Local properties of harmonic maps

Chapter V: The factorization theorem

5.2 Local properties of harmonic maps

We simplify our study of singularities using complex analytic methods. Without the conformal hypothesis, the only local information we have comes from the Hartman- Wintner representation formula. If this is the main tool, then it is also natural to ask about more general solutions to second order semiliinear elliptic systems, rather than just harmonic maps. An analysis of singularities would be related to understanding local behaviour of spherical harmonics.

A substitute for the unique continuation property seems to be a large hurdle. Implicit in the proof of the unique continuation property for minimal maps is the following result (see [GOR73, Lemma 2.10]).

Proposition 5.1.3. Let D ⊂ R² be the unit disk. Suppose 𝑢

1, 𝑢

2 : D → 𝑀 are minimal maps such that, for all open sets 𝐷

1 ⊂ Dcontaining 0, there is an open subset𝐷

2 ⊂ D(possibly not containing0) such that𝑢2(𝐷

2) ⊂𝑢1(𝐷

1). Then there exists an open subset 𝐷^′ ⊂ Dcontaining0such that𝑢²(𝐷^′) ⊂𝑢¹(D).

This result above fails emphatically if we replace minimal maps with harmonic maps, even if𝑀 =R². Indeed, the simple example

𝑢1(𝑥 , 𝑦) =(𝑥 , 𝑥 𝑦) , 𝑢

2(𝑥 , 𝑦) =(𝑥 , 𝑦)

does not satisfy Proposition 5.1.3. On the other hand, our “holomorphic unique continuation property” provides a substitute for Proposition 5.1.3 (see Proposition 5.3.5). This is one of the reasons we expect a more general version of Theorem 5A to be much more delicate, and we defer this investigation to a future project.

Acknowledgements

Many thanks to Vlad Marković for encouragement and sharing helpful ideas. I would also like to thank John Wood and Jürgen Jost for comments on earlier drafts.

Analytic continuation

Until Section 5.4, let 𝑓 : (Σ, 𝜇) → (𝑀 , 𝜈) be an admissible harmonic map with non-zero Hopf differential Φ and let ℎ : Ω₁ → Ω₂ be a holomorphic map as in the statement of Theorem 5A. We treat anti-holomorphic maps in Section 5.3.

Throughout the chapter, we letZ denote the zero locus ofΦ. By restricting, we assumeΩ₁is aΦ-disk.

We use the geometry of the Hopf differential to analytically continueℎ. Let𝑝 ∈Ω₁ be such that Φ(𝑝) ≠ 0, and let 𝑈 ⊂ Ω₁ be an open subset containing 𝑝 such thatΦ ≠ 0 in 𝑈. Given a holomorphic local coordinate 𝑧 in𝑈, we define a local coordinate𝑤onℎ(𝑈)by𝑤=𝑧◦ℎ⁻1. In these coordinates,ℎis given by𝑤(ℎ(𝑧))=𝑧 and

𝑑 𝑓_𝑝 𝜕

𝜕 𝑧

=𝑑 𝑓_ℎ(𝑝) 𝜕

𝜕 𝑤

∈𝑇_𝑓(𝑧)𝑀 ⊗C.

Remark 5.2.1. Here we are viewing𝑑 𝑓 as a map from𝑇Σ →𝑇 𝑀rather than as a section of the endomorphism bundle𝑇^∗Σ⊗ 𝑓^∗𝑇 𝑀.

Choosing𝑧to be a natural coordinate with𝑧(𝑝) =0, we obtain

⟨𝑓_𝑤, 𝑓_𝑤⟩ (𝑤(ℎ(𝑧))) =⟨𝑓_𝑧, 𝑓_𝑧⟩ (𝑧) =1.

Therefore, 𝑤 defines a natural coordinate on ℎ(𝑈). We have proved the following lemma.

Lemma 5.2.2. ℎis a local isometry in theΦ-metric. IfΩ₁is aΦ-disk then so isΩ₂, andℎtakes a natural coordinate𝑧onΩ₁to a natural coordinate𝑤onΩ₂in which 𝑤(ℎ(𝑧)) =𝑧.

The goal of this subsection is to prove the proposition below. In the proof we use the notion of a maximal Φ-disk. See section 5 in [Str84] for a detailed discussion on maximalΦ-disks. LetZdenote the zero set ofΦ(which is isolated).

Proposition 5.2.3. Suppose Ω₁,Ω₂ are Φ-disks with no zeros of Φ and that 𝛾 : [0, 𝐿] → Σ is a curve starting in Ω₁ and that 𝛾 first strikes 𝜕Ω₁ at a point 𝑞. If there is an𝜖 >0such that

min inf

𝑠∈𝛾|Ω 1

,𝑡∈Z

𝑑(𝑠, 𝑡), inf

𝑠∈𝛾|Ω 1

,𝑡∈Z

𝑑(ℎ(𝑠), 𝑡) ≥ 𝜖

then there is a neighbourhood of𝑞 in whichℎ can be analytically continued along 𝛾.

Proof. We can choose an arc on 𝜕Ω₁ centered at 𝑞 on which Φ ≠ 0. We then connect the endpoints via an arc contained insideΩ₁so that the enclosed region𝑈 is a topological disk. We pick these arcs in such a way that

min inf

𝑠∈𝑈 ,𝑡∈Z

𝑑(𝑠, 𝑡), inf

𝑠∈𝑈 ,𝑡∈Z

𝑑(ℎ(𝑠), 𝑡) ≥ 𝜖/2.

The restriction of theΦ-metric to any compact region that does not intersect Zis complete. Asℎis an isometry in theΦ-metric, we can extend it to a mapℎ :𝑈 →𝑈. Therefore, we have a well-defined pointℎ(𝑞).

For every point 𝑝 ∉ Z, there is a maximal radius𝑟_𝑝 such that we can extend any natural coordinate centered at𝑝to aΦ-disk of radius𝑟_𝑝. 𝑟_𝑝does not depend on the initial choice of natural coordinate. If𝑑(𝑠, 𝑡)=𝛿, then

𝑟_𝑠−𝛿 ≤ 𝑟_𝑡 ≤ 𝑟_𝑠+𝛿.

Let 𝑟

0 = min{𝑟_𝑞, 𝑟_ℎ(𝑞)}. Select a point 𝑞^′ ∈ 𝐵_𝑟

0/4(𝑞) ∩Ω₁. This point satisfies 𝑟_𝑞′ ≥ 3𝑟

0/4 and likewise forℎ(𝑞^′). Let𝛿 =𝑑(𝑞, 𝑞^′)and take a natural coordinate𝑧 in aΦ-disk𝐵_𝛿/

2(𝑞^′). We restrict ℎto thisΦ-disk, and as above, we useℎto build a natural coordinate 𝑤 on 𝐵_𝛿/

2(ℎ(𝑞^′)). More precisely, we have a disk 𝐷 ⊂ Cof radius𝛿/2 and two holomorphic maps

𝜑: 𝐷 → 𝐵_𝛿/

2(𝑞^′) , 𝜓: 𝐷 → 𝐵_𝛿/

2(ℎ(𝑞^′))

such that𝑧 =𝜑⁻¹,𝑤 =𝜓⁻¹. We can extend these maps to a larger disk𝐷^′⊂ Cwith radius 3𝑟

0/4. The map

𝑤⁻¹◦𝑧 : 𝐵

3𝑟

0/4(𝑞^′) → 𝐵

3𝑟

0/4(ℎ(𝑞^′)) is a holomorphic diffeomorphism that agrees withℎon𝐵_𝛿/

2(𝑞^′). Since𝐵

𝑟0/2(𝑞) ⊂ 𝐵3𝑟

0/4(𝑞^′), we see we have analytically continuedℎ to the open setΩ₁∪𝐵_𝑟

0/2(𝑞). From conformal invariance, the map 𝑓 ◦ ℎ is harmonic, and hence the Aronszajn theorem [Aro57] implies 𝑓 ◦ℎ= 𝑓 onΩ₁∪𝐵_𝑟

0/2(𝑞). □

Via this result, we often find ourselves in the following situation: either ℎ can be continued along an entire curve𝛾, or we have a segment𝛾^′ ⊂ 𝛾 along whichℎhas been continued but the endpoint ofℎ(𝛾^′)is a zero ofΦ.

We remark that there is no guarantee that the analytic continuation is a diffeomor- phism. It is at least a local diffeomorphism and a local isometry for theΦ-metric.

Harmonic singularities

Toward the proof of the main theorem, we rule out possible pathological behaviour of harmonic maps near rank 1 singularities. We need not delve too deep into the theory of singularities, but we invite the reader to see Wood’s thesis [Woo74] and the paper [Woo77], in which he studies singularities of harmonic maps between surfaces in detail.

Our key tool is the Hartman-Wintner theorem [HW53], which gives a local repre- sentation formula for harmonic maps. Let𝑧 be a holomorphic coordinate centered on a disk centered at 𝑝 ∈ Σ with 𝑧(𝑝) = 0, and let (𝑥

1, . . . , 𝑥_𝑛) be normal (but not necessarily orthogonal) coordinates in a neighbourhood 𝑈 of 𝑓(𝑝) such that 𝑓(𝑝) = 0. According to the Hartman-Wintner theorem, we can write the compo- nents(𝑓1, . . . , 𝑓^𝑛)as

𝑓^𝑘 = 𝑝^𝑘 +𝑟^𝑘,

where 𝑝^𝑘 is a spherical harmonic (a harmonic homogeneous polynomial) of some degree𝑚 <∞and𝑟^𝑘 ∈𝑜(|𝑧|^𝑚). We are allowing 𝑝^𝑘 =∞, which means 𝑓^𝑘 =0.

By permuting the coordinates, we may assume deg𝑝1=min^𝑘deg𝑝^𝑘, and deg𝑝^𝑘 ≥ deg𝑝2 for all 𝑘 ≥ 3. Note deg𝑝1,deg𝑝2 < ∞, for otherwise Sampson’s result [Sam78, Theorem 3] implies 𝑓 takes its image in a geodesic.

Lemma 5.2.4. There does not exist a sequence of points(𝑝_𝑛)^∞

𝑛=1 ⊂ Σconverging to 𝑝 with the property that there exists a (not necessarily conformal) diffeomorphism ℎ_𝑛taking a neighbourhood of𝑝_𝑛to a neighbourhood of 𝑝that leaves 𝑓 invariant.

Proof. Arguing by contradiction, suppose there is such a sequence(𝑝_𝑛)^∞

𝑛=1. Since 𝑓 is an embedding near regular points,𝑝must be a singular point. Choose a coordinate 𝑧 on the source and normal coordinates on the target with 𝑝 = 0, 𝑓(𝑝) = 0. We apply Hartman-Wintner to obtain the formula

𝑓^𝑘 = 𝑝^𝑘+𝑟^𝑘

with the same degree assumptions as above. It is clear that there is at least one 𝑝^𝑘 with deg𝑝^𝑘 =𝑚 > 1,𝑚 ≠∞.

We invoke a result of Cheng [Che76, Lemma 2.4]: there is a𝐶1diffeomorphism from a neighbourhood of 0 inR²to a neighbourhood of 𝑝, taking 0 to 0 in coordinates, and such that

𝑓^𝑘 ◦𝜑(𝑤) = 𝑝^𝑘(𝑤).

As a spherical harmonic of degree 𝑚, the zero set of 𝑝^𝑘 consists of 𝑚 distinct lines going through the origin, arranged so that the angles between two adjacent lines is constant (this is an easy consequence of homogeneity). Notice that in our neighbourhood of 𝑝,

{𝑞 : 𝑓^𝑘(𝑞) = 𝑓^𝑘(𝑝)}={𝜑(𝑤) : 𝑝^𝑘(𝑤) =𝑝^𝑘(0)}.

Therefore, the set{𝑞 : 𝑓^𝑘(𝑞) = 𝑓^𝑘(𝑝)}is collection of𝑚disjoint𝐶1arcs all trans- versely intersecting at the origin. For𝑛large enough, 𝑝_𝑛 lies inside the coordinate chart determined by 𝜑, and hence it lies on one of the arcs. Fixing such a 𝑝_𝑛, we use that ℎ_𝑛 is a diffeomorphism to see that there should be𝑚−1 more curves transversely intersecting the line containing 𝑝_𝑛, and such that 𝑓(𝑞) = 𝑓(𝑝)on those

curves. This is a clear contradiction. □