Harmonic Maps into Surfaceswith an Upper Curvature Bound

in the Sense of Alexandrov

Chikako Mese

1. Introduction

In this paper, we study harmonic maps into metric spaces of curvature boundedfrom above in the sense of Alexandrov. In particular, we are interested in the localbehavior of energy minimizing maps into these surfaces. It is known that smoothenergy minimizing maps between compact surfaces are quasiconformal diffeomor-phisms (see [SY] and [JS]). On the other hand, when the target surface hassingularities, energy minimizing maps can fail to be homeomorphic. In fact, whenthe target has a flat metric with cone singularities, it is known that the only home-omorphic energy minimizing maps are the Teichmuller map with the Teichmullermetric (see [Lei] and [Kuw]). In this paper, we study the collapsing behavior ofenergy minimizing maps when the target surface is a metric space of curvaturebounded from above. We have the following result:

Local Behavior of Energy Minimizing Maps. Let S be a compact topo-logical surface of genus s and let d be a distance function on S which makes (S, d) ametric space of curvature bounded from above by κ and so that the metric topologyis equivalent to the surface topology. Let Σ1 be a compact Riemann surfaces of thesame genus and let f : Σ1 → (S, d) be an energy minimizing map in its homotopyclass. For any P ∈ S, the preimage f−1(P ) is either a point or a connected unionof finitely number of vertical arcs of the Hopf differential of f .

The main technnical tool used in the proof of the above theorem is the orderfunction for energy minimizing maps. For a harmonic function u, the order functionmeasures the order with which u attains the value u(x) at x. It has been importantin the study of harmonic maps into metric spaces of non-positive curvature; Gromovand Schoen [GS] utilized the order function to show regularity of enery minimizingmaps into Riemannian simplical complex of non-positive curvature and Hardt andLin [HL] have used it to study nematic liquid crystals. In this paper, we will developthe order function for energy minimizing maps into metric spaces of curvaturebounded from above by κ.

We will also prove the following theorem which is of independent interest:

Compactness Theorem for Energy Minimizing Maps. Let (M, g) be acompact Riemannian manifold (with or without a boundary) and let di be distancefunctions on X with curvature bounded from above by κ. Assume X is compact withrespect to the metric topology induced by di. Let h : M → X be a continuous mapand let fi : M → (X, di) be energy minimizing maps in the homotopy class of hwith fi = h on ∂M if ∂M 6= ∅. Let δi be the pull back distance function of di underfi, i.e.

δi(·, ·) = di(fi(·), fi(·)).1

2

If the energy of fi is bounded above by K for each i and if di converges uniformlyto a distance function d0, then there exists a subsequence i′ ⊂ i and an en-ergy minimizing map f0 with respect to d0 so that δi′(·, ·) converges uniformly tod0(f(·), f(·)) and the energy of fi′ converges to that f0.

The above theorem leads to the following characterization of energy minimizingmaps between surfaces.

Harmonic Map as a Limit of Diffeomorphisms. Let S be a compacttopological surface of genus s and let d be a distance function on S which makes(S, d) a metric space of curvature bounded from above by κ and so that the metrictopology is equivalent to the surface topology. Let Σ1 be a compact Riemann surfacesof the same genus and φ : Σ1 → S a homeomorphism. Then there exists an energyminimizing map f : Σ1 → (S, d) in the homotopy class of φ, smooth metrics gi onS, and maps fi which are energy minimizing diffeomorphisms with respect to gi sothat the pull back distance function of fi converges uniformly to that of f and theenergy of fi converges to that of f .

The generalization of the classical harmonic map theory when the target is ametric space of curvature bounded from above was initiated by the work of Gromovand Schoen [GS] who developed the general existence and regularity theory for har-monic maps into non-positively curved Riemannian simplicial complexes. Korevaarand Schoen [KS1], [KS2] and also Jost [J] has further generalized the setting inwhich we consider harmonic map theory. The theory has proven to be useful inmany applications. For example, [GS] provides a new approach in the study ofp-adic representations of lattices in noncompact semisimple Lie groups.

The present investigation is motivated by the study of Teichmuller space usingharmonic maps. Gerstenhaber and Rauch [GR] proposed the problem of construct-ing the Teichmuller map by first minimizing the energy of maps from a surfacewith a fixed conformal class to a surface with a metric in a fixed conformal class(thereby obtaining a harmonic map), then maximizing the energy by varying themetric within the conformal class on the target. Further investigations were carriedout by Reich and Strebel [R], [RS], Miyahara [Mi], Leite [Lei], and Kuwert [Kuw].The result pertaining to the local behavior of a energy minimizing map is a directgeneralization of results obtained by Kuwert who studied harmonic maps into sur-faces with conical singularities of non-positive curvature. Recently, Wolf [Wo1],[Wo2], [Wo3], Fischer and Tromba [FT] and Yamada [Y] have used harmonicmaps to investigate Teichmuller spaces.

2. Preliminaries

2.1. Metric Spaces with Curvature Bounded from Above.2.1.1. Definition. First, we review the notion of curvature bounds in a metric

space X. We assume our metric spaces are length spaces, i.e. for each P , Q ∈ X,there exists a curve γPQ such that the length of γPQ is exactly d(P,Q) (which wewill sometimes write as dPQ). We call γPQ a geodesic between P and Q. We thensay that X is an NPC (non-positively curved) space if every point of X is containedin a neighborhood U so that for every P,Q,R ∈ U ,

d2PQτ

≤ (1− τ)d2PQ + τd2

PR − τ(1− τ)d2QR (1)

3

where Qτ is the point on γQR so that dQQτ = τdQR. Note that equality is achievedfor every triplet P,Q,R ∈ R2. More generally, a length space is said to havecurvature bounded from above by κ if

cosh dPQτ≤ sinh(1− τ)κdQR

sinhκdQRcosh dPQ +

sinh τκdQR

sinhκdQRcosh dPR (2)

for κ < 0 and

cos dPQτ≥ sin(1− τ)κdQR

sinκdQRcos dPQ +

sin τκdQR

sinκdQRcos dPR (3)

for κ > 0 and dPQ, dQR, dPR < π√κ. Note that if X is Sκ, a surface of constant

curvature κ, then the equality is achived for (2) when κ > 0 and for (3) when κ < 0for every triplet P,Q,R ∈ Sκ

Proposition 1. Let X be a metric space of curvature bounded from above byκ and let P,Q,R ∈ X. Let Qt be the point on γPQ so that dPQt

= (1− t)dPQ andRs be the point on γPR so that dPRs

= (1− s)dPR. For Q and R sufficiently closeto P , we have the following three cases:

Case 1: If κ = 0 then

d2QtRs

≤ tsd2QR + t(t− s)(d2

PQ − d2PR) + (t− s)2(d2

PR − d2PQ). (4)

Case2: If κ > 0 then

d2QtRs

≤ tsd2QR + t(t− s)(d2

PQ − d2PR) + (t− s)2(d2

PR − d2PQ)

+κts

((1− t)2d2

pQ + (1− s)2d2PR

6

)d2

QR +O3(dQR)

+O3(dPQ − dPR, t− s) (5)

Case 3: If κ < 0 then

d2QtRs

≤ tsd2QR + t(t− s)(d2

PQ − d2PR) + (t− s)2(d2

PR − d2PQ)

−κts

((1− t)2d2

pQ + (1− s)2d2PR

6

)d2

QR +O3(dQR)

+O3(dPQ − dPR, t− s) (6)

Proof. Case 1 has been established in [GS]. We will prove Case 2 when κ = 1.Then general inequality for κ > 0 follows from a scaling argument. Furthermmore,we will omit the proof for Case 3 since it follows from an argument analogous tothe proof for Case 2.

For Q and R sufficiently close to P , using the Law of Cosines in sphericalgeometry, we obtain

cos dQtRs≥ cos tdPQ cos sdPR +

sin tdPQ sin sdPR

sin dPQ sin dPR(cos dQR − cos dPQ cos dPR).

Now we claim that

cos tdPQ cos sdPR − 1 +sin tdPQ sin sdPR

sin dPQ sin dPR(1− cos dPQ cos dPR)

≥t(t− s)d2

PQ

2− s(s− t)d2

PR

2+O3(dPQ − dPR, t− s) (7)

4

To prove inequality (7), we let ε = dPR − dPQ and τ = s− t. Furthermore, we let

φ(ε, τ) = cos tdPQ cos[(t+ τ)(dPQ + ε)]− 1

+sin tdPQ sin[(t+ τ)(dPQ + ε)]

sin dPQ sin(dPQ + ε)(1− cos dPQ cos(dPQ + ε))

and

ψ(ε, τ) =tτd2

PQ

2− (t+ τ)τ(dPQ + ε)2

2.

At ε = 0 and τ = 0, we have the following:∂φ

∂ε=∂ψ

∂ε=∂φ

∂τ=∂ψ

∂τ= 0,

∂2φ

∂τ2=∂2ψ

∂τ2= −d2

PQ,

∂2φ

∂ε∂τ=

∂2ψ

∂ε∂τ= −tdPQ,

∂2φ

∂ε2=

12(csc2 dPQ − cos 2tdPQ csc2 dPQ − t2 csc2 dPQ + t2 cos 2dPQ csc2 dPQ),

∂2ψ

∂τ2= 0.

Sincecsc2 x− cos 2tx csc2 x− t2 csc2 x+ t2 cos 2x csc2 x ≥ 0,

by Taylor seires expansion, we see that

φ(ε, τ)− ψ(ε, τ) ≥ O3(ε, τ)

which is exactly inequality (7).We can also check that

sin tdPQ sin sdPR

sin dPQ sin dPR≥ ts

(1 +

(1− t2)d2PQ + (1− s2)d2

PR

6

).

by Taylor series expansion. Thus

d2QtRs

+O3(dQt,Rs) ≤ tsd2

PQ + ts

((1− t2)d2

PQ + (1− s2)d2PR

6

)d2

QR

+t(t− s)d2PQ + s(s− t)d2

PR

from which inequality (5) follows immdediately.

2.1.2. Isometric Embedding of Metric Spaces of Curvature Bounded from Aboveinto NPC spaces. For any metric space X, we let CX be the cone over X. Topo-logically, CX is defined by

CX = X × [0,∞)/X × 0.

A point in CX is represented by a pair [P, t], P ∈ X and t ∈ [0,∞), and (P, 0)and (Q, 0) represent the same point in CX for all P,Q ∈ X. We endow CX witha distance function D defined by

D2([P, t], [Q, s]) = t2 + s2 − 2ts cos min(d(P,Q), π).

The following is given in [Ser]:

5

Proposition 2. Let (CX,D) be as above. If X has curvature bounded fromabove by 1 then CX is a NPC space. For P,Q ∈ X with d(P,Q) sufficiently small,the geodesic between [P, t] and [Q, s] is given by

τ 7→ [Pτ , tτ ]

where Pτ is the point on the geodesic γPQ in X so that d(P, Pτ ) = τd(P,Q) and

tτ =((1− τ)t2 + τs2 − τ(1− τ)D2([P, t], [Q, s])

)1/2.

We identify X with the set [P, 1] ⊂ CX. Since

limQ→P

D2([P, 1], [Q, 1])d2(P,Q)

= limQ→P

2− 2 cos d(P,Q)d2(P,Q)

= 1,

X can be considered to be isometrically embedded in CX in the sense that thedistance functions d and D coincide infinitesmally on X ⊂ CX. Let [P, t], [Q, s] ∈CX and ε = min1− t, 1− s. Then

D2([P, t], [Q, s]) = t2 + s2 − 2ts cos dPQ

≥ (t− s)2 + ts(1− cos dPQ)

≥ (1− ε)2(1− cos dPQ)

≥ (1− ε)2D2([P, 1], [Q, 1]) (8)

2.1.3. Tangent Cones. For O in X, we define a notion of angle for geodesicsemanating from O. Such a definition is used in [ABN] to define tangent cones(which generalizes tangent spaces of Riemannian manifolds) in metric spaces.

Definition 3. Let γ, σ geodesics in X emanating from point P . Then

α(γ, σ) = limt,s→0

arccoscosκd(γ(t), P ) cosκd(σ(s), P )− κ cos d(γ(t), σ(s))

sinκd(γ(t), P ) sinκd(σ(s), P )

is called the angle between γ and σ.

Remark 4. The limit above exists because X has curvature bounded fromabove by κ.

Definition 5. Let ΛO(X) be the set of all geodesics emanating from the pointO and define an equivalence relation γ ∼ σ if α(γ, σ) = 0. Then ΩO(X) =ΛO(X)/ ∼ is called the space of directions. We denote by Π : ΛO(X) → ΩO(X) thecanonical projection and let α be a distance function in ΩO(X) defined by pushingforward α by Π. The tangent cone TOX is the cone over ΩO(X), i.e. TOX is theset ΩO(X) × [0,∞) identifying all points with zero second coordinate as the pointO. The distance function d0 on TOX is defined by

d20(x, y) = t2 + s2 − 2ts cos α(ξ1, ξ2) (9)

for x = (ξ1, t), y = (ξ2, s) ∈ ΩO(X)× [0,∞).

Remark 6. If X has curvature bounded from above by κ, then (TOX, d0) isan NPC space.

6

2.2. Sobolev Space Theory in Complete Metric Spaces. Let (Ω, g) be acompact Riemannian domain and (X, d) any complete metric space. In [KS1] and[KS2], Korevaar and Schoen develop the space W 1,2(Ω, X). Here we define thisspace and collect some of their results.

A Borel measurable map u : Ω → X is said to be in L2(Ω, X) if for P ∈ X,∫Ω

d2(u(x), P )dµ <∞.

Note that by the triangle inequality, this definition is independent of P chosen. Foru ∈ L2(Ω, X), we can construct an ε-approximate energy function eε : Ωε → Rwhere Ωε = x ∈ Ω : d(x, ∂Ω) > ε by

eε(x) =1ωn

∫S(x,ε)

d2(u(x), u(y))ε2

dσ

εn−1.

where ωn is the area form for the unit sphere S(x, 1). For any Borel measure onthe interval (0,2) satisfying

ν ≥ 0, ν((0, 2)) = 1 and∫ 2

0

λ−2dν(λ) <∞ (10)

we can also define approximate an energy density function νeε(x) : Ω2ε → R byaveraging eε(x). More precisely,

νeε(x) =∫ 2

0

eλε(x)dν(λ).

By setting νeε(x) = 0 for x ∈ Ω− Ω2ε, we can consider νeε(x) to be a L1 functionon Ω and hence it defines linear functional Eu

ε : Cc(Ω) → R. We say u ∈ L2(Ω, X)has finite energy (or that u ∈W 1,2(Ω, X)) if

Eu ≡ supf∈Cc(Ω),0≤f≤1

lim supε→0

Euε (f) <∞.

It can be shown that if u has finite energy, the measures νeε(x)dx convergeweakly a to measure, indepent of the choice of ν, which is absolutely continuouswith respect to the Lebesgue measure. Therefore, there exists a function e(x),which we call the energy density, so that eε(x)dµ e(x)dµ. In analogy to the caseof real valued functions, we often write |∇u|2(x) in place of e(x). In particular,

Eu =∫

Ω

|∇u|2dµ.

Similarly, the directional energy measures |u∗(Z)|2dµ for Z ∈ ΓΩ can also bedefined as the weak* limit of measures Zeεdµ. Here,

Zeε(x) =d2(u(x), u(x(x, ε))

ε2.

where x(x, ε) denotes the flow along Z at time ε, starting at point x. For almostevery x ∈ Ω,

|∇u|2(x) =1ωn

∫Sn−1

|u∗(ω)|2dσ(ω).

This definition of Sobolev space W 1,2(Ω, X) is consistent with the usual definitionwhen X is a Riemannian manifold. Finally, if X has curvature bounded from above

7

by κ, then for any map u ∈ W 1,2(Ω, X), we can also make sense of the notion ofthe pull back metric

π : Γ(T Ω)× Γ(T Ω) → L1(Ω)

defined by

π(V,W ) =14|u∗(V +W )|2 − 1

4|u∗(V −W )|2 for V,W ∈ Γ(T Ω).

Suppose Ω is a surface with conformal parameter z = x+ iy. Then u ∈W 1,2(Ω, X)is said to be conformal if

π

(∂

∂x,∂

∂x

)= π

(∂

∂y,∂

∂y

)and π

(∂

∂x,∂

∂y

)= 0.

λ = π(

∂∂x ,

∂∂x

)is called the conformal factor of u.

A continuous map f : Ω → X is said to be harmonic if it is locally energyminimizing. In other words, each point of x ∈ Ω has a neighborhood so that allcontinuous comparison maps which agree with f outside this neighborhood haveno less energy. The local existence of regularity for energy minimizing maps hasbeen worked out by Schoen and Korevaar in [KS1] for κ ≤ 0 and by Serbinowski[Ser] for κ > 0. We will state their regularity result:

Theorem 7. Let X be a metric space of curvature bounded from above by κ.If f : Ω → X is harmonic, then f is locally Lipschitz continuous.

We will also need the following slightly modified version of the lower semicon-tinuity of energy result of Korevaar and Schoen (Theorem 1.6.1 [KS1]).

Theorem 8 (Lower Semicontinuity of Energy). Let (X, d) and (Xk, dk), k =1, 2, .., be complete metric spaces and uk ∈ W 1,2(Ω, Xk). Suppose that the pullback distance functions δk(·, ·) = dk(uk(·), uk(·)) converge uniformly to δ(·, ·) =d(u(·), u(·)) for u ∈ L2(Ω, X). Then u ∈W 1,2(Ω, X) and its energy density measureEu = lim supε→0E

uε (f) satisfies

Eu(f) ≤ lim infk→∞

Euk(f)

for every f ∈ C∞c (Ω).

Proof. We can essentially follow the proof of Theorem 1.6.1 [KS1]. The slightmodification in their argument that is needed for our case is that equation (1.6ii)of [KS1] follows because of the uniform convergence of δk to δ.

An important tool developed in [KS2] is the mollification of maps in L2(Ω, X)when X is an NPC space. Let η ∈ C∞c (Rn) be a non-negative, monotone, radialtest function of total integral 1 and with compact support in the unit ball. Define

ηε(x) = ε−nη(xε

)For u ∈ L2(Ω, X) and z ∈ Ωε, define u ∗ ηε(z) be the center of mass of u withrespect to ν, the push-forward of the measure ηε(x)dx induced by the exponentialmap expx : TzΩ → Ω. In other words, u ∗ ηε(z) is the point in X which minimizesthe integral

Iu,ν(Q) =∫d2(u(x), Q)dν(x).

8

The existence and the uniqueness of the point minimizing the above integral followsfrom the NPC condition (see Lemma 2.5.1 in [KS1]). The following is Theorem 1.5.2of [KS2]

Theorem 9. Let X be an NPC space and u ∈ L2(Ω, X). Then for ε > 0 small(depending on (Ω, g)), the map u ∗ ηε is Lipschitz in Ωε and the Lipschitz constantonly depends on ε, the L2 oscillation of u, and the Riemannian structure of (Ω, g).Furthermore, there exists C dependeing only on (Ω, g) so that

Eu∗ηε(f) ≤ νEuε (fε) + Cε(µEε(fε))

where f ∈ C∞c (Ω2ε) and

fε(x) = f(x) + ω(f)(x, ε),

ω(f)(x, ε) = max|y−x|≤ε

|f(y)− f(x)|

and the measure µ satisfies equation 10.

We will also need the following in our analysis later on:

Proposition 10. Let X be a metric space of curvature bounded from above by κand f : Ω → X be an energy minimizing map. Fix P ∈ X and let r(x) = d(f(x), P ).For ε > 0, let Bε(x) ⊂ Ω be a geodesic ball of radius ε centered at x. Then for eachx ∈ Ω, there exists a sufficiently small ε > 0 so that for η ∈ C∞c (Bε(x)),∫

(4η)rdx ≥ 2κ∫η|∇f |2 − 2κ

3

∫ηr2|∇f |2. (11)

Proof. When κ = 0, the result follows from the proof of Proposition 2.2 of[GS]. We will prove inequality (11) when κ > 0. The case when κ < 0 follows froman analogous argument. Define a map fη : Ω → X so that the point fη(x) is onthe geodesic from P to f(x) and (1− η(x))d(P, f(x)) = d(P, fη(x)). Let Q = f(x),R = f(y), t = 1 − η(x) and s = 1 − η(y) in inequality (5). Then Qt = fη(x),Rs = fη(y) and

d2(fη(x), fη(y)) ≤ d2(f(x), f(y))− 2(η(x) + η(y))d2(f(x), f(y))

+(η(y)− η(x))(r(x)− r(y)) +O2(η)

+κ(η(x)r2(x) + η(y)r2(y)

3

)d2(f(x), f(y))

Let y = x(x, ε) be the flow starting at point x along unit vector ω ∈ Sn−1 at timeε. By dividing by ε2, integrating over Sn−1 and Ωε and letting ε→ 0, we obtain,∫

|∇fη|2dx ≤∫|∇f |2dx− 2

∫η|∇f |2 −

∫∇η · ∇r +

2κ3

∫ηr2|∇f |2 + 02(η).

Since f is an energy minimizing map we can cancel the left hand side by the firstterm on the right hand side. Finally, replacing η by τη, dividing by τ and lettingτ → 0, we obtain inequality (11).

9

3. The Order Function

In this section, we develop the order function for energy minimizing map intometric spaces of curvature bounded from above. [GS] considered this constructionfor maps into a NPC Riemannian simplicial complex. Here, extra care must betaken to overcome the technical difficulty posed by the more general curvautrebound.

Let (Ω, g) be a compact Riemannian domain, X a metric space of curvaturebounded from above, and f : Ω → X an energy minimizing map. Let Bσ be ageodesic ball of radius σ about some point in Ω and P ∈ X. Define

E(σ) =∫

Bσ

|∇f |2dµ, I(σ) =∫

∂Bσ

d2(f(z), P )dΣ, and ord(σ) =σE(σ)I(σ)

.

Proposition 11. Let X be a metric space with curvature bounded from aboveby κ and let f : Ω → (X, d) be an energy minimizing map. Then limσ→0 ord(σ)exists.

Proof. Let k be the dimension of Ω. The following equality is derived usingonly domain variations and is independent of the curvature of the target space (seefor example, [GS]):

0 = (2− n+O(σ))∫

Bσ

|∇f |2dµ+ σ

∫∂Bσ

|∇f |2dΣ− 2σ∫

∂Bσ

|∂f∂r|2dΣ.

The above equality combined with the fact that

I ′(σ)I(σ)

=k − 1σ

+ (I(σ))−1

∫∂Bσ

∂

∂r(d2(f, P ))dΣ +O(σ)

gives

d

dσlog(I(σ)σE(σ)

)=

(E(σ)

∫∂Bσ

∂∂r (d2(f, P ))dΣ− 2I(σ)

∫∂Bσ

|∂f∂r |

2dΣ)

E(σ)I(σ)+O(σ).

(12)Choose η to approximate the characteristic function of Bσ ⊂ Ω in inequality (11).Then for sufficiently small σ,∫

∂Bσ

∂

∂r(d2(f(x), P )dΣ +

2κ3

∫Bσ

d2(f(x), P )|∇f |2dµ ≥ 2∫

Bσ

|∇f |2dµ. (13)

By Theorem 7, f is locally Lipschitz. Let C = κL2

3 where L is the local Lipschitzconstant of f . Then equality (12), inequality (13), and the fact that | ∂

∂rd(f, P )| ≤|∂f∂r | yields

d

dσlog(eC0σ I(σ)

σE(σ)

)≤

((12

∫∂Bσ

∂∂r (d2(f, P ))dΣ

)2

− 2I(σ)∫

∂Bσ|∂f∂r |

2dΣ)

E(σ)I(σ)

+Cσ2(I(σ))−1

∫∂Bσ

∂

∂r(d2(f, P ))dΣ

= Cσ2

(I ′(σ)I(σ)

− k − 1σ

)≤ Cσ2 ∂

∂σlog I(σ)

10

Thus, if we let

F (σ) = I(σ) exp(C

∫ 1

σ

s2d

dslog I(s)ds

)then

d

dσlogF (σ) = (1− Cσ2)

d

dσlog I(σ)

and consequentlyd

dσlog(eC0σ σE(σ)

F (σ)

)≥ 0.

Therefore,

limσ→0

eC0σ σE(σ)F (σ)

exists. Let K = eC0 E(1)F (1) = eC0 E(1)

I(1) . Then for σ ≤ 1,

σE(σ)F (σ)

≤ K.

We show now that

limσ→0

∫ 1

σ

s2d

dslog I(s)ds <∞.

Let 0 < θ1 < θ2 ≤ 1. Then

I(θ2)− I(θ1) =∫ θ2

θ1

d

dsI(s)ds

=∫ θ2

θ1

(∫∂Bs

2d(f, P )∂

∂rd(f, P )dΣ +

k − 1s

I(s))ds

≤ 2∫ θ2

θ1

∫∂Bs

d(f, P )|∇f |dΣds+k − 1θ1

∫ θ2

θ1

I(s)ds

≤∫ θ2

θ1

∫∂Bs

(1εd2(f, P ) + ε|∇f |2

)dΣds+

k − 1θ1

∫ θ2

θ1

I(s)ds

≤ εE(θ2) +(

1ε

+k − 1θ1

)∫ θ2

θ1

I(s)ds

≤ εE(θ2) +(

1ε

+k − 1θ1

)(θ2 − θ1) sup

s∈[θ1,θ2]

I(s) (14)

Let

φ(θ, n, j) = 1− θ2j(1−θ2n)

1−θ2

2−(

2Kθ−2j(1−θ2n)

1−θ2 +k − 1θ

)(1− θ).

Then

limθ→1

φ(θ, n, j) =12

and limθ→1

∂

∂θφ(θ, n, j) = −1 + k + 2K − jn.

In particular, we see that for n1 < n2 and j1 < j2,

limθ→1

∂

∂θφ(θ, n1, j1) > lim

θ→1

∂

∂θφ(θ, n2, j2)

11

Therefore, there exists θ0 sufficiently close to 1 so that φ(θ0, n, j) > 14 independently

of n and j. Choose j so that θj0 <

14 . Then

1− 12θ

2j(1−θ2n0 )

1−θ20

0 −

2Kθ

−2j(1−θ2n0 )

1−θ20

0 +k − 1θ0

(1− θ0) > θj0 (15)

for any n. Let ε = θ

2j(1−θ2n0 )

1−θ20

02K , θ1 = θ0 and θ2 = 1 in inequality (14). Then for

n = 0, [1− 1

2−(

2K +k − 1θ0

)(1− θ0)

]I(1) ≤ I(θ0)

and by inequality (15), θj0I(1) < I(θ0). Now suppose θj

0I(θk0 ) < I(θk+1

0 ) for k =1, ..., n− 1. Then∫ 1

θn0

s2d

dslog I(s)ds =

n−1∑k=0

∫ θk0

θk+10

s2d

dslog I(s)ds

≤n−1∑k=0

θ2k0

∫ θk0

θk+10

d

dslog I(s)ds

≤n−1∑k=0

θ2k0 log

I(θk0 )

I(θk+10 )

≤n−1∑k=0

θ2k0 log θ−j

0

=n−1∑k=0

log θ−jθ2k0

0

= log θ−j∑n−1

k=0θ2k0

0

= log θ

−2j(1−θ2n0 )

1−θ20

0

Therefore,

logF (1)F (θn

0 )=

∫ 1

θn0

(1− s2)d

dslog I(s)ds

= logI(1)I(θn

0 )−∫ 1

θn0

s2d

dslog I(s)ds

≥ log θ

2j(1−θ2n0 )

1−θ20

0

I(1)I(θn

0 )

and since σE(σ)F (σ) ≤ K for any σ,

θn0E(θn

0 )I(θn

0 )≤ θn

0E(θn0 )

F (θn0 )

θ

−2j(1−θ2n0 )

1−θ20

0 ≤ Kθ

−2j(1−θ2n0 )

1−θ20

0

12

Let ε = θ

2j(1−θ2n0 )

1−θ20

02K , θ1 = θn

0 , and θ1 = θn+10 in inequality (14). Then1− θ

2j(1−θ2n0 )

1−θ20

0

2−

2Kθ

−2j(1−θ2n0 )

1−θ20

0 +k − 1θ0

(1− θ0)

I(θn+10 ) ≤ I(θn

0 )

and by inequality (15),θj0I(θ

n0 ) < I(θn+1

0 ). (16)By induction, inequality (16) holds for all n which in turn inplies that∫ 1

θn0

s2d

dslog I(s)ds ≤ log θ0

−2j(1−θ2n0 )

1−θ20

holds for all n. Letting n→∞, we obtain

C1 = C

∫ 1

θn0

s2d

dslog I(s)ds ≤ C log θ0

−2j

1−θ20 <∞

Therefore

limσ→0

F (σ)I(σ)

= eC1 <∞

and hence

limσ→0

eC0σ σE(σ)I(σ)

= eC1 limσ→0

eC0σ σE(σ)F (σ)

.

For an energy minimizing map f : Ω → X and x ∈ Ω, let

ord(x, σ, f(x)) =σ∫

Bσ(x)|∇f |2dµ∫

∂Bσ(x)d2(f, f(x))dΣ

andOrd(x) = lim

σ→0ord(x, σ, f(x)).

From the proof of Proposition 11, we see that Ord(x) is a upper semicontinuousfunction since it is a decreasing limit of continuous functions.

Lemma 12. Let f : Ω → X be a non-constant energy minimizing map. For anyz ∈ Ω, Ord(z) ≥ 1.

Proof. LetS1 = z ∈ Ω : |∇f | 6= 0

and

S2 = z ∈ Ω : |∇f |2 = limσ→0

1|∂Bσ(z)|

∫∂Bσ

d2(f(x), f(z))σ2

dΣ(x)

By Theorem 4.1 of [Me1], |∇f |2 ≥ −2κL4 locally and thus,

|∇f |2(z) = limσ→0

1|Bσ|

∫Bσ

|∇f |2(x)dµ

everywhere. Therefore, for z ∈ S1 ∩ S2,

limσ→0

σ∫

Bσ(z)|∇f |2dµ∫

∂Bσ(z)d2(f(x), f(z))dΣ

= 1.

13

Since Ord(x) is an upper semicontinuous function, we need just to show that S1∩S2

is a dense set in D. This will be established from the following two claims:

Claim 1 Let f : Ω → X be a non-constant energy minimizing map. Then forevery z ∈ Ω and σ > 0 so that Bσ(z) ⊂ Ω, m(Bσ(z) ∩ S1) > 0.

From Theorem 4.1 of [Me1], |∇f |2 ≥ −2κL2|∇f |2 weakly in Ωε (where L dependson ε). This implies that there exists a constant c so that for each Bδ(x0) ⊂ Ωε

|∇f |2(x0) ≤ c

∫Bδ(x0)

|∇f |2(x)dx.

If m(Bσ(z) ∩ S1) = 0, then for every x0 ∈ Bσ(z), |∇f |2(x0) = 0 and f is constantin Bσ(z). On the other hand, the arguments of Proposition 3.4 of [GS] implies thatif f is constant on an open set, then f is identically constant. This contradicts theassumption of the claim.

Claim 2 Let f : Ω → X be a non-constant energy minimizing map. Thenm(Ω− S2) = 0

Let ω be a unit vector. Then by Theorem 1.9.6 of [KS1],

|u∗(ω)|2(z) = limσ→0

d2(f(z), f(z + σω)σ2

for a.e. z ∈ Ω. Furthermore, equation (1.10v) of [KS1] gives

|∇f |2(z) =1

|Sn−1|

∫Sn−1

|u∗(ω)|2(z)dΣ(ω).

Hence, by the Lebesgue Dominated Convergence Theorem, for a.e. w ∈ D,

|∇f |2(w) = limσ→0

1|∂Bσ(w)|

∫∂Bσ(w)

d2(f(w), f(z))σ2

dΣ(w).

In other words, m(Ω− S2) = 0. This establishes Claim 2.

Thus, for every σ > 0, m(Bσ(z) ∩ S1 ∩ S2) = m(Bσ(z) ∩ S1) > 0. This showsthat Bσ(z)∩S1∩S2 6= ∅ and every z ∈ Ω is a limit of a sequence zi ⊂ S1∩S2.

4. Compactness Theorem for Energy Minimizing Maps

In this section, we prove the compactness theorem for energy minimizing mapsmentioned in the introduction.

Theorem 13. Let (M, g) be a compact Riemannian manifold without a bound-ary and let di be distance functions on X with curvature bounded from above byκ. Assume X is compact with respect to the metric topology induced by d. Leth : M → X be a continuous map and let fi : M → (X, di) be energy minimizingmaps in the homotopy class of h with fi = h on ∂M if ∂M 6= ∅. Let δi be the pullback distance function of di under fi, i.e.

δi(·, ·) = di(fi(·), fi(·)).

14

If the energy of fi is bounded above by K for each i and if di converges uniformlyto a distance function d0, then there exists a subsequence i′ ⊂ i and an en-ergy minimizing map f0 with respect to d0 so that δi′(·, ·) converges uniformly tod0(f(·), f(·)) and the energy of fi′ converges to that f0.

Proof. First we assume that ∂M = ∅. By the result of [Ser], the localLipschitz bound of fi is dependent only on the total energy of fi. Hence thereexists L chosen independently of i so that δi(x, y) = L|x − y| where | · − · | is thedistance function on M (induced by the Riemannian metric g). By the triangleinequality,

|δi(x1, y1)− δi(x2, y2)| ≤ δi(x1, x2) + δ(y1, y2)≤ L(|x1 − x2|+ |y1,−y2|)

and we see that δi : M ×M → R is an equicontinuous family of functions. Hencethere is a subsequence, which we will still denote by δi which converges uniformlyto say δ0. We now construct a map f0 so that

δ0(·, ·) = d0(f0(·), f0(·))Let S be a countable dense subset of X. Since X is compact, for a fixed x ∈ S, thesequence fi(x) converges. By using a diagonalization process, we may assumefi(x) converges to a point, say px, for every x ∈ S. We set f0(x) = px for allx ∈ S. For any x, y ∈ S,

d(px, py) = limi→∞

d(fi(x), fi(y))

= limi→∞

di(fi(x), fi(y))

= L|x− y|by the uniform convergence of di to d0. In particular, this shows that if xk ⊂ Sconverges to z ∈ X, then pxj converges, say to q, and thus we define f0(z) = q. Inthis way, we can define f0(z) for all z ∈ X.

We now show that f0 is an energy minimizing map. The idea for this portionof the proof is derived from Theorem 3.9 of [KS2]. Some modification is needed tohandle the more general curvature bound. Let h be a competitor of f0. Let diEh

(resp. diν E

hε ) be the energy (resp. the ε-approximate energy with measure ν) of h

with respect to the distance function di. Since di → d0 uniformly, given δ > 0,diν E

hε ≤ d0

ν Ehε + δ ≤ d0Eh + 2δ (17)

for sufficiently small ε > 0 and sufficiently large i.Fix di and let CX be the cone over (X, di) (see Section 1.1) and let h : M → CX

be the map defined by h(z) = [h(z), 1]. Note that h(N) ⊂ [P, 1] : P ∈ X andhence h ∈ L2(M,CX) since X is compact. By Theorem 9, we can mollify h toobtain a Lipschitz map h ∗ ηε with the property that

D(h ∗ ηε(z), h(z)) < O(ε) a.e.

where D denotes the distance function of CX andDEh∗ηε ≤ (1 + Cε) D

ν Ehε = (1 + Cε) di

ν Ehε (18)

where DE denotes the energy with respect to the distance function D in CX. Byinequality (8),

(1−O(ε))2D(π (h ∗ ηε)(z1), π (h ∗ ηε)(z1)) ≤ D(h ∗ ηε(z1), h ∗ ηε(z2)).

15

where π is the projection π([P, t]) = [P, 1]. Let hi : M → (X, di) be the map definedby taking the first coordinate of π (h ∗ ηε) (or equivalently the first coordinate ofh ∗ ηε). Then the above inequality and the fact that X is isometrically embeddedin CX implies that

(1−O(ε))2 diEhi ≤ DEh∗ηε .

Combining this with inequality (17) and inequality (18), we obtain

(1−O(ε))2 diEhi ≤ (1 + Cε)( d0Eh + 2δ).

But since fi is the energy minimizing map for the distance function di,

(1−O(ε))2 diEfi ≤ (1 + Cε)( d0Eh + 2δ).

Noting that ε and δ were arbitrarily,

lim supi→∞

diEfi ≤ d0Eh.

Now combining this with the lower semicontinuity result, we obtaind0Ef ≤ lim inf

i→∞diEfi ≤ lim sup

i→∞

diEfi ≤ d0Eh

which shows that f is a energy minimizing as a map into (X, d). We have alsoshown that the energy of fi converges to that of f0.

If ∂M 6= ∅, then let Mk be a compact exhaustion of M . The Lipschitzbound for fi is uniform in each Mk and we can choose a subsequence of δiwhich converges uniformly in Mk each k. By a diagonalization procedure, we maychoose a subsequence, still denoted by δi, which converges uniformly to δ0 inM . Proceeding as above, we define a map f0 so that a δi converges uniformly tod0(f0(·), f0(·)). Again, let h be a competitor for f0. We use the above constructionto obtain a map π (h ∗ ηε). Now note that π (h ∗ ηε) is not a competitor for theDirichlet problem with respect to the distance function di since the mollified maph is not defined near ∂M . On the other hand, by the Bridge Lemma (Lemma 3.12)of [KS2], we can define a map, which we can call gε : M → CX so that π gε = hon ∂M and diEπgε − diEh ≤ O(ε). This proves that f0 is an energy minimizingmap.

Corollary 14. Let fi′ , f0, di′ , and d0 as above. Then the energy density mea-sures and the directional energy density meaures of fi′ converges weakly to those off0.

Proof. Since there is no loss in the total energy of the limit map and becausethe Sobolev and directional energy functionals are lower semicontinuous it followsthat the corresponding measures actually converge weakly.

5. Harmonic Maps into Surfaces

5.1. Conformal Representation of NPC Surfaces. In this section, wewill explain how a distance function induces a conformal structure. We need thefollowing theorem.

Theorem 15. Suppose a compact surface S is endowed with a distance functiond which makes (S, d) into a metric space of curvature bounded from above by κ andthe metric topology of d is equivalent to the surface topology. Then for every P ∈ S,there is a neighborhood U of P and a conformal homeomorphism ψ : D → U from

16

the unit disk D in the complex plane. If λ is the conformal factor of ψ, then log λis a subharmonic function in D.

Thus (S, d) has a coordinate system in which d is represented locally by λ|dz|2.An area of the surface with respect to a distance function d is given by∫

Σ

λ|dz|2,

and is a direct generalization of the area for smooth metrics. We also have thefollowing theorem of Huber [Hu]:

Theorem 16. Let U1, U2 ⊂ S and ψ1 : D → U1, ψ2 : D → U2 be conformalmaps with conformal factors λ1 and λ2. If U1 ∩ U2 is non-empty, then the mapT = ψ−1

2 ψ1 : ψ−11 (U1 ∩ U2) → ψ−1

2 (U1 ∩ U2) is a conformal map. Moreover,

λ2(z) = |T ′(z)|2λ1(z).

The above two theorems gives (S, d) a structure of a Riemann surface. Us-ing the uniformization theorem, there is a Riemannian surface Σ and a conformalhomeomorphism ι : Σ → (S, d). The following can be found in [Me1]:

Theorem 17. Let φ : D → (S, d) be a conformal energy minimizing map. Theconformal factor λ satisfies, ∫

(4η) log λ ≥ −2κ∫ηλ (19)

for every η ∈ C∞c (D).

Therefore, if λ is the conformal factor of the map ι : Σ → (S, d), λ satisfies theinequality (19). Furthermore, by the result of [Ser], there is L so that λ ≤ L.

Given a locally bounded function λ on a domain Ω ⊂ R2, we can define adistance function d by setting

d(w1, w2) = infl(γ) : γ ∈ C

where C is the collection of all smooth curves from w1 to w2 in Ω and

l(γ) =∫

γ

√λds.

We have the following converse to Theorem 15:

Theorem 18. Suppose Σ has a quadratic differential σ that can be expressed inlocal coordinate system by λ|dz|2. If log λ satisfies inequality (19), then there is adistance function d (defined locally by the above construction and extended globally)which makes (Σ, d) locally into a metric space of curvature bounded from above byκ.

Remark 19. Theorem 15 (the existence of isothermal coordinates) was orig-inally proved by Reshetnyak [Re1] in the 1960’s. It is also proved under a moregeneral context by the author in [Me2] using the variational theory of [KS1] andthe minimal surface theory for NPC spaces developed in [Me1]. Theorem 18 is alsoproved in [Me1] and by Huber [Hu].

17

Two metric spaces (X1, d1) and (X2, d2) is said to be infinitesmally isometricif there is a homeomorphism h : X1 → X2 so that the (Korevaar-Schoen) energyof the any f : Ω → (X1, d1) is equal to that of h f : Ω → (X2, d2). Fromthe above discussion, (S, d) and (Σ, dλ) are infinitesmally isometric. Hence, inthe discussions below, we will identify (S, d) with (Σ, σ) where σ is a (possibly non-smooth and degenerate) quadratic form represented locally by λ|dz|2 and λ satisfiesinequality (19). Let λh|dz|2 be the local expression of the hyperbolic metric on Σ.We define the function ρ : Σ → R by λ|dz|2 = ρλh|dz|2.

5.2. Harmonic Map as a Limit of Diffeomorphisms. Let S be a surfaceof genus s and (S, d) be a metric space of curvature bounded from above by κ. LetΣ1,Σ2 be Riemann surface of genus s so that Σ2 is as Σ above.

Theorem 20. Let φ0 : Σ1 → S be a homeomorphism. Then there exists anenergy minimizing map f : Σ1 → (S, d) in the homotopy class of φ.

Proof. We will provide an alternate proof from that given in [KS1]. The nicefeature of this proof is that f is realized as a certain limit of maps fi which arediffeomorphic and energy minimizing with respect to smooth Riemannian metricswith an upper curvature bound of κi (which tends to κ in the limit) on Σ2.

The key is to construct smooth Riemannian metrics on Σ2 which approximateρg and then to use Proposition 13. We do this by taking the appropriate smoothapproximation ρσ of the function ρ. Let u = log ρ and v = log λ.

Let ησ ∈ C∞([0, 1]) be a non-negative function with so that∫C

η(|z|)|dz|2 = 1

and let

ησ = σ−2η(t

σ)

Define

uσ(z) =∫

Σ2

ησ(d(z, ξ))u(ξ)dµg(ξ)

where dg(·, ·) is the distance function and dµg is the volume form for g. Then uσ

has the following properties:

(1) For every ε > 0 there exists σ0 > 0 so that for every z, uσ2(z) − uσ1(z) > −εwhenever σ1 < σ2 < σ0.(2) uσ(z) → u(z)(3) 4g(uσ + v) ≥ −2κeuev − o(σ)(4) For sufficiently small σ, uσ ≤ log 2M where M is the bound for eu

Proof of property (1): Since we can work locally, we assume that euev ≤ L.and thus u+ v + L(x2 + y2) is subharmonic. Therefore, (u+ v + L(x2 + y2))σ2 ≥(u+ v +L(x2 + y2))σ1 whenever σ2 > σ1. Hence uσ2 − uσ1 ≥ (v +L(x2 + y2))σ1 −(v + L(x2 + y2))σ2 ≥ −|(v + L(x2 + y2))σ2 − v| − |(v + L(x2 + y2))σ1 − v|. But thecontinuity of v implies that (v + L(x2 + y2))σ → v + L(x2 + y2) uniformly whichimplies property (1).

18

Proof of property (2): From the proof of property (1), it is clear that for anyε > 0, uσ − u > −ε for sufficiently small σ.

uσ(z)− u(z) =∫

Σ2

ησ(d(z, ξ))u(ξ)dµg(ξ)− u(z)

=∫

Σ2

ησ(d(z, ξ)) (u(ξ)− u(z)) dµg(ξ)

+(∫

Σ2

ησ(d(z, ξ))dµg(ξ)− 1)u(z).

The first term on the right hand side can be made to be less than ε2 for sufficiently

small σ since u is upper semicontinuous. The second term can also be made to beless than ε

2 for sufficiently small σ since g is a smooth metric of constant curvature−1 and we have ∫

Σ2

ησ(d(z, ξ))dµσ(ξ) → 1 (20)

uniformly as σ → 0. Thus, for sufficiently small σ, uσ(z) − u(z) < ε which provesproperty (2).

Proof of property (3):

4guσ(z) =∫4gησ(d(z, ξ))u(ξ)dµg

=∫

1ev4ησ(d(z, ξ))u(ξ)ev|dξ|2

=∫4ησ(d(z, ξ))u(ξ)|dξ|2

≥ −2κ∫ησ(d(z, ξ))euev|dξ|2 −

∫ησ(d(z, ξ))4v(ξ)|dξ|2

= −2κ∫ησ(d(z, ξ))eudµg − 2

∫ησ(d(x, y))dµg(w)

≥ −2κeu − 2κ(∫ησ(d(z, ξ))eudµg − eu)−4gv

+2(1−∫ησ(d(z, ξ))|dξ|2)

= −4gv − 2κeu − o(σ)

Hence, 4(uσ + v) ≥ −2evφ(σ). This proves property (3).Proof of Property (4): Since eu ≤M ,

u(z) = logM∫

Σ2

ησ(d(z, ξ))dµg(ξ) < log 2M

for sufficiently small σ. This ends the proofs of properties (1)-(4).

We note that property (4) and the proof of Theorem 6.1 of [Me1], we see thatρσg has curvature bounded from above by κ+ o(σ).

A theorem of Schoen and Yau [SY] gives the existence of a harmonic diffeo-morphism with respect to the hyperbolic metric g homotopic φ. Thus, without theloss of generality, φ can be assumed to be diffeomorphic with respect to ρσg for all

19

σ. By a theorem of Jost and Schoen [JS], there exists a energy minimizing diffeo-morphism fσ : Σ1 → (Σ2, ρ

σg) in the homotopy class of φ. Let δσ : Σ1 × Σ1 → Rbe defined by

δσ(z1, z2) = dσ(fσ(z1), fσ(z2)),where dσ is the distance function on Σ2 associated with the metric ρσg. By Propo-sition 13, there exists a map f and a sequence σi → 0 so that the pull back distancefunctions δσi

converge uniformly to d(f(·), f(·)) and f minimizes energy with re-spect to the metric ρg.

5.3. Local Analysis. In this section, we will investigate the local behavior ofan energy minimizing map f : Σ1 → (Σ2, σ) where σ = λ|dz|2 is again a (possiblynonsmooth and degenerate) quadratic form with λ satisfying inequality (19). Thelocal analysis for energy minimizing maps into surfaces with cone singularity wascarried out by Kuwert [Kuw]. We are able to duplicate his work in the more generalsetting using the order function of Section 3 and the existence of tangent maps to fat a point (Theorem 21). In this section, we record the result of this local analysis.

An important tool for this study is the Hopf differential Φ defined by locallyby ϕ|dz|2 where

ϕ = π

(∂

∂x,∂

∂x

)− π

(∂

∂y,∂

∂y

)− 2iπ

(∂

∂x,∂

∂y

).

Even when the target of f is nonsmooth, ϕ is holomorphic and Φ is a holomorphicquadratic differential on the Riemann surface Σ1 (see for example [Sch] wherethe proof of this property depends only on the domain variations). When Φ ≡0, f is a conformal map. The local behavior of a conformal energy minimizingmap was analyzed by the author in [Me2]. When this is the case, f is a localhomeomorphism. Hence, we will assume that Φ is not identically zero. Since Φ is aholomorphic differential, it has finite zeroes counting multiplicities. Locally, Φ canbe written as

Φ =(m+ 2

2

)2

zmdz2

where the integer m is the vanishing order of Φ at 0. z is then called the naturalparameter of Φ. The integral curves of the distribution v ∈ TΣ1 : Φ(v, v) ≤ 0is called the vertical trajectory and any arc contained in the vertical trajectory iscalled a vertical arc. The Hopf differential gives a geometric picture of harmonicmaps as the vertical trajectories give the direction of minimal stretch.

By blowing up the map f at the origin, we can construct a tangent map Tfto f . The tangent map was utilized by Gromov and Schoen [GS] in their analysisof harmonic maps into a Riemannian simplicial complex. Because we have gener-alized their construction of the order function, we are also able to duplicate theirconstruction of the tangent map.

Theorem 21. Let f be an energy minimizing map from the unit disk D into(Σ2, d), a locally compact metric space of curvature bounded from above by κ. Con-sider fε to be the rescaling of f into Σ2 with a rescaled distance function. Moreprecisely, let fε : D → (Σ2,

1√I(ε)

d) be defined by fε(z) = f(εz) and let (Tf(0)Σ2, d0)

be the tangent cone of (Σ2, d) at f(0). Then there is a sequence εi → 0 and a homo-geneous map Tf0 : D → Tf(0)Σ2 so that 1√

I(εi)d(fεi(·), fεi(·)) converges uniformly

to d0(Tf0(·), T f0(·)).

20

Proof. This follows by using Proposition 11, Lemma 12, and Theorem 13 andfollowing the outline of the construction of blow up map in Part I of [GS].

Since Σ2 is a surface, the tangent cone Tf(0)Σ2 is isometric to a cone and can beexpressed locally as (D, ds2) where ds2 = α2|w|2(α−1)|dw|2. Hence, we will considerTf as a map from D into (D, ds2). Because (Σ2, d) is an NPC space, the cone isnon-positively curved which means that α ≥ 1. Kuwert [Kuw] has analyzed thebehavior of homogenous harmonic maps into cones.

Lemma 22. Let the reference map, w0 : C → C, be defined as follows:

w0(z) =

z for k = 0(12

(k−

12 z

m+22 + k

12 z

m+22

)) 2m+2

for 0 < k ≤ 1.

Let w : D → (D, ds2) be a homogeneous degree τ harmonic map and assume that wis normalized so that the Hopf differential is given in terms of the natural parame-ters. Then after a suitable rotation, one of the following alternatives hold:

(a) w(z) = w0(z)τ/α where k = 0, τ/α ∈ N(b) w(z) = w0(z)τ/α where 0 < k < 1, τ/α ∈ N and m = 2(τ − 1) ∈ N(c) For l ∈ 1, 2, ...,m+2 there exists ωl ∈ S1 such that w(reiθ0) = |w0(reiθ0)|τ/αωl

for(l − 1

2

)2π

m+2 ≤ θ0 ≤(l + 1

2

)2π

m+2 where k = 1,m = 2(τ − 1) ∈ N and6 (ωl, ωl+1) ≥ π

α .

Using the above, we can classify the tangent maps Tf (after normalizationof depending on the Hopf differential described above) by associating the numberk ∈ [0, 1] refered to as the stretch of f . In fact, given an energy minimizing mapf , k is a well-defined number at each point on the domain of f and describes thelocal behavior of f . This is a consequence of the following lemma:

Lemma 23. Let f be an energy minimizing.(i) All tangent maps of of f at 0 have the same stretch k.(ii) If k(0) < 1 then 0 is an isolated point of f−1(0).

Proof. This lemma follows in our situation because of the construction of theorder function for energy minimizing maps into metric spaces of curvature boundedfrom above. We can follow the argument of Lemma 5 of [Kuw] where Kuwert usesthe order function of [GS] to prove the result in the case when the target map is asurface non-positively curved cone singularities.

Further analysis as in Lemma 6 and Theorem 1 of [Kuw] and the fact that wehave shown the energy minimizing map is a limit of homeomorphisms gives us thefollowing local behavior of f :

Theorem 24. Let f : Σ1 → Σ2 be an energy minimizing map with respect todistance function d which makes (Σ2, d) into metric space of curvature bounded fromabove by κ. For z0 ∈ Σ1 so that Φ(z0) 6= 0, there is a square Qδ = z : |x|, |y| ≤ δgiven in terms of the natural parameter z around z0 so that f−1(f(0)) ∩ Qδ = 0,f−1(f(0)) ∩ Qδ = iy : 0 ≤ y ≤ δ, or f−1(f(0)) ∩ Qδ = iy : −δ ≤ y ≤ δ.Consequently, if h is a homeomorphism and f is the energy minimizing map inits homotopy class, then the preimage f−1(w) for w ∈ Σ2 is either a point or aconnected union of finitely number of vertical arcs of the Hopf differential of f .

21

References

[Ahl] L. Ahlfors. Lectures on Quasiconformal Mappings. D. Van Nostrand Company, Princeton.[FT] A.E. Fischer and A.J. Tromba. A new proof that Teichmuller space is a cell. Trans. Amer.

Math. Soc. 303 (1987), no.1, 257-262.

[GR] M. Gerstenhaber and H.E.Rauch On extremal quasiconformal mapppings I, II. Proc. N.A.S.40 (1954), 808-812, 991-994.

[GS] M. Gromov and R. Schoen. Harmonic maps into singular spaces and p-adic superrigidity

for lattices in groups of rank one. IHES Publications Math’ematiquej 76 (1992), 165-246.[HL] R. Hardt and F.H.Lin. it Harmonic maps into round cones and singularites of nematic liquid

crystals. Math. Z. 213 (1993) no.4, 575–593.[HK] W.K. Hayman and P.B. Kennedy. Subharmonic functions, vol. 1. Academic Press, London,

1976.

[Hu] A. Huber, Zum potentialtheoretischen Aspekt der Alexandrowschen Flachentheorie. Com-

ment. Math. Helv. 34 (1960), 99-126.[J] J. Jost. Generalized harmonic maps between metric spaces. Geometric analysis and the

calculus of variation (1996), 143-174.[JS] J. Jost and R. Schoen. On the existence of harmonic diffeomorphims between surfaces.

Inventiones Mathematica 66 (1982), 353-359.

[KS1] N. Korevaar and R. Schoen. Sobolev spaces and harmonic maps for metric space targets.Communications in Analysis and Geometry 1 (1993), 561-659.

[KS2] N. Korevaar and R. Schoen. Global existence theorem for harmonic maps to non-locally

compact spaces. Communications in Analysis and Geometry 5 (1997), 333-387.[Kuw] E. Kuwert. Harmonic maps between flat surfaces with conical singularities. Mathematische

Zeitschrift 221 (1996) 421-430.

[Lei] M. Leite. Harmonic mappings of surfaces with respect to degenerate metrics. AmericanJournal of Math. 110 (1988), 399-412.

[R] E. Reich. On the variational principle of Gerstenhaber and Rauch. Ann. Acad. Sc. Fennicae

Ser. A 10 (1985), 469-475.[RS] E. Reich and K. Strebel. On the Gerstenhaber-Rauch principle. Israel J. Math. 57 (1987),

89-100.[Me1] C. Mese. Curvature of minimal surfaces in singular spaces. to appear in Communications

in Analysis and Geometry.

[Me2] C. Mese. The structure of singular spaces of dimension 2. to appear in Manuscripta Math-ematica.

[Mi] Y. Miyahamra. On some properties of a Teichmuller mapping. TRU Math. 4 (1968), 36-43.

[Re1] Y.G. Reshetnyak. Geometry IV Non-regular Riemann geometry. Translated by E. Primrose.Encyclopedia of Mathematical Science, 70. Springer-Berlag, Berlin, 1993.

[Re2] On the isoperimetric property of two dimensional manifolds of curvature not greater than

K. Vestn. Leningr. Univ, 16 no. 19 (1961), 58-76.[Sch] R. Schoen. Analytic aspects of the harmonic map problem. Seminar in non-linear partial

differential equations. MSRI vol 2 (S.S. Chern, editor). Springer, Heidelberg and New York,

1985.[SY] R. Schoen and S.T.Yau. On univalent harmonic maps between surfaces. Inventiones Math-

ematica 44 (1978), 265-278.

[Ser] T. Serbinowski. Harmonic maps into metric spaces with curvature bounded above. Ph.D.thesis. University of Utah, 1995.

[Wo1] M. Wolf. Harmonic maps from surfaces to R-trees. Math. Z. 218 (1995) no.4 577-593.[Wo2] M. Wolf. High energy degeneration of harmonic maps between surfaces and rays in Te-

ichmuller space. Topology 30 (1991). no.4, 517-540.[Wo3] M. Wolf. The Teichmuller theory of harmonic maps. Journal of Differential Geometry 29

(1989) no.2, 449-479.

[Y] S. Yamada. Weil-Peterson convexity of the energy functional on classical and universal

Teichmuller spaces. Journal of Differential Geometry 51 (1999) no.1.

Box 5657, Department of Mathematics, Connecticut College, 270 Mohegan Ave.,New London, CT 06320

E-mail address: cmes@conncoll.edu