Harmonic Maps into the Model Space for theWeil-Petersson Metric

Georgios Daskalopoulos1

Brown Universitydaskal@math.brown.edu

Chikako Mese2

Johns Hopkins Universitycmese@math.jhu.edu

Last Revision: September 12, 2013

Abstract

We investigate the behavior of harmonic maps into the met-ric completion of Teichmuller space with respect to the Weil-Petersson metric. In this paper, we study harmonic maps from asmooth Riemannian domain to a space modeling the normal spaceto a boundary strata. The main result is that the singular set ofa harmonic map into this model space is of Hausdorff codimen-sion 2. This regularity result will be used in the sequel [DMW] tostudy rigidity properties of Teichmuller space.

1 Introduction

Let Tg,n denote the Teichmuller space of a genus g Riemann surfaceswith n punctures and 3g−3 +n > 0. Endowed with the Weil-Peterssonmetric, Tg,n is a smooth Riemannian manifold of non-positive sectionalcurvature. In fact, Tg,n is a Kahler manifold whose curvature tensor isstrongly negative in the sense of Siu (cf. [Siu]). The curvature conditionsuggests that one can apply harmonic maps theory to study Tg,n. Theobstacle though is that Tg,n is not metrically complete. To address thisissue, one can take the metric completion T g,n of Tg,n. Though T g,nis no longer a Riemannian manifold, Yamada [Ya1] observed that it isan NPC space (complete metric space of non-positive curvature in thesense of Alexandrov). Thus, the study of harmonic maps into T g,n fitsin the framework of the harmonic map theory developed in [KS1] and

1supported by research grant from the Simons Foundation2supported by research grant NSF DMS-1105599

1

[KS2]. Using this, we study rigidity problems for Teichmuller space andits mapping class group by the following program:

1. Construct an equivariant harmonic map u : M → T g,n from asmooth Riemannian manifold into the Weil-Petersson completionof Teichmuller space.

2. Show that u maps most points of M to Tg,n ⊂ T g,n. (More specif-ically, one needs to prove that u maps into Tn,g except on a set ofHausdorff dimension 2.)

3. Use a Bochner’s formula to prove rigidity.

Step 1 follows from [DW]. Step 3, addressed in detail in [DMW], canbe accomplished using Schumacher’s calculation (cf. [Sch],[Wo]) of thecomplexified curvature of the Weil-Petersson metric along with Bochnerformulas (cf. [Siu], [JoYa], [MSY]).

The purpose of this paper is to develop the the necessary tools toaddress the main step 2. More precisely, we want to understand thebehavior of harmonic maps into a neighborhood the boundary ∂Tg,n =T g,n\Tg,n.

The boundary ∂Tg,n is stratified by lower dimensional Teichmullerspaces with each stratum being totally geodesic. Furthermore, in aneighborhood N of a boundary point, the Weil-Petersson metric can beapproximated up to higher order by a simpler model space. Indeed, Nis asymptotically a product U×V (cf. [Ya2], [DW], [Wo], [Wol2], [DM3])where the smooth manifold U is an open subset of a lower dimensionalTeichmuller space along with the Weil-Petersson metric and V is anopen subset of Y = H × . . . × H where H (referred to as the modelspace) is the metric completion of the half-plane

H = (ρ, φ) ∈ R2 : ρ > 0

with respect to the metric ds2 = 4dρ2 + ρ6dφ2. The metric completionis constructed by identifying the axis ρ = 0 to a single point P0 andsetting

H = H ∪ P0.

The major difficulty in studying harmonic maps into T g,n is thatthe target space is not locally compact. Specifically, a geodesic ballcentered at a boundary point of T g,n is not compact. This property is

2

captured by the model space H at the point P0. Indeed, the geodesicball of radius r centered at P0,

Br(P0) = (ρ, φ) ∈ Rn : 0 < ρ < r ∪ P0 ⊂ H,

is not compact. The crucial ingredient in understanding the behaviorof harmonic maps into T g,n is understanding the behavior of harmonicmaps into H. The main result of this paper is the following.

Theorem 1 (Main Theorem) If u : Ω→ H is a harmonic map froma Riemannian domain, then u maps to the interior H of H except on aset of Hausdorff codimension 2.

In our previous paper [DM1], we analyzed harmonic maps into

H2 = H+ tH−/ ∼, (1)

where H+ and H− denotes two distinct copies of H and ∼ indicatesthat the point P0 from each copy is identified as a single point. The keyto proving Theorem 1 is to show that a harmonic map into H can beclosely approximated by a harmonic map into H2. Indeed, we introduceof a new coordinate system (%, ϕ) in H and show that the harmonic mapequations of a map into H with respect to the coordinates (%, ϕ) are closeto the harmonic map equation H2 with respect to coordinates (ρ, φ) (cf.Section 4).

In the sequel [DMW], we use Theorem 1 to prove that a harmonicmap u : BR(x0) → T g,n maps to the interior Tg,n of T g,n except ona set of Hausdorff codimension 2. Indeed, using the coordinates onthe neighborhood N of the boundary, we can write u = (V, v) whereV : B1(0)→ R2j and v = (v1, . . . , vm) : B1(0)→ H×. . .×H. The mapsV , v1, . . . , vm are not harmonic maps, but they are only approximatelyharmonic since the Weil-Petersson metric and the Weil-Petersson con-nection are asymptotically a product near the boundary. Applying thetheory of approximately harmonic maps that we developed in [DM4] andcombining it with Theorem 1, we complete the program outlined aboveto prove rigidity statements for Teichmuller space and its mapping classgroup in [DMW].

3

2 Preliminaries

For the sake of simplicity, we replace the metric for the model space by

gH(ρ, φ) =

(1 00 ρ6

). (2)

Note that 4dr2 + r6dθ2 is isometric dρ2 + ρ6dφ2 via the change of co-ordinates ρ = 2r, φ = θ

8 . The Christoffel symbols with respect to thismetric is

Γρρρ = 0 Γφφφ = 0

Γρρφ = 0 Γφρφ =3ρ

Γρφφ = −3ρ5 Γφρρ = 0.

(3)

Let dH be the distance function induced by the metric gH. The metriccompletion is constructed by identifying the axis ρ = 0 to a single pointP0 and setting

H = H ∪ P0.

The distance function dH induced by gH is extended to H by settingdH(Q,P0) = ρ for Q = (ρ, φ) ∈ H. The following facts are easy to check(cf. [DW]):

(1) The Riemann surface (H, gH) is geodesically convex.(2) The complete metric space (H, d) is an NPC space.(3) The space (H, d) is not locally compact.

For a map v : (Ω, g)→ (H, d) from a bounded Riemannian domain,let the function |∇v|2 be the energy density as defined in [KS1]. Theenergy of v is

Ev =∫

Ω|∇v|2dµ.

The map v is said to be harmonic if it is energy minimizing with respectto all finite energy maps with the same trace (cf. [KS1]). A harmonicmap u : (Ω, g) → (H, d) have the following important monotonicityformula. Given x0 ∈ Ω and σ > 0 such that Bσ(x) ⊂ Ω, identify x0 = 0via normal coordinates and let

Eu(σ) :=∫Bσ(0)

|∇u|2dµ and Iu(σ) :=∫∂Bσ(0)

d2(u, u(x))dΣ.

4

There exists a constant c > 0 depending only on the C2 norm of themetric on g (with c = 0 when g is the standard Euclidean metric) suchthat

σ 7→ ecσ2 σ Eu(σ)Iu(σ)

is non-decreasing. As a non-increasing limit of continuous functions,

Ordu(x0) := limσ→0

ecσ2 σ Eu(σ)Iu(σ)

is an upper semicontinuous function and Ordu(x0) ≥ 1. The valueOrdu(x0) is called the order of u at x0. (See Section 1.2 of [GS] with[KS1] and [KS2] justify various technical steps.) The following theoremsays that most point are order 1 points.

Theorem 2 ([DM2] Theorem 16) If u : (Ω, g) → (H,d) is a har-monic map from a Riemannian domain, then

dimH(S0(u)) ≤ n− 2

whereS0(u) = x ∈ B1(0) : Ordu(x) > 1.

The homogeneous cordinates (ρ,Φ) of H is defined by setting

Φ = ρ3φ.

It can be easily seen that the metric gH is invariant under the scaling

ρ→ λρ, Φ→ λΦ.

Thus, the distance function of H is homogeneous of degree 1 under thisscaling. More precisely, for P given by (ρ,Φ) in homogeneous coor-dinates if P 6= P0 and λ ∈ (0,∞), we denote by λP the point givenby

(λρ, λΦ) and λP0 = P0.

Thend(λP, λQ) = λd(P,Q).

We note that in the original coordinates (ρ, φ) of H, we have

λ(ρ, φ) = (λρ,1λ2φ). (4)

5

Given a harmonic map u : (Ω, g) → (H, d), the homogeneous coordi-nates can be used to define blow up maps of u at x0 ∈ Ω. More precisely,we write

u = (uρ, uΦ)

in coordinates (ρ,Φ). After identifing x0 = 0 via normal coordinates,we define

uσ = (uσρ, uσΦ) : B1(0)→ (H, d)

by setting

uσρ(x) = µ−1(σ)uρ(σx) and uσΦ(x) = µ−1(σ)uΦ(σx) (5)

where

µ(σ) =

√Iu(σ)σn−1

. (6)

The choice of the scaling constant µ(σ) implies that

Iuσ(1) =∫∂B1(0)

d2(uσ, P0)dΣ = 1. (7)

Theorem 3 ([DM2] Corollary 15) Let u : (BR(0), g) → (H,d) isa harmonic map with u(0) = P0 and Ordu(0) = 1. There exists asequence of blow up maps uσi of u at 0 converging locally uniformlyin the pullback sense (cf. [KS2] Definition 3.3) to a linear functionu∗ : B1(0)→ R. In particular,

d(uσi(·), uσi(·))→ d∗(u∗(·), u∗(·)) uniformly on compact sets.

We consider the following normalization of the blow up maps and itslimit in Theorem 3. By composing with a rotation if necessary, we mayassume that uσi converges in the pullback sense to the linear function

u∗(x) = Ax1 (8)

for some constant A > 0. Furthermore, let

ci =uσiφ(1) + uσiφ(−1)

2

and define an isometry Tci : H→ H by setting

Tci(P0) = P0 and Tci(ρ, φ) = (ρ, φ− ci).

6

Since (Tci uσi)φ(1) = (Tci uσi)φ(−1), we can assume that the sequenceuσi of blow up maps in Theorem 3 satisfy

uσiφ(1) = uσiφ(−1), ∀i = 1, 2, . . . . (9)

Below, we will assume that u∗ and uσi satisfy the normalization as-sumptions (8) and (9).

Definition 4 An arclength parameterized geodesic

γ = (γρ, γφ) : (−∞,∞)→ H

is said to be symmetric if

γρ(s) = γρ(−s) and γφ(s) = −γφ(−s).

A homogeneous degree 1 map

l : B1(0)→ H

is said to be a symmetric homogeneous degree 1 map if

l(x) = γ(Ax)

for some A > 0 and some symmetric geodesic. We call the number γφ(1)the address of a symmetric geodesic.

Lemma 5 Let u : (BR(x0), g) → (H, d) be a harmonic map such thatu(x0) = P0 and Ordu(x0) = 1. Furthermore, let u∗ and uσi be as inTheorem 3 normalized such that (8) and (9) are satisfied. Then thereexists a sequence of symmetric geodesics γσi with the address γσiφ(1)→∞ and a symmetric homogeneous degree 1 maps lσi : B1(0)→ H givenby

lσi(x) = γσi(Ax1)

such thatlimi→∞

supBr(0)

d(uσi , lσi) = 0

for any r ∈ (0, R).

Proof. By [DM1] Lemma 8 and the normalizations (8), (9), thereexists a sequence σi → 0, homogeneous degree 1 maps l′σi : B1(0)→ Hdefined by

l′σi(x) =

(Ax1, φσi) x1 > 0

P0 x1 = 0(Ax1,−φσi) x1 < 0

7

such that, for any r ∈ (0, 1),

limi→∞

supBr(0)

d(uσi , l′σi) = 0.

We claim that φσi → ∞. Indeed, if this claim is not true, then (asubsequence of) φσi converges to φ∞. Thus, l′σi converges uniformly toa homogeneous degree 1 map

l′σi(x) =

(Ax1, φ∞) x1 > 0

P0 x1 = 0(Ax1,−φ∞) x1 < 0

andlimi→∞

supBr(0)

d(uσi , l′∞ R) = 0.

Since uσi is a harmonic map, so is l′∞. By the maximum principle

d(P0, P ) = d(l′∞(0), P ) 6= sup d(l′∞, P ) = supt∈[0,1]

d((At,±φ∞), P )

Taking P = (ρ, 0) for ρ sufficiently large, this contradiction proves theclaim. Set γσi to be the symmetric geodesic and with address γσiρ(1) =φσi . Since φσi →∞, we have that

limi→∞

supB1(0)

d(lσi , l′σi) = 0.

q.e.d.

3 The foliation of H by symmetric geodesics

In the previous section (cf. Lemma 5), we showed that if a harmonicmap hits the boundary P0 of H, then its blow ups behave much like asymmetric geodesic. Motivated by this observation, we introduce newcoordinates

(s, t) for H (10)

by foliating it by a family of symmetric geodesics. More precisely, weconsider

c = (cρ, cφ) : (−∞,∞)× (−∞,∞)→ H (11)

8

satisfying the following:

• s 7→ ct(s) = c(s, t) is a unit speed symmetric geodesic, (12)• c(0, 1) = (1, 0), (13)• s 7→ cρ(s, t) increasing, lim

s→±∞cφ(s, t) = ±∞, (14)

• t 7→ cρ(s, t) increasing, limt→−∞

cρ(0, t) = 0, limt→∞

cρ(0, t) =∞,(15)

•∣∣∣∣∂c∂t (±1, t)

∣∣∣∣ = 1, ∀t ∈ (−∞,∞). (16)

We remark that the above conditions imply that

limt→−∞

cρ(±1, t) = 1 and limt→−∞

cφ(±1, t) =∞.

For each t ∈ (0,∞), consider the vector field

s 7→ Xt(s) :=∂c

∂t(s, t) along s 7→ c(s, t) (17)

and setJt(s) = J(s, t) = |Xt(s)|. (18)

Note that the symmetry of s 7→ c(s, t) implies that for all t ∈ (0,∞),

cφ(0, t) = 0, J ′t(0) = 0 and < Xt(0),∂c

∂s(0, t) >= 0. (19)

Sine it is generated by a family of geodesics, Xt is a Jacobi field andsatisfies the differential equations

X ′′t (s) +Kt(s)Xt(s) = 0 (20)

whereKt(s) = K(s, t) = − 6

c2ρ(s, t)

(21)

is the Gauss curvature of H at c(s, t). The initial conditions are

Xt(0) = (Jt(0), 0) and X ′t(0) = (0, 0). (22)

Indeed, the first initial condition above follows immediately from the or-thogonality condition of (19). Furthermore, the second initial conditionabove follows from

< X ′t(s),∂c

∂s>=<

∂2c

∂s∂t,∂c

∂s>=

12∂

∂t

∣∣∣∣∂c∂s∣∣∣∣2 = 0 (23)

9

and

< X ′t(0),∂c

∂t(0, t) > = <

∂2c

∂s∂t(0, t),

∂c

∂t(0, t) >

=12∂

∂s

∣∣∣∣∂c∂t∣∣∣∣2 (0, t)

=12∂

∂sJ2t (s)|s=0 = Jt(0)J ′t(0) = 0.

For each t, Xt is orthogonal to the geodesic s 7→ c(s, t) at s = 0 (cf.(19)). Hence it is orthogonal for all s, i.e.

< Xt(s),∂c

∂s(s, t) >=<

∂c

∂t,∂c

∂s> (s, t) = 0. (24)

Equation (20) implies

J ′′t (s) = −K(s, t)Jt(s)+Jt(s)−3(∣∣X ′t(s)∣∣2 |Xt(s)|2− < X ′t(s), Xt(s) >2

).

Combining (23) and (24), we conclude that both X ′t(s) and Xt(s) areperpendicular to ∂c

∂s . By the two-dimensionality of H, X ′t(s) and Xt(s)are parallel, and hence∣∣X ′t(s)∣∣2 |Xt(s)|2 =< X ′t(s), Xt(s) >2 .

The above two equalities along with (16) and (19) give us the differentialequation along with boundary conditions

J ′′t +KJt = 0, Jt(1) = 1, J ′t(0) = 0. (25)

The goal for the rest of this section is to derive estimates for Jt(s) =∣∣∣∂c∂t (s, t)∣∣∣. These estimates are then used in the next section to deriveestimates for the metric gH. We start with the following lemma.

Lemma 6 Let c(s, t) be given by (11). There exist constants m, b ∈(0, 1) such that for s ∈ [−1, 1], we have

m|s|+ bcρ(0, t) ≤ cρ(s, t) ≤ |s|+ cρ(s, 0)

and6

(|s|+ cρ(0, t))2 ≤ −K(s, t) ≤ 6

(m|s|+ bcρ(0, t))2 .

Moreover, we can choose m arbitrarily close to 1 (by choosing b suffi-ciently close to 0).

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Proof. Let γ : (−∞,∞) → H be the unit speed symmetricgeodesic satisfying

γ(0) = (1, 0) and γ′(0) = (0, 1). (26)

We first claim that

σ < γρ(σ) ≤ σ + 1 ∀σ ∈ [0,∞). (27)

Indeed, for a fixed σ ∈ [0,∞), consider the function

f : (0,∞)→ R+, f(ρ) = d((ρ, 0), γ(σ)).

Since ρ 7→ (ρ, 0) is a geodesic, f(ρ) is a strictly convex function. Fur-thermore, the geodesic γ intersects the line φ = 0 perpendicularly at(1, 0) by (26). Thus, ρ = 1 is the unique minimum of the function f(ρ).In particular,

σ = d(γ(σ), (1, 0)) = f(1) < limσ→0

f(σ) = d(P0, γ(σ)) = γρ(σ).

This is the first inequality of (27). We now prove the second inequality.The second inequality is true at σ = 0 since γρ(0) = 1. Now assumeσ ∈ (0,∞). Note that the slope of the secant line connecting the points(σ, γρ(σ)) and (0, 1) on the graph the function s 7→ γρ(s) is

γρ(σ)− 1σ

.

Since γρ(s) is convex, this quantity is less than γ′ρ(σ) < |γ′(σ)| = 1.This immediately implies the second inequality of (27) and ends theproof of the claim.

The inequalities of (27) implies that for a fixed m ∈ (0, 1), we canchoose b ∈ (0, 1) sufficiently small such that

mσ + b < γρ(σ) ≤ σ + 1, ∀σ ∈ [0,∞).

For each t ∈ (0,∞), we rescale (cf. (4)) the curve s 7→ ct(s) =c(s, t), s ∈ [−1, 1] to define the unit speed geodesic

ct(σ) : [0, ctρ(0)−1]→ H, ct(σ) :=1

ctρ(0)ct(ctρ(0)σ).

Since ct(0) = (1, 0) and ∂ct

∂s (0) = (0, 1), we conclude by (26) that ct(σ) =γ(σ). Consequently,

mσ + b ≤ ctρ(σ) ≤ σ + 1, ∀σ ∈ (0, ctρ(0)−1].

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Multiply through by ctρ(0) to obtain

mctρ(0)σ + ctρ(0)b ≤ ctρ(ctρ(0)σ) ≤ ctρ(0)σ + ctρ(0), ∀σ ∈ (0, ctρ(0)−1].

Letting s = ctρ(0)σ, we conclude

ms+ ctρ(0)b ≤ ctρ(s) ≤ s+ ctρ(0), ∀s ∈ [0, 1].

By (21), we obtain

−6(ms+ ctρ(0)b

)−2≤ K(s, t) ≤ −6

(s+ ctρ(0)

)−2, ∀s ∈ [0, 1].

By symmetry, the same argument applies for s ∈ [−1, 0]. q.e.d.

Lemma 7 For c(s, t) given by (11), let

at = cρ(0, t) and Jt(s) = J(s, t) =∣∣∣∣∂c∂t (s, t)

∣∣∣∣ .Then there exist constants m, b ∈ (0, 1), β ∈ (0, 1) such that for s ∈[−1, 1], we have ∣∣∣∣ ∂∂s log J(s, t)

∣∣∣∣ ≤ 3βmm|s|+ ab

and

(m|s|+ atb)3β

(m+ atb)3β≤ J(s, t) ≤ e2 (|s|+ at)

3

(1 + at)3

Moreover, m and β can be chosen arbitrarily close to 1 by choosing bsufficiently close to 0.

Proof. Fix t ∈ (0,∞) and omit the subscript t for simplicity. Withthis notation, we rewrite (25) as

J ′′ +KJ = 0, J(1) = 1, J ′(0) = 0. (28)

Let m, b ∈ (0, 1) be as in Lemma 6 and let β > 1 be a number satisfying

3β(3β − 1)m2 = 6.

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The functions

j1(s) =(s+ a)3

(1 + a)3

and

j2(s) =(ms+ ab)3β

(m+ ab)3β(29)

are the solutions to the differential equation

j′′1 + k1j1 = 0, k1 = − 6(s+ a)2

andj′′2 + k2j1 = 0, k2 = − 6

(ms+ ab)2 (30)

respectively with boundary conditions

j1(1) = 1 = j2(1).

For s ∈ [0, 1], Lemma 6 implies

−k1 ≤ −K ≤ −k2.

Hence

(J(s)j′1(s)− J ′(s)j1(s))′ ≤ 0 ≤ (J(s)j′2(s)− J ′(s)j2(s))′. (31)

Since J ′(0) = 0 (cf. (28)), we have

J(0)j′1(0)− J ′(0)j1(0) = J(0)j′1(0)

andJ(0)j′2(0)− J ′(0)j2(0) = J(0)j′2(0)

which in turn implies by (31) that

J(s)j′1(s)− J ′(s)j1(s) ≤ J(0)j′1(0) (32)

and0 ≤ J(0)j′2(0) ≤ J(s)j′2(s)− J ′(s)j2(s). (33)

In particular,

0 ≤ ∂

∂slog

j2(s)J(s)

, ∀s ∈ [0, 1]. (34)

13

This implies

J ′(s)J(s)

≤ j′2(s)j2(s)

=3βm

ms+ ab, ∀s ∈ [0, 1]. (35)

Furthermore, J ′′ = −KJ > 0 implies that J ′(s) is increasing. SinceJ ′(0) = 0 (cf. (28)), we have that J ′(s) ≥ 0 for s ∈ [0, 1]. Thiscompletes the proof of the first estimate of the lemma for s ∈ [0, 1]. Bysymmetry, the same argument applies for s ∈ [−1, 0].

Integrating (34) in the interval [s, 1] and noting that J(1) = j2(1),we obtain

(ms+ ab)3β

(m+ ab)3β= j2(s) ≤ J(s), ∀s ∈ [0, 1]. (36)

Since J ′(s) ≥ 0 for s ∈ [0, 1], we have that J(0) ≤ J(s) and

∂

∂s

(log

j1(s)J(s)

)=

j′1(s)J(s)− J ′(s)j1(s)j1(s)J(s)

≤ j′1(0)J(0)j1(s)J(s)

(by (32))

≤ j′1(0)j1(s)

=3a2

(s+ a)3∀s ∈ [0, 1]. (37)

Integrating this inequality in the interval [s, 1] and noting that J(1) =j1(1), we obtain

J(s)j1(s)

≤ exp

(−3a2

2

(1

(1 + a)2− 1

(s+ a)2

))≤ e2, ∀s ∈ [0, 1]

and hence

J(s) ≤ e2j1(s) = e2 (s+ a)3

(1 + a)3. (38)

Combining (36) and (38), we obtain the second inequality of the lemmafor s ∈ [0, 1]. By symmetry, the same argument applies for s ∈ [−1, 0].q.e.d.

Lemma 8 For c(s, t) given by (11), let

at = cρ(0, t) and J(s, t) =∣∣∣∣∂c∂t (s, t)

∣∣∣∣ .14

Then there exists C > 0 such that for (s, t) ∈ [−1, 1]× (−∞, 1], we have

C−1(|s|+ at)3 ≤ J(s, t) ≤ C(|s|+ at)3.∣∣∣∣∂J∂s (s, t)∣∣∣∣ ≤ C(|s|+ at)2,∣∣∣∣∣∂2J

∂s2(s, t)

∣∣∣∣∣ ≤ C(|s|+ at),∣∣∣∣∣∂3J

∂s3(s, t)

∣∣∣∣∣ ≤ C.

Proof. Fix t ∈ (−∞, 1] and omit the subscript t for simplicity. Wecan assume that b

m < 1 and β ∈ (0, 1) close to 1 in Lemma 7. Thus, fors ∈ [0, 1], we have ∣∣∣∣ ∂∂s log J

∣∣∣∣ ≤ 3βmb

mb |s|+ a

≤3βmb

|s|+ a

and

b3β (|s|+ a)3β

23β≤b3β

(mb |s|+ a

)3β23β

≤ (m|s|+ ab)3β

23β≤ J ≤ e2 (|s|+ a)3 .

Combining the above, the first two estimates follow immediately. TheGauss curvature estimate of Lemma 6 states

0 < −K ≤6b2(

mb |s|+ a

)2 ≤ 6b2

(|s|+ a)2 .

Combining this with first inequality of the lemma and (28), we obtainimplies the third estimate. Finally, differentiate (28) with respect to sto obtain

J ′′′ = −K ′J −KJ ′.Furthermore, since c′ρ ≤ |c′| = 1, we have

|K ′| =∣∣∣∣∣ ∂∂s

(−6

c2ρ(s, t)

)∣∣∣∣∣ =12

c3ρ(s, t)

∣∣∣c′ρ∣∣∣ ≤ 12c3ρ(s, t)

.

Lemma 6 implies

12c3ρ(s, t)

≤ 12(m|s|+ bcρ(0, t))3

≤12b3

(|s|+ cρ(0, t))3.

Combining this with the first two estimates and Lemma 6 imply thefourth estimate. q.e.d.

15

Lemma 9 For c(s, t) given by (11), let

at = cρ(0, t) and Jt(s) = J(s, t) =∣∣∣∣∂c∂t (s, t)

∣∣∣∣ .Then there exists constant C > 0 such that for (s, t) ∈ [−1, 1]×(−∞, 1],∣∣∣∣∂J∂t (s, t)

∣∣∣∣ ≤ C(|s|+ at)5,∣∣∣∣∣ ∂2J

∂t∂s(s, t)

∣∣∣∣∣ ≤ C(|s|+ at)4,∣∣∣∣∣ ∂3J

∂t∂s2(s, t)

∣∣∣∣∣ ≤ C(|s|+ at)3.

Proof. Throughout the proof, C will denote a constant that isindependent of s or t. For simplicity, we omit the subscript t and denotethe t-derivative by a dot. By the assumption that t ∈ (−∞, 1], we canapply Lemma 8 and also to obtain the inequality a = cρ(0, t) ≤ 1 (cf.(13) and (15)).

Differentiate (21) with respect to t to obtain

K =∂

∂t

(− 6c2ρ

)=

12cρc3ρ

. (39)

By (15) and the definition of J , we have 0 ≤ cρ ≤ J. Thus, Lemma 6and Lemma 8 imply

0 ≤ K ≤ C. (40)

Differentiate (28) with respect to t and note (16) and the symmetry ofJ to obtain

J ′′ = −KJ − KJ, J(±1) = 0, J ′(0) = 0. (41)

If J achieves a negative local minimum at s ∈ (−∞,∞), then 0 ≤J ′′(s) = −K(s)J(s)− K(s)J(s) < 0, a contradiction. Thus, the bound-ary condition J(±1) = 0 implies that

0 ≤ J(s), ∀s ∈ [−1, 1]. (42)

By (28) and (41),(J ′J − JJ ′)′ = KJ2.

16

Hence (40) and Lemma 8 imply that there exists a constant C0 > 0sufficiently large such that

(J ′J − JJ ′)′ < 14C0(s+ a)6 in [−1, 1]. (43)

Furthermore, Lemma 8 and the fact that a ≤ 1 implies there exists aconstant C1 > 0 sufficiently large such that

(s+ a)6 ≤ C1eC0(s+a)2J2(s), ∀s ∈ [−1, 1]. (44)

For constants C0 and C1 in (43) and (44) respectively, we claim

J(s) ≤ C1eC0(s+a)2a2J(s), ∀s ∈ [−1, 1]. (45)

Indeed, suppose (45) is not true. Then since

J(±1) = 0 < C1eC0(1+a)2a2J(±1),

there exists s0 ∈ (−1, 1) such that

J(s0) = C1eC0(s0+a)a2J(s0).

By the symmetry of J and J , we can assume s0 ∈ (0, 1]. Let σ0 be thesmallest number in [0, s0) such that

C1eC0(s+a)2a2J(s) < J(s), ∀s ∈ (σ0, s0). (46)

By the definition of σ0 ∈ [0, s0), we have the following two cases:

(i) σ0 = 0 and C1eC0(σ0+a)2a2J(σ0) ≤ J(σ0)

or(ii) σ0 ∈ (0, s0) and C1e

C0(σ0+a)2a2J(σ0) = J(σ0).

For case (i), the boundary conditions of (28) and (41) imply

J ′(σ0)J(σ0)− J(σ0)J ′(σ0) = J ′(0)J(0)− J(0)J ′(0) = 0. (47)

For case (ii), the assumption that J(σ0) = C1e−C(σ0+a)2a2J(σ0) and

(46) implies that J(s)− C1eC(s+a)2a2J(s) is increasing at σ0. Thus

C1eC0(σ0+a)2a2J ′(σ0)

≤ C1eC0(σ0+a)2a2J ′(σ0) + 2C2(σ0 + a)C1e

C0(σ0+a)2a2J(σ0)

= (C1eC0(s+a)2a2J(s))′

∣∣∣s=σ0

≤ J ′(σ0).

17

Combining this inequality and equality J(σ0) = C1eC(σ0+a)2a2J(σ0), we

obtain

C1eC0(σ0+a)2a2J ′(σ0)J(σ0) ≤ C1e

C(σ0+a)2a2J(σ0)J ′(σ0),

or more simply,

J ′(σ0)J(σ0)− J(σ0)J ′(σ0) ≤ 0. (48)

Integrating (43) in the interval (σ0, s) and noting (47) for case (i) and(48) for case (ii) respectively, we obtain

a2J ′(s)J(s)− a2J(s)J ′(s) < 2a2C0(s+ a)7, ∀s ∈ (σ0, s0). (49)

Furthermore,

a2(s+ a)6 ≤ C1eC0(s+a)2a2J2(s) (by (44))

< J(s)J(s). ∀s ∈ (σ0, s0). (by (46))

The above two inequalities imply that for s ∈ (σ0, s0),(log

CeC0(s0+a)2a2J

J

)′=a2J ′(s)J(s)− a2J(s)J ′(s)

J(s)J(s)− 2C0(s+ a) < 0.

(50)Integrating in the interval (σ0, s0), we obtain

1 = log

(CeC0(s0+a)2 a

2J(s0)J(s0)

)< log

(C1e

C0(σ0+a)2 a2J(σ0)J(σ0)

)≤ 1,

a contradiction. This completes the proof of the claim (45). Combinedwith (42), we obtain

0 ≤ J(s) ≤ Ca2J(s), ∀s ∈ [−1, 1]. (51)

Lemma 7 and (51) imply the first estimate of the lemma.By (41), Lemma 6, the first estimate, (40) and Lemma 8,

|J ′′(s)| = | −KJ − KJ | ≤ C(s+ a)3, ∀s ∈ [−1, 1].

Since J ′(0) = 0, integration implies

|J ′(s)| ≤ C(s+ a)4, ∀s ∈ [−1, 1].

The above two inequalities are the third and second estimates of thelemma. q.e.d.

18

Lemma 10 For c(s, t) given by (11),

|cρ| ≤ C(s+ a)3,

|cρ| ≤ C(s+ a)5

and|cρ| ≤ C(s+ a)7

where c, c indicates the first and second derivatives of c with respect tot and c indicates the third derivative of c with respect to t.

Proof. Throughout this proof, we define the function c′φ and cφ byexpressing c′ and c as

c = cρ∂

∂ρ+cφc3ρ

∂

∂φ.

with respect to the orthonormal basis ∂∂ρ ,∂∂φ

c3ρ. Thus we obtain√

(c′ρ)2 + (c′φ)2 = |c′| = 1.

and √c2ρ + c2

φ = |c| = J. (52)

Note that with this convention, c′φ and cφ are not the s- and t-derivativesof the coordinate function cφ of c. In fact, there differ from these deriva-tives by a factor of c3

ρ =∣∣∣ ∂∂φ ∣∣∣.

The first inequality of the lemma follows directly from (52) andLemma 8. We now proceed with the proof of the second and the thirdinequality. Since < c, c′ >≡ 0 by (24), we obtain

< c, c′ >=∂

∂t< c, c′ > − < c, c′ >= −

∂∂s < c, c >

2= −JJ ′.

Additionally, (52) implies

< c, c >=∂∂t < c, c >

2= JJ.

Since

< c′,∂

∂ρ>= c′ρ, <

c

J,∂

∂ρ>=

cρJ,

< c′,∂

∂φ>= c3

ρc′φ <

c

J,∂

∂φ>=

c3ρcφ

J,

19

we can express ∂∂ρ and ∂

∂φ in terms of the orthonormal basis c′, cJ (cf.(24)) by

∂

∂ρ= c′ρc

′ +cρJ

c

Jand

∂

∂φ= c3

ρc′φc′ +

c3ρcφ

J

c

J.

Combining the above identities, we obtain

< c,∂

∂ρ> = < c, c′ρc

′ +cρJ

c

J>

= c′ρ < c, c′ > +cρJ2

< c, c >

= −c′ρJJ ′ +cρJ

J,

< c,∂

∂φ> = < c, c3

ρc′φc′ +

c3ρcφ

J

c

J>

= c3ρc′φ < c, c′ > +

c3ρcφ

J2< c, c >

= −c3ρc′φJJ

′ +c3ρcφJ

J

and

< c′,∂

∂ρ> = < c′, c′ρc

′ +cρJ

c

J>

= c′ρ < c, c′ > +cρJ2

< c′, c >

=cρJ2

∂∂s < c, c >

2

=cρJ′

J.

By (3), we obtain

∇ ∂∂t

∂

∂ρ= cρ∇ ∂

∂ρ

∂

∂ρ+cφc3ρ

∇ ∂∂φ

∂

∂ρ

=cφc3ρ

Γφρφ∂

∂φ

=3cφc4ρ

∂

∂φ

20

and

∇ ∂∂t

∂

∂φ= cρ∇ ∂

∂ρ

∂

∂φ+cφc3ρ

∇ ∂∂φ

∂

∂φ

= cρΓφρφ

∂

∂φ+cφc3ρ

Γρφφ∂

∂ρ

=3cρcρ

∂

∂φ− 3cφc2

ρ

∂

∂ρ.

Thus,

cρ =∂

∂t< c,

∂

∂ρ>

= < c,∂

∂ρ> + < c,∇ ∂

∂t

∂

∂ρ>

= −c′ρJJ ′ +cρJ

J+

3c2φ

cρ, (53)

cφ =∂

∂t< c,

∂

∂φ>

= < c,∂

∂φ> + < c,∇ ∂

∂t

∂

∂φ>

= −c3ρc′φJJ

′ +c3ρcφJ

J+ 3c2

ρcρcφ − 3cρcφc2ρ (54)

and

c′ρ =∂

∂t< c′,

∂

∂ρ>

= < c′,∂

∂ρ> + < c′,∇ ∂

∂t

∂

∂ρ>

=cρJ′

J+

3(cφ)2

cρ. (55)

Thus, applying Lemma 8 and Lemma 9 to the right hand side of (53),we obtain the first inequality of the lemma. Differentiating (53) withrespect to t and applying (54) and (55), we obtain the second estimate.q.e.d.

21

Lemma 11 For c(s, t) given by (11), let

at = cρ(0, t) and Jt(s) = J(s, t) =∣∣∣∣∂c∂t (s, t)

∣∣∣∣ .Then there exists constant C > 0 such that for (s, t) ∈ [−1, 1]×(−∞, 1],∣∣∣∣∣∂2J

∂t2(s, t)

∣∣∣∣∣ ≤ C(|s|+ at)7,∣∣∣∣∣ ∂3J

∂s∂t2(s, t)

∣∣∣∣∣ ≤ C(|s|+ at)6,∣∣∣∣∣ ∂4J

∂s2∂t2(s, t)

∣∣∣∣∣ ≤ C(|s|+ at)5.

Proof. Differentiate (41) and note (16) to obtain

J ′′ = −KJ − KJ − 2KJ , J ′(0) = 0, J(1) = 0. (56)

Differentiate (39) to obtain

K = −36c2

ρ

c4ρ

+12cρc3ρ

. (57)

Applying Lemma 10, we thus obtain∣∣∣K∣∣∣ ≤ C(s+ a)2. (58)

Thus, by (58), Lemma 8, (40) and Lemma 9 imply that there existsC0 > 0 such that

|KJ + 2KJ | ≤ C0(s+ a)5 in [−1, 1]. (59)

By Lemma 6, there exists a constant C1 > 0 such that

−K ≤ C1

(s+ a)2(60)

With C0 as in (59) and C1 as in (60), suppose that there exists s ∈(−1, 1) such that J(s) < −C0

C1(s+a)7. Since J(±1) = 0, this implies that

J achieves local minimum J(s) < −C0C1

(s+a)7 at some point s ∈ (−1, 1).Thus, (60) implies −K(s)J(s) < −C0(s + a)5. By (56) and (59), weconclude 0 ≤ J ′′(s) < 0. This contradiction proves

−C(s+ a)7 ≤ J , ∀s ∈ [−1, 1]. (61)

22

By (28), (56), (59) and Lemma 8,

(J ′J − JJ ′)′ = (KJ + 2KJ)J ≤ C(s+ a)8. (62)

By comparing (62) to (43), we observe that by following the proof of(45), we obtain

J(s) ≤ C1e−C0(s+a)2a4J, ∀s ∈ [−1, 1]. (63)

Combined with (61), we obtain

−(s+ a)7 ≤ J ≤ Ca4J.

Thus, Lemma 7 and (51) imply the first estimate of the lemma. By(56), Lemma 6, the first estimate and (59),

|J ′′(s)| = | −KJ − KJ − 2KJ | ≤ C(s+ a)3, ∀s ∈ [−1, 1].

Since J ′(0) = 0, integration implies

|J ′(s)| ≤ C(s+ a)4, ∀s ∈ [−1, 1].

The above two inequalities are the third and second estimates of thelemma. q.e.d.

Lemma 12 For c(s, t) given by (11), let

at = cρ(0, t) and Jt(s) = J(s, t) =∣∣∣∣∂c∂t (s, t)

∣∣∣∣ .Then there exists constant C > 0 such that for (s, t) ∈ [−1, 1]×(−∞, 1],∣∣∣∣∣∂3J

∂t3(s, t)

∣∣∣∣∣ ≤ C(|s|+ at)9,∣∣∣∣∣ ∂4J

∂s∂t3(s, t)

∣∣∣∣∣ ≤ C(|s|+ at)8,∣∣∣∣∣ ∂5J

∂s2∂t3(s, t)

∣∣∣∣∣ ≤ C(|s|+ at)7.

23

Proof. Differentiate (56) and note (16) to obtain (with the thirdderivative of J and K with respect to t denoted by J and K respectively)

J ′′ = −KJ − KJ − 2KJ − 2KJ , J ′(0) = 0, J(1) = 0. (64)

Differentiate (57) to obtain

K =144c3

ρ

c5ρ

− 144cρcρc4ρ

+12cρc3ρ

. (65)

Applying Lemma 10, we thus obtain

|K| ≤ C(s+ a)4 in [−1, 1]. (66)

Thus, (66), Lemma 8, (58), Lemma 9, (40) and Lemma 11 imply thatthere exists C0 > 0 such that

|KJ + 2KJ + 2KJ | ≤ C0 ≤ (s+ a)7 in [−1, 1]. (67)

Next, combining (28) and (64), we obtain

(J ′J − J ′J)′ =(KJ + 2KJ + 2KJ

)J ≤ C(s+ a)10 in [−1, 1]. (68)

By comparing (67) to (59) and (68) to (62), we observe that the followsfrom the proof of Lemma 11 q.e.d.

4 Metric estimates of the model space

In this section, we use the family of symmetric geodesics c(s, t) given inthe previous section to define a new coordinate system for H. We thengive estimates of the metric gH and Christoffel symbols represented interms these new coordinates. The motivation of using c(s, t) is thatblow ups of a harmonic map are closely approximated by symmetrichomogeneous degree 1 maps (cf. Lemma 5). With the estimates derivedin this section, we derive estimates of harmonic maps in Section 5 below.

Given a symmetric homogeneous degree 1 map l : B1(0) → H de-fined in Definition 4, let

l(x) = (ls(x), lt(x))

24

be the expression of l with respect to the coordinates (s, t). Then bythe construction of the coordinates (s, t),

∃t∗ > 0 such that lt(x) ≡ t∗. (69)

For simplicity, apply a linear change of variables

(s, t) 7→ (%, ϕ) = (s, t− t∗) (70)

such that in coordinates (%, ϕ),

l(x) = (l%(x), lϕ(x)) = (Ax1, 0).

We goal is to compare the geometry of our space H to the spaceH2 studied in [DM1]. Recall that H2 is constructed by taking twocopies H+ and H− of H and identifying the point P0 (cf. (1)). Con-sider coordinates on H2\P0 by first applying the change of variables(ρ, φ) 7→ (−ρ, φ) to obtain new coordinates for H−. Thus, we then havecoordinates

(ρ, φ) ∈ R\0 ×R (71)

for H2\P0 with the property that ρ > 0 implies (ρ, φ) ∈ H+ andρ < 0 implies (ρ, φ) ∈ H−. The metric and the Christoffel symbols ofH2\P0 expressed in the coordinates (ρ, φ) are(

1 00 ρ6

)(72)

andΓρρρ = 0 Γφφφ = 0

Γρρφ = 0 Γφρφ =3ρ

Γρφφ = −3ρ5 Γφρρ = 0.

(73)

We want to compare the above metric expression and the Christoffelsymbols to that of of gH in terms of our new coordinates given in (70).By construction (cf. (12), (18) and (24)), the metric gH with respect tocoordinates (%, ϕ) is

gH(%, ϕ) =

(1 00 J 2(%, ϕ)

)where we set

J (%, ϕ) :=∣∣∣∣∂c∂t (%, ϕ+ t∗)

∣∣∣∣ = J(ϕ, %+ t∗).

25

We will assume (%, ϕ) ∈ [−1, 1]×(−∞, 1−t∗] in order to apply Lemma 8,Lemma 9, Lemma 11 and Lemma 12. For simplicity, introduce thefunction

℘ = ℘(%, ϕ)

given by℘ = |%|+ cρ(0, ϕ+ t∗) = |s|+ at. (74)

Here one should observe that in H2, |ρ| is the distance of a point (ρ, φ) tothe singular point P0. Recall that P0 is the point defined by identifyingthe entire line ρ = 0 as a single point. In coordinates (%, ϕ) of H, theline % = 0 plays the role of the singular point P0. Indeed, % = 0 hasthe geometric property that the family of geodesic lines ϕ = const., issqueezed together at % = 0 as const.→ −∞. The distance of the point(%, ϕ) to % = 0 is |%|. Thus, ℘ is the distance to % = 0 modulothe error of at. Note that by (15), we have that at = cρ(0, t) → 0 ast→ −∞. Therefore, we observe that for a fixed ϕ,

limt∗→−∞

cρ(0, ϕ+ t∗) = 0.

To summarize, the line % = 0 is asymptotically the singular point P0

in H2 and the parameter ℘ is asymptotically the parameter ρ in H, i.e.distance to the line % = 0.

In [DM1], we considered the subset of H2 consisting of geodesic lines;more specifically, in terms of the coordinates (ρ, φ) and a fixed φ0 > 0,we define the subset

H2[φ0] := (ρ, φ) ∈ H2 : |φ| ≤ φ0.

Note that H2[φ0] is totally geodesic since the boundary lines φ = φ0

and φ = −φ0 are images of geodesic lines. Define an analogous totallygeodesic subset of H; more specifically, in terms of the coordinates (%, ϕ)and a fixed ϕ0 > 0, define the subset

H[ϕ0] := (%, ϕ) ∈ H : |ϕ| ≤ ϕ0.

Implicit in this definition is that H[ϕ0] depends on the choice of t∗ sincethe coordinates (%, ϕ) depends on t∗ (cf. (70)). The estimates derivedbelow for the metric gH and Christoffel symbols expressed with respectto the coordinates (%, ϕ) show that H[ϕ0] geometrically approximatesH2[φ0]. We shall make this statement more precise (in terms of har-monic maps) in Section 5.

26

By Lemma 8, there exists a constant C > 0 such that (compare thisto (72))

C−1

(1 00 ℘6

)≤(

1 00 J 2(%, ϕ)

)≤ C

(1 00 ℘6

). (75)

Furthermore, by Lemma 8, Lemma 9, Lemma 11, Lemma 12 and thefact that (s, t) 7→ (%, ϕ) is a linear change of variables (and thus ∂

∂s = ∂∂% ,

∂∂t = ∂

∂ϕ) yield the derivative estimates of J ; i.e. there exists C > 0such that∣∣∣∣∂J∂% (%, ϕ)

∣∣∣∣ ≤ C℘2,

∣∣∣∣∣∂2J∂%2

(%, ϕ)

∣∣∣∣∣ ≤ C℘,∣∣∣∣∣∂3J∂%3

(%, ϕ)

∣∣∣∣∣ ≤ C.∣∣∣∣∂J∂ϕ (%, ϕ)∣∣∣∣ ≤ C℘5,

∣∣∣∣∣ ∂2J∂ϕ∂%

(%, ϕ)

∣∣∣∣∣ ≤ C℘4,

∣∣∣∣∣ ∂3J∂ϕ∂%2

(%, ϕ)

∣∣∣∣∣ ≤ C℘3∣∣∣∣∣∂2J∂ϕ2

(%, ϕ)

∣∣∣∣∣ ≤ C℘7,

∣∣∣∣∣ ∂3J∂ϕ2∂%

(%, ϕ)

∣∣∣∣∣ ≤ C℘6,

∣∣∣∣∣ ∂4J∂ϕ2∂%2

(%, ϕ)

∣∣∣∣∣ ≤ C℘5∣∣∣∣∣∂3J∂ϕ3

(%, ϕ)

∣∣∣∣∣ ≤ C℘10,

∣∣∣∣∣ ∂4J∂ϕ3∂%

(%, ϕ)

∣∣∣∣∣ ≤ C℘9,

∣∣∣∣∣ ∂5J∂ϕ3∂%2

(%, ϕ)

∣∣∣∣∣ ≤ C℘8.

(76)By standard computation, Christoffel symbols with respect to the coor-dinates (%, ϕ) are

Γ%%% = 0 Γϕϕϕ =12∂

∂ϕlogJ 2 =

∂

∂ϕlogJ

Γ%%ϕ = 0 Γϕ%ϕ =12∂

∂%logJ 2 =

∂

∂%logJ

Γ%ϕϕ = −12∂

∂%J 2 = −

(∂J∂%

)J Γϕ%% = 0.

(77)Thus, we can apply (76) to obtain the following Christoffel symbols es-timates: there exists C > 0 such that (compare this to (73) and itsderivatives)

∣∣∣Γ%ϕϕ∣∣∣ (%, ϕ) ≤ C℘5∣∣∣Γϕ%ϕ∣∣∣ (%, ϕ) ≤ C

℘∣∣∣Γϕϕϕ∣∣∣ (%, ϕ) ≤ C℘2∣∣∣∣ ∂∂%Γ%ϕϕ

∣∣∣∣ (%, ϕ) ≤ C℘4

27

∣∣∣∣∣ ∂2

∂%2Γ%ϕϕ

∣∣∣∣∣ (%, ϕ) ≤ C℘3

∣∣∣∣ ∂∂ϕΓ%ϕϕ

∣∣∣∣ (%, ϕ) ≤ C℘7∣∣∣∣∣ ∂2

∂ϕ2Γ%ϕϕ

∣∣∣∣∣ (%, ϕ) ≤ C℘6

∣∣∣∣∣ ∂2

∂ϕ∂%Γ%ϕϕ

∣∣∣∣∣ (%, ϕ) ≤ C℘5. (78)

5 Estimates for harmonic maps

In [GrSc], Gromov and Schoen introduced the notion of a totally geodesicsubset being essentially regular. Loosely speaking, this means a har-monic map into this subset can be well-approximated by homogeneousdegree 1 maps. In this section, we show that H resembles H2 in a crucialway: the totally geodesic subspace H[ϕ0] of H (introduced in Section 5)and its counterpart H2[φ0] (introduced in [DM1]) of H2 satisfy a prop-erty which is similar to being essentially regular. More precisely, weprove that harmonic maps into H[ϕ0] are regular in a sense given byTheorem 16 below. We remark that so far we are unable to prove thatH2[φ0] or H[ϕ0] is essentially regular in the strict sense of [GrSc]. How-ever, the weaker notion that we prove in Theorem 16 is sufficient forour applications.

For a harmonic map w : (B1(0), g)→ (H, dH), write

w = (w%, wϕ)

in our new coordinates (%, ϕ) of H. For simplicity, set ℘w to be thecomposition of ℘ defined in (74) and the map w; i.e.

℘w := ℘(w%, wϕ) = |wρ|+ c(0, wϕ + t∗). (79)

Fix R ∈ [34 , 1). The map w is locally Lipschitz continuous (cf. [KS1]

Theorem 2.4.6); in particular,

|∇w| =(|∇w%|2 + |J (wρ, w%)|2 |∇wϕ|2

) 12 ≤ L in BR(0)

for some constant L that depends on the dimension n of the domain, Rand the total energy of w. Thus, (75) and (79) imply

|∇w%| ≤ L and ℘3w|∇wϕ| ≤ L in BR(0). (80)

28

The harmonic map equations are (cf. (77))

4w% = −Γ%ϕϕ(w%, wϕ)|∇wϕ|2 (81)

and

4wϕ = −(2Γϕ%ϕ(w%, wϕ)∇wϕ · ∇w% + Γϕϕϕ(w%, wϕ)|∇wϕ|2

)= −

(2∂J∂% (w%, wϕ)∇wϕ · ∇w% + ∂J

∂ϕ (w%, wϕ)|∇wϕ|2)

J (w%, wϕ)

= −

(2∇J (w%, wϕ) · ∇wϕ − ∂J

∂ϕ (w%, wϕ)|∇wϕ|2)

J (w%, wϕ). (82)

Thus, the derivative estimate of the metric (76), the Christoffel symbolsestimates (78) and Lipschitz estimates (80) imply

|4w%| ≤ C℘5w|∇wϕ|2 ≤

CL2

℘w

|4wϕ| ≤ C(℘−1w |∇w%||∇wϕ|+ ℘2

w|∇wϕ|2)≤ CL2

℘4w

. (83)

in BR(0). Define

wΥ := w%+w2ϕ

2Γ%ϕϕ(w%, wϕ) and wΦ := J (w%, wϕ)wϕ−

12∂J∂ϕ

(w%, wϕ)w2ϕ.

We are now in position to prove the main result of this section (Lemma 14below). We first need the following definition.

Definition 13 A smooth Riemannian metric g on BR(0) ⊂ Rn is saidto be normalized if the standard Euclidean coordinates (x1, . . . , xn) arenormal coordinates of g. The metric gs for s ∈ (0, R] on B1(0) is definedby

gs(x) = g(sx).

Lemma 14 Let R ∈ [12 , 1), E0 > 0, A0 > 0 and a normalized metric

g on BR(0) be given. Then there exists C0 > 0 such that if φ0 > 0,s ∈ (0, 1] and w : (BR(0), gs)→ H[ϕ0] is a harmonic map with

Ew ≤ E0 and |w%|L∞(B 15R16

(0)) ≤ A0,

then|J (w%, wϕ)∇wϕ|L∞(B 15R

16(0)) ≤ C0φ0,

29

and|4wΥ|L∞(B 15R

16(0)) ≤ C0φ

20.

Proof. Through out the proof, C0 will denote a generic constantdependent ony on E0, A0 and g. The assumption on the bounds of |wρ|and |wφ| imply that

|wΦ|L∞(BR(0)) = |w3ρwφ|L∞(BR(0)) ≤ Cφ0. (84)

Since the harmonic map equation (82) implies

2∇J (w%, wϕ) · ∇wϕ + J (w%, wϕ)4wϕ −∂J∂ϕ

(w%, wϕ)|∇wϕ|2 = 0,

we have

4wΦ = 4(J (w%, wϕ)wϕ −

12∂J∂ϕ

(w%, wϕ)w2ϕ

)(85)

= 4J (w%, wϕ)wϕ + 2∇J (w%, wϕ) · ∇wϕ + J (w%, wϕ)4wϕ

−124∂J∂ϕ

(w%, wϕ)w2ϕ −∇

∂J∂ϕ

(w%, wϕ) · ∇wϕwϕ

−∂J∂ϕ

(w%, wϕ)4wϕwϕ −∂J∂ϕ

(w%, wϕ)|∇wϕ|2

= 4J (w%, wϕ)wϕ

−124∂J∂ϕ

(w%, wϕ)w2ϕ −∇

∂J∂ϕ

(w%, wϕ) · ∇wϕwϕ

−∂J∂ϕ

(w%, wϕ)4wϕwϕ.

We now estimates the terms on the right hand side of (85). First,

4J (w%, wϕ) =∂J∂%4w% +

∂2J∂%2|∇w%|2 +

∂J∂ϕ4wϕ +

∂2J∂ϕ2|∇wϕ|2

+2∂2J∂ϕ∂%

∇w% · ∇wϕ. (86)

By the metric derivative estimates (76), the Lipschitz estimate (80) andthe bounds of the Laplacians (83), the five terms on the right hand sidehas a bound in BR(0) of ∣∣∣∣∂J∂% 4w%

∣∣∣∣ =

∣∣∣∣∣C℘2w

CL2

℘w

∣∣∣∣∣30

≤ CL2℘w∣∣∣∣∣∂2J∂%2|∇w%|2

∣∣∣∣∣ ≤ CL2℘w∣∣∣∣∂J∂ϕ4wϕ∣∣∣∣ ≤ C℘5CL

2

℘4w

≤ CL℘2∣∣∣∣∣∂2J∂ϕ2|∇wϕ|2

∣∣∣∣∣ ≤ CL℘2w∣∣∣∣∣ ∂2J

∂ϕ∂%∇w% · ∇wϕ

∣∣∣∣∣ ≤ CL2.

Thus, we conclude

4J (w%, wϕ) ≤ C0 in BR(0). (87)

where C0 denotes a generic constant dependent ony on E0, R and ℘w.Similarly,∣∣∣∣4∂J∂ϕ (w%, wϕ)

∣∣∣∣ =

∣∣∣∣∣ ∂2J∂ϕ∂%

4w% +∂3J∂ϕ∂%2

|∇w%|2 +∂2J∂ϕ24wϕ +

∂3J∂ϕ3|∇wϕ|2

+2∂3J∂ϕ2∂%

∇w% · ∇wϕ

∣∣∣∣∣≤ C0 in BR(0), (88)

∣∣∣∣∇∂J∂ϕ (w%, wϕ) · ∇wϕ∣∣∣∣ =

∣∣∣∣∣ ∂2J∂%∂ϕ

(w%, wϕ)∇w% · ∇wϕ +∂2J∂ϕ2

(w%, wϕ)|∇wϕ|2∣∣∣∣∣

≤ C℘4L2

℘3+ C℘7L

2

℘6

≤ C0 in BR(0) (89)

and ∣∣∣∣∂J∂ϕ (w%, wϕ)4wϕ∣∣∣∣ ≤ C℘5

w

CL2

℘4w

≤ C0 in BR(0). (90)

By (87), (88), (89) and (90),

|4wΦ| ≤ C0ϕ0 in BR(0). (91)

31

Thus, by (84), (91) and elliptic regularity, for any α ∈ (0, 1)

|wΦ|C1,α(B 15R16

(0)) ≤ C0

(|4wΦ|L∞(BR(0)) + |wΦ|L∞(BR(0))

)≤ C0φ0. (92)

Since

∇wΦ = J (w%, wϕ)∇wϕ +∂J∂%

(w%, wϕ)∇w%wϕ +∂J∂ϕ

(w%, wϕ)∇wϕwϕ,

we have

|J (w%, wϕ)∇wφ| ≤∣∣∣∣∂J∂% (w%, wϕ)∇w%wϕ

∣∣∣∣+∣∣∣∣∂J∂ϕ (w%, wϕ)∇wϕwϕ∣∣∣∣+|∇wΦ|.

By Lemma 8, Lemma 9 and (80), the first two terms are bounded byC0φ0. By (92), the third term is also bounded by C0φ0. Thus, we obtainfirst estimate of the lemma. Combined with (75), we also obtain

℘3w|∇wϕ| ≤ C0ϕ0 in B 15R

16(0). (93)

Next, we compute

4wΥ = 4w% + Γ%ϕϕ(w%, wϕ)|∇wϕ|2

+wϕΓ%ϕϕ(w%, wϕ)4wϕ +w2ϕ

2∂

∂ϕΓ%ϕϕ(w%, wϕ)4wϕ

+w2ϕ

2∂2

∂ϕ2Γ%ϕϕ(w%, wϕ)|∇wϕ|2 +

w2ϕ

2∂

∂%Γ%ϕϕ(w%, wϕ)4w%

+w2ϕ

2∂2

∂%2Γ%ϕϕ(w%, wϕ)|∇w%|2 + wϕ

∂

∂%Γ%ϕϕ(w%, wϕ)∇w% · ∇wϕ

+w2ϕ

2∂2

∂%∂ϕΓ%ϕϕ(w%, wϕ)∇w% · ∇wϕ. (94)

The harmonic map equation (81) implies that the first two terms of (94)cancel. Combining with (78), (83), (93) and the Christoffel symbolsestimates in the previous section, we obtain bounds for the other termson the right hand side of (94). More precisely,∣∣∣wϕΓ%ϕϕ(w%, wϕ)4wϕ

∣∣∣ ≤ Cϕ0℘5w

((℘−1

w |∇w%||∇wϕ|+ ℘2w|∇wϕ|2

)≤ Cϕ2

0℘wL2,∣∣∣∣∣w2

ϕ

2∂

∂ϕΓ%ϕϕ(w%, wϕ)4wϕ

∣∣∣∣∣ ≤ Cϕ20℘

7w(w%, wϕ)

(℘−1w |∇w%||∇wϕ|+ ℘2

w|∇wϕ|2)

32

≤ Cϕ20℘

3wL

2,∣∣∣∣∣w2ϕ

2∂2

∂ϕ2Γ%ϕϕ(w%, wϕ)|∇wϕ|2

∣∣∣∣∣ ≤ Cϕ20℘

6w|∇wϕ|2

≤ Cϕ20L

2,∣∣∣∣∣w2ϕ

2∂

∂%Γ%ϕϕ(w%, wϕ)4w%

∣∣∣∣∣ ≤ Cϕ20℘

4w · ℘5

w|∇wϕ|2

≤ Cϕ20℘

3wL

2,∣∣∣∣∣w2ϕ

2∂2

∂%2Γ%ϕϕ(w%, wϕ)|∇w%|2

∣∣∣∣∣ ≤ Cϕ20℘

3w|∇w%|2

≤ Cϕ20℘

3wL

2,∣∣∣∣wϕ ∂∂%Γ%ϕϕ(w%, wϕ)∇w% · ∇wϕ∣∣∣∣ ≤ Cϕ0℘

4w|∇w%||∇wϕ|

≤ Cϕ20℘wL

2,∣∣∣∣∣w2ϕ

2∂2

∂%∂ϕΓ%ϕϕ(w%, wϕ)∇w% · ∇wϕ

∣∣∣∣∣ ≤ Cϕ20℘

5w|∇w%||∇wϕ|

≤ Cϕ20℘

2wL

2.

In summary, we have shown that there exists a constant C0 > 0 de-pending only on R the total energy of w such that

|4wΥ| ≤ C0ϕ20 in B 15R

16(0) (95)

which is the second estimate of the lemma. q.e.d.

Definition 15 We say that a map l = (lρ, lφ) : B1(0) → H2 is analmost affine map if lρ(x) = ~a · x+ b for ~a ∈ Rn and b ∈ R, i.e. lρ is anaffine function.

Theorem 16 Let R ∈ [12 , 1), E0 > 0, A0 > 0 and a normalized metric

g on BR(0) (cf. Definition 13) be given. There exist C > 0 and α > 0with the following property:

For φ0 > 0 and s ∈ (0, 1], if w = (wρ, wφ) : (BR(0), gs) → H[ϕ0] isa harmonic map with

Ew ≤ E0 and |wρ|L∞(B 15R16

(0)) ≤ A0,

33

then

supBr(0)

d(w, l) ≤ Cr1+α supBR(0)

d(w,L) + Crφ20, ∀r ∈ (0,

R

2]

where l = (lρ, lφ) : B1(0)→ H is the almost affine map given by

lρ(x) = wρ(0) +∇wρ(0) · x, lφ(x) = wφ(x)

and L : B1(0)→ H2 is any almost affine map.

Proof. Lemma 14 is analogous to [DM1] Lemma 8. With this, thetheorem follows from the proof of [DM1] Theorem 10. q.e.d.

With Lemma 14, we are in position to prove the main theorem (cf.Theorem 18) of this section. We first start with the following definition.

Definition 17 For a constant E0 > 0 and a normalized metric g onB1(0) (cf. Definition 13), define F [E0, g] to be a set of harmonic mapsu : (B1(0), gσ)→ H2 with σ ∈ (0, 1], u(0) = P0 and Eu ≤ E0.

Theorem 18 Let E0 > 0, A > 0, δ0 > 0 and a normalized metric g onB1(0) be given. There exists σ0 > 0 and D0 > 0 such that if σ ∈ (0, σ0)and u : (B1(0), gσ)→ H2 ∈ F [E0, g] (cf. Definition 17) satisfies

supB1(0)

d(u, l) < D0 (96)

where l(x) = (Ax1, 0) when written in terms of coordinates (%, ϕ) of(70), then

s−1 supBs(0)

|u% −Ax1| < δ0.

Furthermore, for L = (%, ϕ) ∈ H2 : ϕ = 0, we have

lims→0

s−1 supBs(0)

d(u,L) = 0. (97)

Proof. The statement and the proof of [DM1] Lemma 14 can bemodified to the present setting by simply replacing ε30 by C℘(ε0, φ0).Thus, a straightforward modification of the proof of [DM1] Lemma 16and [DM1] Theorem 17 implies the assertion. q.e.d.

34

6 Proof of the main theorem

In this section, we provide the proof of Theorem 1. We first prove

Theorem 19 Let g be a normalized metric on B1(0) (cf. Definition 13)and u : (B1(0), g) → H be a harmonic map. If u(0) = P0, thenOrdu(0) 6= 1

Proof. Assume on the contrary that Ordu(0) = 1 and u(0) =P0. By the normalization (7), the energy of the blow up map uσ :(B1(0), gσ) → H of u at x0 = 0 is bounded by 2 for sufficiently smallσ > 0. Lemma 5 implies that by choosing an the appropriate t∗ (cf.(69)) to define coordinates (%, ϕ) (cf. (70)) and rotating if necessary,uσ ∈ F [2, g] satisfies assumption (96) of Theorem 18 for some sufficientlysmall σ > 0. Thus, (97) implies that for L = (%, ϕ) ∈ H2 : ϕ = 0, wehave

lims→0

d(P0, s−1L) = lim

s→0s−1d(P0,L) = lim

s→0s−1d(uσ(0),L) = 0.

This is a contradiction. Indeed, L does not contain the point P0, andhence

lims→0

d(P0, s−1L) =∞.

q.e.d.

Proof of Theorem 1. By [DM2] Theorem 16,

dimHx ∈ Ω : Ordu(x) > 1 ≤ n− 2.

Thus, Theorem 19 implies the assertion. q.e.d.

The motivation of this paper is in the study of harmonic maps intothe metric completion T g,nof the Teichmuller space with respect to theWeil-Petersson metric. With this in mind, we now extend Theorem 19to the following slightly more general setting. Let Hk−j (resp. Hk−j)be the product space of (k − j)-copies of H (resp. H). Let

h = gH ⊕ . . .⊕ gH and d = dh

denote the product metric on Hk−j and the distance function on Hk−j

induced by h respectively. Define

P0 := (P0, . . . , P0) ∈ Hk−j.

35

The boundary ∂Hk−j of Hk−j the set of points (P 1, . . . , P k−j) ∈ Hk−j

such that at least one of P 1, . . . , P k−j is equal to P0. Define

λ(P 1, . . . , P k−j) := (λP 1, . . . , λP k−j).

Given a map v : B1(0)→ Hk−j , define its blow up maps

vσ : B1(0)→ H

by setting

Iv(σ) =∫∂Bσ(0)

d2(v, P0)dΣ, ν(σ) =

√Iv(σ)σn−1

.

andvσ : B1(0)→ Hk−j

, vσ(x) = ν−1(σ)v(σx).

A harmonic map u : B1(0)→ T g,n can be written locally as u = (V, v)where V : B1(0)→ R2j and v : B1(0)→ Hk−j . The maps V , v are notharmonic maps, but they are only approximately harmonic since theWeil-Petersson metric and the Weil-Petersson connection are asymptot-ically a product near the boundary. This will be explained in greaterdetail in the upcoming paper [DMW]. For now, we will prove the follow-ing theorem concerning a map v : B1(0)→ Hk−j that is not necessarilyharmonic but that shares important properties (cf. (i) and (ii) below)with it (with C = 0 in inequality (98) below).

Theorem 20 There does not exists a map v : B1(0) → Hk−j withv(0) = P0 satisfying the following:

(i) There exists a sequence σi → 0 such that vσi converges locallyuniformly in the pullback sense to a linear function.

(ii) There exists σ0 > 0 such that for any σ ∈ (0, σ0] and a harmonicmap w : (B1(0), gσ)→ H, there exists a constant C > 0 such that

supB 15R

16(0)d2(vσ, w) ≤ c0

∫∂BR(0)

d2(vσ, w)dΣgσ + Cσ2. (98)

Proof. After a rotation of the domain if necessary, we may as-sume that the sequence vσi converges locally uniformly in the pullbacksense to a linear function L(x) = Ax1. By applying [DM1] Lemma 7

36

to B±1 (0) = x ∈ B1(0) : x1 ≥ (≤) 0 and following the argument ofLemma 5, we can reach the same conclusion of Lemma 5 with vσi replac-ing uσi . As discussed in [DM1] Remark 18, Assumption (ii) can takethe place of harmonicity in [DM1] Lemma 16 and [DM1] Theorem 17.Theorem. Thus, we can also do this in Theorem corofinduct above. Nowwe can argue as in proof of Theorem 19 to obtain the result. q.e.d.

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