ A, ˙Φ) ⊥ therefore reduces to solving µ ( ρ ( X )) = − d D (cid:63) d D X − [[ X, Φ] , (cid:63) Φ] = Q (4.10)for X . Equation (4.10) implies in particular that X is diagonal and traceless; thus we maywrite X = 12 i v (cid:18) − (cid:19) . (4.11)After so doing, (4.10) becomes a scalar equation for v ,∆ v − v (e u + e − u | P | ) + 8e − u Im( P ˙ P ) = 0 . (4.12)We note the striking similarity between (4.12) and (4.4); in fact, replacing ˙ P → i ˙ P and v → − ˙ u in (4.12) gives exactly (4.4). This suggests that we combine ˙ u (the metric variation)and v (the infinitesimal gauge transformation to put the tangent vector in gauge) into thesingle complex function F = ˙ u − i v, (4.13)which we call the complex variation , which then satisfies the inhomogeneous linear equation (cid:0) ∆ − u + e − u | P | ) (cid:1) F + 8e − u P ˙ P = 0 . (4.14)As with (4.4) above, when working on the entire compact surface C the equation (4.14)uniquely determines the complex function F . We write F ( φ, ˙ φ ) for this unique global solution D. DUMAS AND A. NEITZKE determined by ( φ, ˙ φ ) ∈ T φ B when it is necessary to distinguish it from other local solutionsof the same equation.4.4. Calculating the norm.
Using the calculations above we can now determine an explicitintegral expression for g φ ( ˙ φ, ˙ φ ) in terms of P , ˙ P , u , and F .The first step is to calculate ρ ( X ) in unitary gauge. We find ρ ( X ) = ( B, Ψ) where B = i2 (cid:18) d v − d v (cid:19) , Ψ = ψ − ψ † , ψ = i v (cid:18) − u P e u (cid:19) d z. (4.15)Now ( ˙ A, ˙Φ) ⊥ = ( ˙ A, ˙Φ) − ( B, Ψ) is orthogonal to ( B, Ψ), hence the hyperk¨ahler norm of theassociated tangent vector to the moduli space M is g φ ( ˙ φ, ˙ φ ) = (cid:107) ( ˙ A, ˙Φ) ⊥ (cid:107) = (cid:107) ( ˙ A, ˙Φ) (cid:107) − (cid:107) ( B, Ψ) (cid:107) . (4.16)Now we need only to substitute the expressions for ( ˙ A, ˙Φ) from (4.6) and ( B, Ψ) from (4.15)and simplify. Two observations will be useful in doing this. First, if Ξ = ξ − ξ † ∈ Ω ( su ( E ))where ξ is expressed in unitary gauge as ξ = f ( z )d z for f a matrix-valued function, then | Ξ | = 4 tr( f ¯ f t ) d x d y. (4.17)Second, if β = (cid:18) θ − θ (cid:19) , with θ a scalar 1-form, then we have | β | = | (cid:63)β | = 2 θ ∧ (cid:63)θ. (4.18)Using (4.17) to simplify | ˙Φ | and (4.18) to simplify | ˙ A | , we find (cid:107) ( ˙ A, ˙Φ) (cid:107) = (cid:90) C | ˙ A | + | ˙Φ | = 12 (cid:107) d ˙ u (cid:107) + (cid:90) C (cid:16) e u ˙ u + e − u | P ˙ u − ˙ P | (cid:17) d x d y, (4.19)where (cid:107) θ (cid:107) = (cid:82) C θ ∧ (cid:63)θ . Proceeding similarly for (cid:107) ( B, Ψ) (cid:107) using (4.15), we have (cid:107) ( B, Ψ) (cid:107) = 12 (cid:107) d v (cid:107) + (cid:90) C (cid:0) u + e − u | P | ) v (cid:1) d x d y. (4.20)Subtracting (4.20) from (4.19) we obtain g φ ( ˙ φ, ˙ φ ) = (cid:90) C (cid:16) u ˙ u + 4e − u | P ˙ u − ˙ P | − u + e − u | P | ) v (cid:17) d x d y + 12 (cid:107) d ˙ u (cid:107) − (cid:107) d v (cid:107) . (4.21)Next we integrate by parts on C to replace (cid:107) d ˙ u (cid:107) and (cid:107) d v (cid:107) by − (cid:82) C ( ˙ u ∆ ˙ u )d x d y and − (cid:82) C ( v ∆ v )d x d y respectively, and substitute for ∆ ˙ u and ∆ v using the differential equations(4.3) and (4.12). A few terms cancel and we are left with g φ ( ˙ φ, ˙ φ ) = (cid:90) C (cid:16) − u | ˙ P | − − u ˙ u Re( P ˙ P ) − − u v Im( P ˙ P ) (cid:17) d x d y (4.22)or more compactly, g φ ( ˙ φ, ˙ φ ) = (cid:90) C − u (cid:16) | ˙ P | − Re(
F P ˙ P ) (cid:17) d x d y. (4.23)As a reassuring consistency check, note that g φ is indeed a Hermitian metric, i.e. g φ (i ˙ φ, i ˙ φ ) = g φ ( ˙ φ, ˙ φ ): one sees this easily from (4.23), since changing ˙ φ → i ˙ φ leads to F → i F and ˙ P → i ˙ P . Abusing notation, we often write integrals over C with the integrand expressed in a local coordinate andframe for E . SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 9
The same is not true of (4.19) by itself: it holds only once we subtract the pure gauge part(4.20).To sum up the results of this section, and restate the formula (3.1) for g sf in the samelocal coordinates, we have: Theorem 2.
For any quadratic differential φ ∈ B (cid:48) and tangent vector ˙ φ ∈ T φ B (cid:48) = B withrespective coordinate expressions φ = P ( z ) d z and ˙ φ = ˙ P ( z ) d z , the norm of ˙ φ in thehyperk¨ahler metric g is given by (4.23), where u and F are the solutions of (4.3) and (4.14).The norm of the same tangent vector in the semiflat metric g sf is g sf φ ( ˙ φ, ˙ φ ) = (cid:90) C | P | − | ˙ P | d x d y. (4.24) (cid:3) The goal of the next three sections is to gain some control over the integral expressions(4.23) and (4.24) by studying the behavior of the functions u and F . We will see that thesefunctions are well-approximated by u ≈
12 log | P | ,F ≈
12 ˙
PP , (4.25)at points that are not too close to the zeros of φ . It is easy to check that substituting theseapproximations directly into (4.23) yields exactly the semiflat integral (4.24). Bounding thedifference g − g sf thus reduces to understanding the error in the approximations.5. Exponential decay principle
We now develop a criterion for solutions to certain elliptic PDE on regions in the plane todecay exponentially fast as we move away from the boundary of the region. The method isstandard—combining the maximum principle with the known behavior of the eigenfunctionsof the Laplacian—and the results in this section are surely not new. A similar methodwas used in [12], for example, to derive the exponential decay results for (4.3) that we willgeneralize in § Theorem 3.
Let Ω = {| z | < R } be a disk in C , and for z ∈ Ω let ρ ( z ) = d ( z, ∂ Ω) = R − | z | denote the distance to the boundary of this disk. Suppose that w ∈ C (Ω) ∩ C ( ¯Ω) satisfies(∆ − k ) w = g (5.1)where k, g ∈ C ( ¯Ω), k (cid:62)
4, and suppose that for every γ < A ( γ )such that g obeys the exponential decay condition | g | < A ( γ )e − γρ . (5.2)Then, for any γ <
4, there exist constants K ( γ ) and A (cid:48) ( γ ), such that w obeys the exponentialdecay condition | w | < K ( γ )( M + A (cid:48) ( γ ))e − γρ , (5.3)where M = sup ∂ Ω | w | . Moreover, given any γ (cid:48) > γ , A (cid:48) ( γ ) can be chosen to be equal to A ( γ (cid:48) ).The proof will rely on the following maximum principle. Lemma 4.
Let Ω be a bounded region in C , and let w, v ∈ C (Ω) ∩ C ( ¯Ω) with v (cid:62) w satisfies (∆ − k ) w = g, (5.4)and that v satisfies (∆ − k ) v = − ¯ g, (5.5)where k, g, k, ¯ g ∈ C ( ¯Ω) are functions such that k (cid:62) k > | g | (cid:54) ¯ g. (5.6)If | w | (cid:54) v on ∂ Ω, then | w | (cid:54) v on ¯Ω. Proof.
First we claim w (cid:54) v , or equivalently that v − w (cid:62) v − w achieves its minimum at a point p , and it suffices to show that the minimumvalue is nonnegative. If p ∈ ∂ Ω then this is true by the hypothesis that | w | (cid:54) v on ∂ Ω. If w ( p ) (cid:54) v − w )( p ) (cid:62) v is everywhere nonnegative. Thus the remaining caseis that w ( p ) > p is an interior local minimum of v − w , hence ∆( v − w )( p ) (cid:62)
0. Thenwe find0 (cid:54) ∆( v − w )( p ) = k ( p ) v ( p ) − k ( p ) w ( p ) − ¯ g ( p ) − g ( p ) using (5.4), (5.5) (cid:54) k ( p )( v − w )( p ) − ¯ g ( p ) − g ( p ) because w ( p ) > k (cid:62) k (cid:54) k ( p )( v − w )( p ) because ¯ g + g (cid:62) k > v − w )( p ) (cid:62) − w (cid:54) v . However, this follows by applyingthe argument above to the function w (cid:48) = − w , which satisfies (∆ − k ) w (cid:48) = g (cid:48) , where g (cid:48) = − g .Since | w (cid:48) | = | w | (cid:54) v on ∂ Ω and | g (cid:48) | = | g | (cid:54) ¯ g , the necessary hypotheses still hold in thiscase. (cid:3) In the proof of Theorem 3 we will use Lemma 4 to reduce to the case where k is constantand where g and w are both radially symmetric eigenfunctions of the Laplacian. In prepa-ration for doing this, we recall the properties of those eigenfunctions and relate them to theexponential decay behavior under consideration.The modified Bessel function of the first kind I is the unique positive, even, smoothfunction on R such that ∆ I ( | z | ) = I ( | z | ) (5.7)and I (0) = 1. Thus the function I ( c | z | ) /I ( cR ) is the solution to the Dirichlet problem for(∆ − c ) on the disk | z | < R with unit boundary values.The function I ( x ) satisfies (see e.g. [1, Section 9.7.1]) I ( x ) ∼ (2 πx ) − e x , (5.8)where f ∼ g means that f ( x ) /g ( x ) → x → ∞ . It follows that, if a function f satisfiesan exponential decay condition f < A e − γρ (5.9)for some γ >
0, then for any ˜ γ < γ we have f < ˜ A I (˜ γ | z | ) I (˜ γR ) (5.10) SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 11 for some ˜ A ( A, γ, ˜ γ ) linear in A . Conversely, if we have (5.10) and γ ≤ ˜ γ then we get (5.9)for some A ( ˜ A, γ, ˜ γ ) linear in ˜ A . Proof of Theorem 3.
Suppose we are given constants γ < γ (cid:48) <
4. The function g obeys | g | < A ( γ (cid:48) )e − γ (cid:48) ρ (5.11)and thus | g | < ˜ A I ( γ | z | ) I ( γR ) (5.12)for some ˜ A ( A, γ, γ (cid:48) ) linear in A = A ( γ (cid:48) ). For w satisfying (5.1) and g satisfying (5.12) wewill show that | w | < ˜ K ( M + ˜ A ) I ( γ | z | ) I ( γR ) (5.13)for some ˜ K ( γ ). Once this is achieved we can pass back from (5.13) to the desired (5.3) usingthe exponential bound on I discussed above. Moreover, since ˜ A depends linearly on A ( γ (cid:48) ),we can choose A (cid:48) = A in (5.3), at the cost of possibly rescaling ˜ K .Define v = B I ( γ | z | ) (5.14)for a constant B >
0. Note that (∆ − γ ) v = 0. We will determine a value of B so thatLemma 4 can be applied to v and w on Ω. Specifically, we must ensure that:(i) | w | (cid:54) v on ∂ Ω and(ii) (∆ − v (cid:54) −| g | on Ω,so that, in the notation of Lemma 4, we can take ¯ g = − (∆ − v and k = 4.First we consider (i). The function v is constant on ∂ Ω and equal to
B I ( γR ). Since M = sup ∂ Ω | w | , it suffices to choose B (cid:62) MI ( γR ) . (5.15)Now we turn to (ii). We have (∆ − v = − (16 − γ ) v (5.16)which we have written in this way to emphasize that (16 − γ ) >
0. With the given bound(5.12) on | g | , the desired inequality (ii) follows if B I ( γ | z | )(16 − γ ) (cid:62) ˜ A I ( γ | z | ) I ( γR ) , (5.17)or equivalently B (cid:62) ˜ A − γ I ( γR ) . (5.18)Using (5.15) and (5.18) it is easy to verify that B = max (cid:18) , − γ (cid:19) · ( M + ˜ A ) · I ( γR ) (5.19) Adjusting the constants when converting between exponential and Bessel bounds is necessary due to the x − factor in the expansion of I , with the relevant observation being that x e − γx = O (e − ˜ γx ) for all ˜ γ < γ whereas of course x e − γx (cid:54) = O (e − γx ). satisfies both conditions (i)-(ii), and then by Lemma 4 we find | w | (cid:54) v on D , which is thedesired bound (5.13) with ˜ K = max (cid:18) , − γ (cid:19) . (5.20) (cid:3) Estimates for the density u As in Section 4 above, let u = u ( φ ) be the solution of the self-duality equation (4.3) on thecompact Riemann surface C for a given holomorphic quadratic differential φ ∈ B expressedlocally as φ = P ( z ) d z . It was shown by Minsky in [12] (see also [14, Lemma 2.2]) that u isapproximated by log | P | up to an error that decays exponentially in the distance from thezeros of φ . Building on Minsky’s results (and following a similar outline to [3, Section 5.4]),we establish the following estimate which gives a slightly faster exponential decay rate: Theorem 5.
Fix φ ∈ B and assume M ( φ ) >
1. For any γ <
4, there exist constants A ( γ )and b ( γ ) such that the density u = u ( φ ) satisfies (cid:12)(cid:12)(cid:12)(cid:12) u ( z ) −
12 log | P ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) < A ( γ )e − γr ( z ) (6.1)for all z ∈ C with r ( z ) > b ( γ ). The constants A ( γ ) and b ( γ ) can be taken to depend onlyon γ and the topological type of C .Furthermore, under the same hypotheses we have the C estimate (cid:12)(cid:12)(cid:12)(cid:12) ∇ φ ( u −
12 log | P | )( z ) (cid:12)(cid:12)(cid:12)(cid:12) φ < A ( γ )e − γr ( z ) (6.2)where ∇ φ and | v | φ denote, respectively, the gradient and the norm of a tangent vector withrespect to the metric | φ | .To prove this, we will first establish some rough bounds on u . These will allow us to applyTheorem 3 to the equation satisfied by u − log | P | .6.1. Rough bounds.
Let e σ | d z | be the Poincar´e metric on C of constant (Gaussian)curvature −
4. In general, the Gaussian curvature of a metric e u | d z | is given by K = − e − u ∆ u (see e.g. [2, Section 1.5]); therefore, the equation K = − σ = 4e σ , (6.3)which is equation (4.3) with φ = 0.Now for the solution u = u ( φ ) of (4.3) associated to a general quadratic differential φ = P d z , we have the following lower bounds in terms of σ and P : Lemma 6.
We have u − σ (cid:62) C , and u − log | P | (cid:62) C \ φ − (0). Proof.
Using (4.3) and (6.3) we have∆( u − σ ) = 4(e u − e σ − e − u | P | ) (cid:54) u − e σ ) . (6.4)At a minimum of u − σ , we have ∆( u − σ ) (cid:62) u − e σ (cid:62) u − σ (cid:62) C . SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 13
The lower bound on u − log | P | is similar. Using that log | P | is harmonic on C \ φ − (0),we find that ∆( u −
12 log | P | ) = 4(e u − e − u | P | ) . (6.5)Since log | P ( z ) | → −∞ as z approaches a zero of φ , while u is smooth on the entire surface C , the difference u − log | P | has a minimum on C \ φ − (0). At such a minimum we have∆( u − log | P | ) (cid:62)
0, which gives e u − e − u | P | (cid:62)
0. This implies u − log | P | (cid:62) (cid:3) Complementing these lower bounds on u , we have the following rough comparison to thesingular flat metric | φ | . Recall that the radius r ( z ) is the distance from z to φ − (0) withrespect to the metric | φ | . Lemma 7 (Minsky [12, Lemma 3.2] ). Fix φ ∈ B and assume M ( φ ) >
1. Let z ∈ C be apoint with r ( z ) (cid:62)
1. Then u ( z ) − log | P ( z ) | (cid:54) M where M is a constant depending onlyon the topological type of C . (cid:3) Note that Minsky’s bound is more general, giving an upper bound at any point z dependingonly on the topological type of C and on the | φ | -radius R of an embedded disk centered at z that contains no zeros of φ . The hypotheses of the lemma above give such a disk of definiteradius (in fact, one can take R = ), resulting in the bound stated above that only dependson the topological type.6.2. Exponential bounds.
Proof of Theorem 5.
We start with the C bound (6.1). Consider a local coordinate ζ about z in which φ = d ζ . Allowing this coordinate chart to be immersed, rather than embedded,we can take it to be defined on | ζ | < r ( z ) with z corresponding to ζ = 0. While the boundaryof this disk touches the zero set of φ (by definition of r ), if we consider D = {| ζ | < r ( z ) − } then the image of this disk in C consists of points satisfying the hypotheses of Lemma 7.Therefore, by Lemma 6 and Lemma 7 we have 0 (cid:54) u ( ζ ) (cid:54) M for all ζ ∈ D .In this coordinate system we have P ( ζ ) ≡ u = 4e u − − u = 8 sinh(2 u ) . (6.6)Similarly, in this coordinate the difference | u − log | P || reduces to | u | .The function 8 sinh(2 x ) /x has a removable singularity at x = 0; let f denote its extensionto a smooth function on R , which satisfies f ( x ) (cid:62)
16 for x (cid:62)
0. Since u (cid:62) − k ) u = 0 (6.7)where k = (cid:112) f ( u ), and thus k (cid:62)
4. Now Theorem 3 applies to u on D with g ≡ ρ = r ( z ) −
1, giving | u | (cid:54) A ( γ )e − γ ( r ( z ) − (6.8)for all γ <
4. Absorbing the e γ factor into the multiplicative constant we obtain the desiredbound (6.1) in terms of e − γr ( z ) .Given this C bound, the corresponding C bound (6.2) follows by standard elliptic theoryapplied to (6.6), as shown in e.g. [3, Corollary 5.10]. (cid:3) Estimates for the complex variation F Next we turn to the complex variation F = F ( φ, ˙ φ ) associated to φ ∈ B (cid:48) and ˙ φ ∈ T φ B .We will see that this function is exponentially close to
12 ˙ φφ =
12 ˙ PP . Specifically, we have: Theorem 8.
Fix φ ∈ B and assume M ( φ ) >
1. Also fix ˙ φ . For any γ <
4, there existconstants A ( γ ) and b ( γ ) such that the function F ( φ, ˙ φ ) satisfies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F −
12 ˙ φφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < A ( γ ) (cid:107) ˙ φ (cid:107) e − γr ( z ) (7.1)for all z ∈ C with r ( z ) > b ( γ ).To prove this, we proceed as in §
6, first deriving some rough bounds, and then improvingthem to exponential bounds using Theorem 3.7.1.
Rough bounds.
We begin with some notation related to metrics on C . If η is a logdensity on C , with associated K¨ahler metric e η | d z | , and if φ ∈ B has local expression φ = P d z , we denote by | φ | η = e − η | P | : C → R (7.2)the pointwise norm function of φ with respect to this metric, and by (cid:107) φ (cid:107) η = sup C | φ | η (7.3)the associated sup-norm. Finally, we let∆ η = e − η ∆ (7.4)denote the Laplace-Beltrami operator of the metric e η | d z | .Recall from § σ denotes the density of the Poincar´e metric on C of curvature − Lemma 9.
The complex variation F satisfies sup | F | (cid:54) (cid:107) ˙ φ (cid:107) σ . Proof.
Rewriting (4.14) in terms of the Laplace-Beltrami operator ∆ u it becomes:(∆ u − K ) F = − G, (7.5)where K = 8(1 + | φ | u ) ,G = 8 ¯ φ ˙ φ e u | d z | . (7.6)Note that K > G is a well-defined complex scalar function on C which satisfies | G | = 8 | φ | u | ˙ φ | u . (7.7)Considering a maximum and a minimum of each of the real and imaginary parts of F , whichexist by compactness, we find from (7.5) thatsup | F | (cid:54) sup 2 K − | G | , (7.8)and we have K − | G | = | ˙ φ | u (cid:18) | φ | u | φ | u (cid:19) (cid:54) | ˙ φ | u . (7.9) SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 15
Finally, by Lemma 6, we have u (cid:62) σ . Therefore | ˙ φ | u (cid:54) | ˙ φ | σ andsup | F | (cid:54) sup 2 K − | G | (cid:54) sup | ˙ φ | σ = (cid:107) ˙ φ (cid:107) σ . (7.10) (cid:3) We will also need the following lower bound on the pointwise norm | φ | σ . Recall r ( z )denotes the | φ | -distance from z to φ − (0). Lemma 10.
Let φ ∈ B (cid:48) and suppose (cid:107) φ (cid:107) σ (cid:62)
1. There exists a constant δ depending onlyon the ray R + φ with the following property: If z ∈ C satisfies r ( z ) >
1, then | φ | σ ( z ) (cid:62) δ . Proof.
Let Z = φ − (0). First suppose that (cid:107) φ (cid:107) σ = 1. For any positive radius r , a uniformlower bound on | φ | σ ( z ) for z with r ( z ) (cid:62) r follows immediately from compactness of C andof the unit ball in B .Using that d tφ = t / d φ , the same argument shows that for (cid:107) φ (cid:107) σ (cid:62) | φ | σ ( z ) when r ( z ) is greater than a fixed positive multiple of (cid:107) φ (cid:107) / σ .Thus to complete the argument it suffices to consider the case when r ( z ) is small comparedto (cid:107) φ (cid:107) / σ . That is, we consider a point z in a disk of radius (cid:15) (cid:107) φ (cid:107) / σ about one of the zeros.Equivalently, if we let φ = (cid:107) φ (cid:107) − σ φ , then z lies in a disk of φ -radius (cid:15) about a zero of φ .Assume that (cid:15) is small enough (depending on the ray) so that there is only one zero of φ inthis disk, and that the disk is identified with | ζ | < R by a coordinate function ζ such that φ = ζ d ζ . We work in this coordinate system for the rest of the proof.Write the Poincar´e metric of C on this disk as e σ | d ζ | . Then, using compactness of theunit ball in B again, we have e σ (cid:54) M for a uniform constant M .The φ -distance from 0 to ζ is proportional to | ζ | / , and thus the φ -distance is propor-tional to (cid:107) φ (cid:107) / σ | ζ | / , with universal constants in both cases. The hypothesis that r ( z ) (cid:62) | ζ ( z ) | (cid:62) c (cid:107) φ (cid:107) − / σ for a constant c >
0. Using that φ = (cid:107) φ (cid:107) σ ζ d ζ , at sucha point z we have | φ | σ ( z ) = e − σ ( z ) (cid:107) φ (cid:107) σ | ζ ( z ) | (cid:62) M − (cid:107) φ (cid:107) σ ( c (cid:107) φ (cid:107) − / σ ) = M − c (cid:107) φ (cid:107) / σ (7.11)Since we assumed (cid:107) φ (cid:107) σ (cid:62)
1, this gives the desired lower bound with δ = M − c . (cid:3) Exponential bounds.
Proof of Theorem 8.
Define f = F −
12 ˙ φφ , (7.12)so that our goal is to give an exponentially decaying upper bound on | f | at a point z . Asin the proof of Theorem 5 we first choose an immersed coordinate chart | ζ | < r ( z ) where φ = d ζ and z corresponds to ζ = 0.Using (4.14), after a bit of algebra we find that in this coordinate system f satisfies theequation (∆ −
16 cosh(2 u )) f = − φφ sinh(2 u ) . (7.13)Let (cid:15) >
0, and restrict attention to the smaller disk Ω = {| ζ | (cid:54) (1 − (cid:15) ) r ( z ) } . Assume that r ( z ) > (cid:15) − . Then all points of Ω are at distance at least 1 from φ − (0), and Lemma 10 gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ φφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | ˙ φ | σ | φ | σ (cid:54) δ − (cid:107) ˙ φ (cid:107) σ (7.14) throughout Ω. Now fix γ <
4, let b ( γ ) denote the constant from Theorem 5, and assumethat r ( z ) > (cid:15) − b ( γ ). Then Theorem 5 applies to u at each point of Ω, giving | u ( ζ ) | < A ( γ )e − γr ( ζ ) . (7.15)Combining this with the bound (7.14) on ˙ φφ , we find that the right hand side of (7.13) isbounded above by A (cid:48) ( γ ) (cid:107) ˙ φ (cid:107) σ e − γr ( ζ ) (7.16)for some constant A (cid:48) ( γ ).Let ρ denote the function on Ω that gives the φ -distance to ∂ Ω. Since r ( z ) > ρ ( z ) we canreplace (7.16) by A (cid:48) ( γ ) (cid:107) ˙ φ (cid:107) σ e − γρ . (7.17)When combined with the fact that 16 cosh(2 u ) (cid:62)
16, this exponential decay of the inhomo-geneous term of (7.13) implies that the solution f is also exponentially decaying; specifically,applying Theorem 3 we have for any γ < | f | (cid:54) K ( γ )( M + A (cid:48)(cid:48) ( γ ) (cid:107) ˙ φ (cid:107) σ )e − γρ (7.18)where M = sup ∂ Ω | f | . We have | f | (cid:54) | F | + (cid:12)(cid:12)(cid:12) ˙ φφ (cid:12)(cid:12)(cid:12) , and therefore Lemma 9 and (7.14) give M (cid:54) (1 + δ − ) (cid:107) ˙ φ (cid:107) σ . Substituting this value for M into (7.18) and evaluating at ζ = 0 (i.e. at z ), where ρ = (1 − (cid:15) ) r ( z ), we obtain | f ( z ) | (cid:54) A (cid:48)(cid:48)(cid:48) ( γ ) (cid:107) ˙ φ (cid:107) σ e − γ (1 − (cid:15) ) r ( z ) (7.19)for a constant A (cid:48)(cid:48)(cid:48) ( γ ) depending on the ray R + φ . Since (cid:15) was arbitrary, this gives the desiredbound. (cid:3) Holomorphic variations
In our analysis of C near we will exploit the following basic observation: if D ⊂ C is adisk containing exactly one zero of φ , then any holomorphic quadratic differential ˙ φ on D can be realized as ˙ φ = L X φ for some holomorphic vector field X on D . This fact allows usto construct an explicit solution of the complex variation equation (4.14) on D , using thefollowing: Theorem 11.
Given a quadratic differential φ = P ( z ) d z , solution u of (4.3), and holomor-phic vector field X = χ ( z ) ∂∂z , let ψ X = L X φ = ( χP z + 2 χ z P ) d z (8.1)and define a complex scalar function F X by F X e u | d z | = L X (e u | d z | ) (8.2)or equivalently in local coordinates F X = χ z + 2 χu z . (8.3)Then F = F X satisfies the complex variation equation (4.14) with ˙ φ = ψ X . SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 17
Proof.
We begin by noting that the self-duality equation (4.3) is natural with respect tobiholomorphic maps, i.e. if Φ is such a map then the log density of the pullback metricΦ ∗ (e u | d z | ) satisfies the equation for the pullback differential Φ ∗ φ . The real vector field X + ¯ X has a local flow which consists of holomorphic maps, and hence gives rise to a local 1-parameter family of solutions for the corresponding family of pullback quadratic differentials.Taking the derivative of this family of solutions at t = 0 we find that the Lie derivative ofe u | d z | with respect to X + ¯ X gives a solution of the variation equation (4.4) for˙ φ = L X + ¯ X φ. (8.4)Specifically, if we define ˙ u by ˙ u e u | d z | = L X + ¯ X (e u | d z | ) (8.5)then ˙ u and ˙ φ satisfy (4.4). The expression (8.5) is equivalent to saying that ˙ u is the Rie-mannian divergence of the vector field X + ¯ X with respect to the metric e u | d z | .Now, recall that (4.14) is equivalent to the separate equations (4.4) for ˙ u = Re( F ) and(4.12) for v = − Im( F ), and that these two equations are related by the substitutions ˙ u → − v and ˙ φ → i ˙ φ .For a real tensor T we have Re( L X T ) = L X + ¯ X T , and hence Re( F X ) is exactly ˙ u as definedby (8.5), which we have seen satisfies (4.4) with ˙ φ = L X + ¯ X φ . Because φ is holomorphic wein fact have L X + ¯ X φ = L X φ = ψ X . Hence Re( F X ) satisfies the desired equation.Because L i X = i L X we have − Im( F X ) = Re( F i X ) which therefore satisfies (4.4) with˙ φ = L i X + ¯ iX φ = L i X φ = i ψ X . Using the substitutions noted above, this is equivalent toIm( F X ) satisfying (4.12). (cid:3) Exactness
Given quadratic differentials φ and ˙ φ on a compact Riemann surface C , recall that ourultimate goal is to bound the difference∆( φ, ˙ φ ) := g φ ( ˙ φ, ˙ φ ) − g sf φ ( ˙ φ, ˙ φ ) . (9.1)Though the integrals defining g φ and g sf φ were previously written in terms of densities (scalarmultiples of d x d y ), using the orientation of C we can convert the integrand to a differential2-form which we denote by δ . Also recall that this integrand depends on the density u and complex function F , respectively satisfying (4.3) and (4.14). Explicitly, by taking thedifference of the integral expressions (4.23)-(4.24) we find δ ( φ, ˙ φ, u, F ) = (cid:32) − u ( | ˙ P | − Re(
F P ˙ P )) − | ˙ P | | P | (cid:33) d x ∧ d y, (9.2)where ˙ φ = ˙ P d z and as usual φ = P d z . Thus if u ( φ ) and F ( φ, ˙ φ ) denote the uniquesolutions to (4.3) and (4.14) on a compact Riemann surface C for given φ and ˙ φ , then wehave ∆( φ, ˙ φ ) = (cid:90) C δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) . (9.3)As mentioned in § δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ ))over the region C near near the zeros of φ involves approximating δ in that region by an exactform. The key to this approximation is that δ ( φ, ˙ φ, u, F ) itself is exact whenever ˙ φ and F are obtained from φ and u using a holomorphic vector field as in Theorem 11: Lemma 12.
Let φ = P d z be a quadratic differential and u a log density satisfying (4.3),both on a domain U ⊂ C . Let X = χ∂ z be a holomorphic vector field on U . Let ˙ φ = L X φ and F = F X as in Theorem 11. Then δ ( φ, ˙ φ, u, F ) = d β , where β = (cid:0) e − u − | P | − (cid:1) (cid:0) | P | (cid:63) d | χ | + | χ | (cid:63) d | P | (cid:1) . (9.4) Proof.
Substituting ˙ φ = L X φ as given by (8.1) and F = F X from (8.2) into (9.2), we obtainan explicit formula in terms of χ , P , and u : δ ( φ, L χ φ, u, F X ) = (cid:32) − u (cid:16) | χP z | + 2 | χ z P | + 3 Re( χ ¯ χ ¯ z P z ¯ P ) − χ ¯ χP ¯ P ¯ z u z ) − χ ¯ χ ¯ z P ¯ P u z ) (cid:17) − | χP z | | P | − χ ¯ χ ¯ z ¯ P P z ) | P | − | χ z | | P | (cid:33) d x ∧ d y. (9.5)Now we consider β . For a holomorphic function f , we have (cid:63) d | f | = (cid:63) d( f ¯ f ) = (cid:63) (cid:0) f z ¯ f d z + f ¯ f ¯ z d¯ z (cid:1) = − i (cid:0) f z ¯ f d z − f ¯ f ¯ z d¯ z (cid:1) = 2 Im( f z ¯ f d z ) . (9.6)Using this, we find that β = 2 Im( ˜ β ) where˜ β = (e − u − | P | − ) (cid:0) | P | χ z ¯ χ + | χ | P z ¯ P (cid:1) d z. (9.7)It is then straightforward to calculate d β = 2 Im( ¯ ∂ ˜ β ) in terms of P and χ , and to verify thatit is equal to (9.5); in the latter step, it is useful to recall Im( c d¯ z ∧ d z ) = 2 Re( c ) d x ∧ d y forany complex scalar c . We omit the details of this lengthy but elementary calculation. (cid:3) Exponential asymptotics
In this section we prove Theorem 1. To do so we return to considering a compact Riemannsurface C and the ray { φ = tφ } t ∈ R + generated by φ ∈ B (cid:48) . Write φ = P d z . Let ˙ φ =˙ P d z ∈ T φ B (cid:48) = B . Fix some γ < z , . . . , z n denote the zeros of φ , and let D i denote an open disk centered on z i of | φ | -radius R = 12 M ( φ ) = 12 t M ( φ ) . (10.1)This is the largest φ -radius for which the sets D i are disjoint, embedded disks. Note that D i can also be described as the disk about z i of | φ | -radius M ( φ ), and in particular the set D i is independent of t .Since we are considering asymptotic statements as t → ∞ , and since by hypothesis M ( φ ) >
0, we may assume when necessary that R is larger than any given constant.Let C near = (cid:83) i D i and C far = C \ C near . Then we have∆( φ, ˙ φ ) = (cid:90) C far δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) + (cid:88) i (cid:90) D i δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) , (10.2)and we will bound these terms separately. The “far” region.
For any z ∈ C far we have r ( z ) (cid:62) R . Assume R is large enough so thatTheorem 5 and Theorem 8 apply. Then we have u ≈ log | P | and F ≈
12 ˙ PP with respectiveerrors bounded by a ( γ )e − γR and a ( γ ) (cid:107) ˙ φ (cid:107) σ e − γR for some constant a ( γ ). If these approximate SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 19 equalities were exact, then δ would vanish identically; that is, by direct substitution into thedefinition (9.2), we find that δ (cid:32) φ, ˙ φ,
12 log | P | ,
12 ˙ PP (cid:33) = 0 . (10.3)To handle the situation at hand, we will strengthen this to show that δ is pointwise smallwhen u and F are only near log | P | and
12 ˙ PP (respectively).Again by substitution into (9.2), we find that for any scalar functions w and µ we have δ (cid:32) φ, ˙ φ,
12 log | P | + w,
12 ˙ PP + µ (cid:33) = (cid:32) | ˙ P | | P | (e − w − − − w | P | Re( P ¯˙ P µ ) (cid:33) d x ∧ d y. (10.4)Now assume that | w | <
1, so that 2 | e − w − | (cid:54) c | w | (10.5)and (cid:12)(cid:12)(cid:12) − w | P | − Re( P ¯˙ P µ ) (cid:12)(cid:12)(cid:12) (cid:54) c | ˙ P || µ | (10.6)for a constant c >
0. Using these estimates with (10.4) gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ (cid:16) φ, ˙ φ, log | P | + w,
12 ˙ PP + µ (cid:17) d x ∧ d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) c (cid:32) | ˙ P | | P | | w | + | ˙ P || µ | (cid:33) . (10.7)If we furthermore assume R >
1, then Lemma 10 applies to φ throughout C far , giving auniform lower bound on e − σ | P | . Substituting this into the previous bound, we can nowbound δ relative to the hyperbolic area form as follows: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ (cid:32) φ, ˙ φ,
12 log | P | + w,
12 ˙ PP + µ (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) c (cid:16) | ˙ φ | σ | w | + | ˙ φ | σ | µ | (cid:17) e σ | d z | . (10.8)Here c is a constant, but not the same constant as in (10.7). We already observed thaton C far the integrand δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) has the form (10.4) with | w | (cid:54) a ( γ )e − γR and | µ | (cid:54) a ( γ ) (cid:107) ˙ φ (cid:107) σ e − γR . Thus (cid:12)(cid:12)(cid:12) δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) (cid:12)(cid:12)(cid:12) (cid:54) c (cid:48) ( γ ) | φ | σ e − γR e σ | d z | on C far , (10.9)for a constant c (cid:48) ( γ ). Integrating (10.9), and using that the σ -area of C far is bounded and | ˙ φ | σ (cid:54) (cid:107) ˙ φ (cid:107) σ , we obtain (cid:90) C far δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) = O (cid:16) (cid:107) ˙ φ (cid:107) σ e − γR (cid:17) (10.10)with the implicit constant depending only on c (cid:48) ( γ ) from (10.9). The “near” region.
Next we consider the integral over one of the disks D i in (10.2).Identify D i with a disk {| z | < R } in C , using a coordinate z in which φ | D i = z d z .On D i there is a unique holomorphic vector field X = χ∂ z such that ˙ φ = L X φ = L X ( z d z );explicitly, if we write ˙ φ = (cid:88) n a n z n d z , (10.11) then χ = (cid:88) n a n n + 1 z n . (10.12)By Theorem 11, the associated function F X defined by F X e u ( φ ) | d z | = L X (e u ( φ ) | d z | ) sat-isfies (4.14) on D i , which is the same equation satisfied by F ( φ, ˙ φ ). We will show that F X and F ( φ, ˙ φ ) are in fact exponentially close on D i .First we consider the restrictions of these functions to ∂D i , which is far from the zeros of φ , allowing the estimates of the previous sections to be applied. By Theorem 8 we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( φ, ˙ φ ) −
12 ˙ PP (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( (cid:107) ˙ φ (cid:107) σ e − γR ) on ∂D i . (10.13)Turning to F X = χ z + 2 χu z , note that the equation ˙ φ = L X φ gives χ z = ˙ P − χP z P = ˙ P P − χ∂ z (log | P | ) (10.14)and thus F X = χ z + 2 χu z = ˙ P P + 2 χ∂ z (cid:18) u −
12 log | P | (cid:19) . (10.15)By the C bound from Theorem 5 we have ∂ z (cid:18) u −
12 log | P | (cid:19) = O (e − γR ) on ∂D i . (10.16)Next we need a bound on | χ | on ∂D i . For this, note that for any z ∈ D i , χ ( z ) dependslinearly on ˙ φ , and scales as t − . Thus, for t > | χ ( z ) | < c ( z ) (cid:107) ˙ φ (cid:107) σ forsome c ( z ), and since the closure of D i is compact we can take this constant to be independentof z , i.e. on D i we have | χ | = O ( (cid:107) ˙ φ (cid:107) σ ) . (10.17)Now combining (10.16) and (10.17), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F X −
12 ˙ PP (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( (cid:107) ˙ φ (cid:107) σ e − γR ) on ∂D i . (10.18)Then by (10.13) and (10.18) we find that the function µ : D i → C defined by µ = F X − F ( φ, ˙ φ ) (10.19)satisfies µ = O ( (cid:107) ˙ φ (cid:107) σ e − γR ) on ∂D i . (10.20)Because F X and F ( φ, ˙ φ ) both satisfy the linear inhomogeneous equation (4.14), their dif-ference µ satisfies the associated homogeneous equation, which has the form (∆ − k ) µ = 0for an everywhere positive function k (compare (7.5)); this implies that | µ | has no interiormaximum. Thus | µ | achieves its maximum on ∂D i , and (10.20) gives µ = O ( (cid:107) ˙ φ (cid:107) σ e − γR ) on D i . (10.21) SYMPTOTICS OF HITCHIN’S METRIC ON THE HITCHIN SECTION 21
Next we use this estimate on µ to estimate the integral of δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) over D i .We have δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) = δ ( φ, ˙ φ, u ( φ ) , F X + µ )= δ ( φ, ˙ φ, u ( φ ) , F X ) + 4e − u ( φ ) Re( µP ¯˙ P ) d x ∧ d y. (10.22)Since e − u ( φ ) | P | (cid:54) | − u ( φ ) Re( µP ¯˙ P ) | (cid:54) | µ || ˙ P | (10.23)and the bound (10.21) gives (cid:90) D i − u ( φ ) Re( µP ¯˙ P ) d x ∧ d y = O ( (cid:107) ˙ φ (cid:107) σ e − γR ) . (10.24)Considering the other term on the right hand side of (10.22), by Lemma 12 the form δ ( φ, ˙ φ, u, F X ) is exact, so we can use Stokes’s theorem to reduce to a boundary term: (cid:90) D i δ ( φ, ˙ φ, u, F X ) = (cid:90) ∂D i (cid:0) e − u − | P | − (cid:1) (cid:0) | P | (cid:63) d | χ | + | χ | (cid:63) d | P | (cid:1) . (10.25)It just remains to show that this boundary term is exponentially small. Fix some γ (cid:48) with γ < γ (cid:48) <
4. By Theorem 5, we have (e − u − | P | − ) = O (e − γ (cid:48) R ) on ∂D i . Next, using theestimate (10.17), the fact that P scales as t , and the fact that the coordinate radius of D i scales as t / , we have (cid:90) ∂D i (cid:0) | P | (cid:63) d | χ | + | χ | (cid:63) d | P | (cid:1) = O ( t (cid:107) ˙ φ (cid:107) σ ) . (10.26)Using this in (10.25) gives (cid:90) D i δ ( φ, ˙ φ, u, F X ) = O ( t (cid:107) ˙ φ (cid:107) σ e − γ (cid:48) R ) = O ( (cid:107) ˙ φ (cid:107) σ e − γR ) , (10.27)where in the last equality we use the fact that R → ∞ as t → ∞ by (10.1).Now we have bounded the integrals of both terms in (10.22); combining these bounds weconclude (cid:90) D i δ ( φ, ˙ φ, u ( φ ) , F ( φ, ˙ φ )) = O ( (cid:107) ˙ φ (cid:107) σ e − γR ) . (10.28) Summing up.
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Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicago, IL [email protected]