aa r X i v : . [ m a t h . DG ] O c t A FACTORIZATION THEOREM FOR HARMONIC MAPS
NATHANIEL SAGMAN
Abstract.
Let f be a harmonic map from a Riemann surface to a Riemannian n -manifold.We prove that if there is a holomorphic diffeomorphism h between open subsets of thesurface such that f ◦ h = f , then f factors through a holomorphic map onto anotherRiemann surface. If such h is anti-holomorphic, we obtain an analogous statement.For minimal maps, this result is well known and is a consequence of the theory ofbranched immersions of surfaces due to Gulliver-Osserman-Royden. Our proof relies onvarious geometric properties of the Hopf differential. Introduction
Let Σ be a Riemann surface with a C conformal Riemannian metric µ , and let M be asmooth n -manifold, n ≥
2, equipped with a C Riemannian metric ν . Both manifolds areassumed to not have boundary. Harmonic maps f : (Σ , µ ) → ( M, ν ) are solutions of thesecond order semilinear elliptic equation τ ( f, µ, ν ) = trace µ ∇ µ ∗ ⊗ f ∗ ν df = 0 . On closed manifolds, τ ( f, µ, ν ) = 0 arises as the Euler-Lagrange equation of the DirichletEnergy functional for the metrics µ, ν . Under fairly general compactness and curvatureassumptions on Σ and M , harmonic maps exist in any non-trivial homotopy class.A harmonic map f : (Σ , µ ) → ( M, ν ) is admissible if its image is not contained in ageodesic. There is a viewpoint that while admissible harmonic maps are abundant in manycontexts, they also reveal rigid geometric properties of the spaces on which they live. Theresult of this paper is another instance of this phenomenon. It connects local behaviour ofa harmonic map to the global complex geometry of the underlying Riemann surface.
Theorem 1.1.
Suppose f : (Σ , µ ) → ( M, ν ) is an admissible harmonic map, and there isa conformal diffeomorphism h : Ω → Ω between open subsets of Σ such that f ◦ h = f on Ω . If h is holomorphic, then there is a Riemann surface (Σ , µ ) , a holomorphic map π : Σ → Σ , and a harmonic map f : (Σ , µ ) → ( M, ν ) such that π (Ω ) = π (Ω ) and f factors as f = f ◦ π . If h is anti-holomorphic, Σ is a Klein surface and π is dianalytic. Among other solutions to geometrically flavoured PDEs, Theorem 1.1 has been knownfor minimal harmonic maps and pseudoholomorphic curves since the 1970s. Osserman in[Oss70] and Gulliver in [Gul73] studied singularites of the Douglas and Rado solutions tothe Plateau problem. The only possible singularities are branch points, which are separatedinto so-called true branch points and false branch points. Osserman ruled out true branchpoints and made progress toward the non-existence of false branch points in [Oss70], andGulliver showed there are no false branch points in [Gul73]. Together their work proves thatthe Douglas and Rado solutions are immersed. For an exposition of the Plateau problem,see Chapters 4 and 6 of [CM11].
Curiously, very few properties specific to minimal surfaces come into play in [Gul73],but rather qualities shared by a larger class of surfaces. This prompted a deeper study ofbranched immersions of surfaces, which was carried out by Gulliver-Osserman-Royden in[GOR73]. A version of Theorem 1.1 holds for the maps considered in [GOR73]. In the nextsubsection we describe their theory of branched immersions of surfaces and how minimalmaps fit into the framework.Aside from connections to the Plateau problem, the result of Gulliver-Osserman-Roydenhas other applications. We would like to highlight the work of Moore in [Moo06] and[Moo17], where he studies moduli spaces of minimal surfaces. A map f is somewhereinjective if there is a regular point p such that f − ( f ( p )) = p . Moore uses Theorem 1.1 forminimal maps to show that a closed minimal map in an n -manifold, n ≥
3, is not somewhereinjective if and only if it factors through a conformal branched covering map. The sameresult holds for pseudoholomorphic curves [MS12, Proposition 2.5.1], whose moduli spacesare an active field of study.If (Σ , µ ) is closed with genus at least 2 and (
M, ν ) has negative curvature, then Σ musthave genus at least 2. The described results for minimal surfaces thus show the somewhereinjective condition is generic, for it is very rare for a closed Riemann surface to admit aholomorphic map onto another Riemann surface with non-abelian fundamental group.In [Moo06], [Moo17], [MS12], the somewhere injective condition plays a role in varioustransversality arguments. With this in mind, Theorem 1.1 should be an essential tool inunderstanding the distribution of somewhere injective harmonic maps in certain modulispaces of harmonic maps.In a different inquiry, Jost and Yau proved a version of Theorem 1.1 in [JY83] for harmonicmaps to K¨ahler manifolds, using it as a tool in their study of deformations of Kodairasurfaces. Their work has played a role in the development of the theories of K¨ahler manifoldsand Higgs bundles. See the survey of Jost [Jos08] for more information.1.1. Minimal surfaces.
Loosely following the exposition of Moore in section 4 of [Moo06],we explain how the proof of Theorem 1.1 for minimal maps is deduced from the results in[GOR73]. Let f : (Σ , µ ) → ( M, ν ) be a C map. A point p ∈ Σ is a branch point if df ( p ) = 0. We say a branch point is a good branch point of order m − z on Σ and ( x , . . . , x n ) on M such that f is described by the equations x = Re z m , x = Im z m , x k = η k ( z ) , k ≥ , where η k ∈ o ( | z | m ). Note that m = 1 implies we have a regular point. Remark 1.2.
These conventions could be a source of confusion. In [GOR73], Gulliver-Osserman-Royden refer to “good branch points” as simply “branch points.” This causes noharm in their work, but we should distinguish here.In [GOR73], a branched immersion is a map from a surface that is regular everywhereapart from an isolated set of good branch points. For clarity we refer to such a map as agood branched immersion. In this paper, a minimal map is a weakly conformal harmonicmap. Gulliver-Osserman-Royden use the representation formula of Hartman and Wintner[HW53] to show that a minimal map is a good branched immersion (see Propositions 2.2and 2.4 in [GOR73]). In fact, using this same formula, Micallef and White recover finercoordinate expressions for minimal surfaces (see [MW95, Theorem 1.]).An order m − p of a good branched immersion is ramified of order r − r is the maximal non-negative integer such that there is a disk U centered at p on which f FACTORIZATION THEOREM FOR HARMONIC MAPS 3 factors through a branched covering of degree r . If r = m , p is called a false branch point,and true otherwise. We say f is unramified if r = 0. We now recast one of the key resultsof [GOR73]. Theorem 1.3 (Proposition 3.19 in [GOR73]) . Let Σ be a C surface, M a C manifold,and f : Σ → M a C good branched immersion with the unique continuation propertyand no true branch points. Then there is a C surface Σ , a C good branched immersion π : Σ → Σ , and an unramified C good branched immersion f : Σ → M such that f = f ◦ π . We do not define the unique continue property of Gulliver-Osserman-Royden (see [GOR73,page 757]), but remark that minimal maps have this property (see [GOR73, Lemma 2.10]).The minimal case is essentially handled in [GOR73, Proposition 3.24]. If a map is con-formal, one can dispense of the hypothesis that there are no true branch points, and theobjects π , Σ , and f all have the same regularity as f apart from at branch points andimages of branch points.To prove Theorem 1.3, Gulliver-Osserman-Royden define a relation ∼ on Σ as follows.(1) If p and p are regular points for f , p ∼ p if there exists open sets Ω i containing p i , and an orientation preserving C map h : Ω → Ω such that f ◦ h = f on Ω .(2) If one of p or p is a branch point, then in any pair of neighbourhoods Ω i containing p i there exists neighbourhoods Ω ′ i ⊂ Ω i of p i consisting of only regular points suchthat for all p ′ ∈ Ω ′ \{ p } there exists p ′ ∈ Ω ′ \{ p } such that p ′ ∼ p ′ , and for all p ′ ∈ Ω ′ \{ p } there exists p ′ ∈ Ω ′ \{ p } such that p ′ ∼ p ′ .Gulliver-Osserman-Royden show that this is an equivalence relation and define the quotient π : Σ → Σ . They prove Σ has the structure of a C manifold and the map f : Σ → M is defined by setting f ([ p ]) = f ( p ). Ramification leads to equivalent points, so f isunramified.When Σ is a Riemann surface and Σ and M are equipped with metrics so that f isminimal, we impose that h is holomorphic. Following the proof of [GOR73, Proposition3.24], one can show that the transition maps on Σ are holomorphic away from the branchpoints and extend holomorphically via the removeable singularities theorem. One checks incoordinates that the map f is minimal with respect to the conformal metric on Σ obtainedvia uniformization. The existence of a map h as in Theorem 1.1 amounts to saying someclasses under ∼ are not singletons. The minimal case follows directly. Remark 1.4.
Pertaining to the Plateau problem, Gulliver-Osserman-Royden prove a ver-sion of Theorem 1.1 holds for surfaces with particular boundary data. The argument demon-strates that if a false branch point exists, then one can lower the area by passing througha holomorphic map onto another surface. The solutions of Douglas and Rado minimize thearea relative to the boundary data, so this cannot occur.Gulliver-Osserman-Royden do not consider orientation reversing maps in the definitionof ∼ , but their construction can be modified to allow for this. In this situation, we may endup with a mapping onto a non-orientable surface. Moore notes this in [Moo06], althoughhis context is slightly different from ours. Since we could not locate a formal proof in theliterature, we explain the necessary adjustments in subsection 4.3.1.2. Harmonic maps vs. minimal maps.
To prove Theorem 1.1 in the holomorphiccase, we follow the blueprint of Gulliver-Osserman-Royden. That is, we define an equiva-lence relation on Σ and take the quotient as our candidate for the surface Σ . However, NATHANIEL SAGMAN it is not obvious how one should define ∼ . The difficulty comes from the singularities ofharmonic maps, in that(i) harmonic maps can have rank 1 singularities, which do not occur in the theory ofGulliver-Osserman-Royden, and(ii) branch points are not good branch points. At best, we can combine the Hartman-Wintner formula with [Che76, Lemma 2.4] to see that near a branch point p of order m − C coordinate z on the source and a C ∞ coordinate on the targetsuch that p f ( p )
0, and f may be expressed f = ( f , . . . , f n ) with f = p , f k = p k + r k , k ≥ , where p is a spherical harmonic of degree m , p k is a spherical harmonic of degreeat least m , and r k ∈ o ( | z | m ).To overcome these difficulties, we exploit the geometry of a holomorphic quadratic differ-ential known as the Hopf differential. In some sense, the Hopf differential treats rank 1and 2 points on an equal footing. Thus, if we define ∼ in terms of a condition on the Hopfdifferential, in theory we shouldn’t encounter any difficulties due to rank 1 singularities.In practice this is mostly true–at some points we need to refer to the Hartman-Wintnerformula. As for (ii), the Hopf differential defines a “natural coordinate” for the harmonicmap near a branch point, in which the geometry can be more easily probed. At a falsebranch point, we see ramification behaviour similar to that displayed by minimal maps.The only missing piece of Gulliver-Osserman-Royden’s theory is the unique continuationproperty. In Proposition 2.6, we show that analytic continuation of natural coordinates forthe Hopf differential induces a continuation of h . Using this proposition, we establish a“holomorphic unique continuation property” (Proposition 3.5).1.3. Future directions.
It is tempting to conjecture that some version of Theorem 1.1should hold without the hypothesis that h is conformal. The main motivation would be toimprove our understanding of somewhere injective harmonic maps. We would like to pointout that, in view of the example below, we cannot expect the map π to be holomorphicwith respect to a complex structure on Σ . Example 1.5.
Let (Σ , µ ) be a closed hyperbolic surface and f : (Σ , µ ) → ( M, ν ) atotally geodesic map. Fix a smooth surface Σ of genus at least and a homotopy class ofmaps f : π (Σ) → π (Σ ) with degree at least . Any C metric µ yields a unique harmonicmap π : (Σ , µ ) → (Σ , µ ) in the homotopy class f . One can then find many diffeomorphisms h : Ω → Ω between open subsets of Σ such that f ◦ h = f , and by construction f factorsas f = f ◦ π . Generically, the surface (Σ , µ ) will not admit a holomorphic map onto anyRiemann surface of genus at least . We simplify our study of singularities using complex analytic methods. Without theconformal hypothesis, the only local information we have comes from the Hartman-Wintnerrepresentation formula. If this is the main tool, then it is also natural to ask about moregeneral solutions to second order semiliinear elliptic systems, rather than just harmonicmaps. An analysis of singularities would be related to understanding local behaviour ofspherical harmonics.A substitute for the unique continuation property seems to be a large hurdle. Implicit inthe proof of the unique continuation property for minimal maps is the following result (see[GOR73, Lemma 2.10]).
FACTORIZATION THEOREM FOR HARMONIC MAPS 5
Proposition 1.6.
Let D ⊂ R be the unit disk. Suppose u , u : D → M are minimal mapssuch that, for all open sets D ⊂ D containing , there is an open subset D ⊂ D (possiblynot containing ) such that u ( D ) ⊂ u ( D ) . Then there exists an open subset D ′ ⊂ D containing such that u ( D ′ ) ⊂ u ( D ) . This result above fails emphatically if we replace minimal maps with harmonic maps,even if M = R . Indeed, the simple example u ( x, y ) = ( x, xy ) , u ( x, y ) = ( x, y )does not satisfy Proposition 1.6. On the other hand, our “holomorphic unique continuationproperty” provides a substitute for Proposition 1.6 (see Proposition 3.5). This is one of thereasons we expect a more general version of Theorem 1.1 to be much more delicate, and wedefer this investigation to a future project.1.4. Acknowledgements.
Many thanks to Vlad Markovic for encouragement and sharinghelpful ideas. I would also like to thank John Wood and J¨urgen Jost for comments onearlier drafts. 2.
Harmonic Maps from Riemann Surfaces
We give background on harmonic maps. The content is standard and can be found inany text on harmonic maps. We then discuss analytic continuation and singularities.2.1.
Harmonic maps.
Throughout, we let (Σ , µ ) denote a Riemann surface with a C conformal metric, and ( M, ν ) an n -manifold, n ≥
3, with a C Riemannian metric. A C map f : (Σ , µ ) → ( M, ν ) gives a pullback bundle f ∗ T M , and the derivative df defines asection of the endomorphism bundle T ∗ Σ ⊗ f ∗ T M . We denote by ∇ the connection on thetensor bundle T ∗ Σ ⊗ f ∗ T M induced by the Levi-Civita connections ( ∇ µ ) ∗ and ∇ f ∗ ν = f ∗ ∇ ν on T ∗ Σ and f ∗ T M respectively.
Definition 2.1.
The tension field of a C map f : (Σ , µ ) → ( M, ν ) is the section of f ∗ T M given by τ = τ ( f, µ, ν ) = trace µ ∇ df. The map f is harmonic if τ = 0.In a local conformal coordinate z = x + iy , the tension field is given by τ = | µ | − (cid:16) ∇ f ∗ ν ∂∂x df (cid:16) ∂∂x (cid:17) + ∇ f ∗ ν ∂∂y df (cid:16) ∂∂y (cid:17)(cid:17) and hence τ ( f, µ, ν ) = 0 defines a conformally invariant semilinear elliptic equation of secondorder. If µ is C α and ν is C β , then the harmonic map is C γ , where γ = min { α + 2 , β + 1 } .We are also allowing α, β = ∞ or ω . Therefore, our harmonic maps are at least C . Definition 2.2.
A harmonic map f : (Σ , µ ) → ( M, ν ) is minimal if it is also weaklyconformal. That is, conformal in the sense of distributions.As for the complex theory, we set ( f ∗ T M ) C = f ∗ T M ⊗ C to be the complexification ofthe pullback bundle and extend f ∗ ν linearly. Given a local coordinate z = x + iy , define ∂∂z = 12 (cid:16) ∂∂x − i ∂∂y (cid:17) , ∂∂z = 12 (cid:16) ∂∂x + i ∂∂y (cid:17) . NATHANIEL SAGMAN
Using this coordinate, we have locally defined sections of ( f ∗ T M ) C given by f z = df (cid:16) ∂∂z (cid:17) , f z = df (cid:16) ∂∂z (cid:17) . Definition 2.3.
The Hopf differential of a harmonic map f : (Σ , µ ) → ( M, ν ) is theholomorphic quadratic differential Φ on Σ specified by the family of local expressions h f z , f z i f ∗ ν dz , where z ranges over local coordinates for Σ.It follows from the definitions that f is minimal if and only if Φ = 0. If Φ does not vanishidentically, the zeros of Φ are independent of the parametrization and discrete. If Φ( p ) = 0,then near p we can find a holomorphic coordinate z such thatΦ( z ) = dz . If Φ( p ) = 0, there is a coordinate z such thatΦ( z ) = z n dz . Such coordinates are called natural coordinates for Φ.The quadratic differential Φ induces a singular flat metric: the Φ-metric. Locally, ifΦ = φ ( z ) dz , then the metric tensor is | φ ( z ) || dz | . The singular points are the zeroes of Φ. A disk of radius r centered at a point p in the Φ-metric shall be called a Φ-disk and written B r ( p ). The induced distance function is denoted d ( · , · ). Although we work with different differentials in the course of the paper, the use ofthis notation in context should be clear.2.2. Analytic continuation.
Until Section 4, let f : (Σ , µ ) → ( M, ν ) be an admissibleharmonic map with non-zero Hopf differential Φ and let h : Ω → Ω be a holomorphicmap as in the statement of Theorem 1.1. We treat anti-holomorphic maps in Section 4.Throughout the paper, we let Z denote the zero locus of Φ. By restricting, we assume Ω is a Φ-disk.We use the geometry of the Hopf differential to analytically continue h . Let p ∈ Ω besuch that Φ( p ) = 0, and let U ⊂ Ω be an open subset containing p such that Φ = 0 in U .Given a holomorphic local coordinate z in U , we define a local coordinate w on h ( U ) by w = z ◦ h − . In these coordinates, h is given by w ( h ( z )) = z and df p (cid:16) ∂∂z (cid:17) = df h ( p ) (cid:16) ∂∂w (cid:17) ∈ T f ( z ) M ⊗ C . Remark 2.4.
Here we are viewing df as a map from T Σ → T M rather than as a sectionof the endomorphism bundle T ∗ Σ ⊗ f ∗ T M .Choosing z to be a natural coordinate with z ( p ) = 0, we obtain h f w , f w i ( w ( h ( z ))) = h f z , f z i ( z ) = 1 . Therefore, w defines a natural coordinate on h ( U ). We have proved the following lemma. Lemma 2.5. h is a local isometry in the Φ -metric. If Ω is a Φ -disk then so is Ω , and h takes a natural coordinate z on Ω to a natural coordinate w on Ω in which w ( h ( z )) = z . FACTORIZATION THEOREM FOR HARMONIC MAPS 7
The goal of this subsection is to prove the proposition below. In the proof we use thenotion of a maximal Φ-disk. See section 5 in [Str84] for a detailed discussion on maximalΦ-disks. Let Z denote the zero set of Φ (which is isolated). Proposition 2.6.
Suppose Ω , Ω are Φ -disks with no zeros of Φ and that γ : [0 , L ] → Σ is a curve starting in Ω and that γ first strikes ∂ Ω at a point q . If there is an ǫ > suchthat min (cid:8) inf s ∈ γ | Ω1 ,t ∈Z d ( s, t ) , inf s ∈ γ | Ω1 ,t ∈Z d ( h ( s ) , t ) (cid:9) ≥ ǫ then there is a neighbourhood of q in which h can be analytically continued along γ .Proof. We can choose an arc on ∂ Ω centered at q on which Φ = 0. We then connect theendpoints via an arc contained inside Ω so that the enclosed region U is a topological disk.We pick these arcs in such a way thatmin (cid:8) inf s ∈ U,t ∈Z d ( s, t ) , inf s ∈ U,t ∈Z d ( h ( s ) , t ) (cid:9) ≥ ǫ/ . The restriction of the Φ-metric to any compact region that does not intersect Z is complete.As h is an isometry in the Φ-metric, we can extend it to a map h : U → U . Therefore, wehave a well-defined point h ( q ).For every point p
6∈ Z , there is a maximal radius r p such that we can extend any naturalcoordinate centered at p to a Φ-disk of radius r p . r p does not depend on the initial choiceof natural coordinate. If d ( s, t ) = δ , then r s − δ ≤ r t ≤ r s + δ. Let r = min { r q , r h ( q ) } . Select a point q ′ ∈ B r / ( q ) ∩ Ω . This point satisfies r q ′ ≥ r / h ( q ′ ). Let δ = d ( q, q ′ ) and take a natural coordinate z in a Φ-disk B δ/ ( q ′ ).We restrict h to this Φ-disk, and as above, we use h to build a natural coordinate w on B δ/ ( h ( q ′ )). More precisely, we have a disk D ⊂ C of radius δ/ ϕ : D → B δ/ ( q ′ ) , ψ : D → B δ/ ( h ( q ′ ))such that z = ϕ − , w = ψ − . We can extend these maps to a larger disk D ′ ⊂ C withradius 3 r /
4. The map w − ◦ z : B r / ( q ′ ) → B r / ( h ( q ′ ))is a holomorphic diffeomorphism that agrees with h on B δ/ ( q ′ ). Since B r / ( q ) ⊂ B r / ( q ′ ),we see we have analytically continued h to the open set Ω ∪ B r / ( q ). From conformalinvariance, the map f ◦ h is harmonic, and hence the Aronszajn theorem [Aro57] implies f ◦ h = f on Ω ∪ B r / ( q ). (cid:3) Via this result, we often find ourselves in the following situation: either h can be continuedalong an entire curve γ , or we have a segment γ ′ ⊂ γ along which h has been continued butthe endpoint of h ( γ ′ ) is a zero of Φ.We remark that there is no guarantee that the analytic continuation is a diffeomorphism.It is at least a local diffeomorphism and a local isometry for the Φ-metric. NATHANIEL SAGMAN
Harmonic singularities.
Toward the proof of the main theorem, we rule out possiblepathological behaviour of harmonic maps near rank 1 singularities. We need not delve toodeep 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 surfacesin detail.Our key tool is the Hartman-Wintner theorem [HW53], which gives a local representationformula for harmonic maps. Let z be a holomorphic coordinate centered on a disk centeredat p ∈ Σ with z ( p ) = 0, and let ( x , . . . , 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 components ( f , . . . , f n ) as f k = p k + r k where p k is a spherical harmonic (a harmonic homogeneous polynomial) of some degree m < ∞ and r k ∈ o ( | z | m ). We are allowing p k = ∞ , which means f k = 0.By permuting the coordinates, we may assume deg p = min k deg p k , and deg p k ≥ deg p for all k ≥
3. Note deg p , deg p < ∞ , for otherwise Sampson’s result [Sam78, Theorem 3]implies f takes its image in a geodesic. Lemma 2.7.
There does not exist a sequence of points ( p n ) ∞ n =1 ⊂ Σ converging to p withthe property that there exists a (not necessarily conformal) diffeomorphism h n taking aneighbourhood of p n to a neighbourhood of p that leaves f invariant.Proof. Arguing by contradiction, suppose there is such a sequence ( p n ) ∞ n =1 . Since f is anembedding near regular points, p must be a singular point. Choose a coordinate z on thesource 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 withdeg p k = m > m = ∞ .We invoke a result of Cheng [Che76, Lemma 2.4]: there is a C diffeomorphism from aneighbourhood of 0 in R to a neighbourhood of p , taking 0 to 0 in coordinates, and suchthat 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 goingthrough the origin, arranged so that the angles between two adjacent lines is constant (thisis 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 of m disjoint C arcs all transverselyintersecting at the origin. For n large enough, p n lies inside the coordinate chart determinedby ϕ , and hence it lies on one of the arcs. Fixing such a p n , we use that h n is a diffeo-morphism to see that there should be m − p n , and such that f ( q ) = f ( p ) on those curves. This is a clear contradiction. (cid:3) Holomorphic Factorization
Throughout this section, we continue to assume h : Ω → Ω is a holomorphic diffeomor-phism. Following the structure of Section 3 in [GOR73], we prove Theorem 1.1 holds forsuch h (although the technical details of our proofs are for the most part quite different). FACTORIZATION THEOREM FOR HARMONIC MAPS 9
The equivalence relation.Definition 3.1.
Given p , p ∈ Σ, we define a relation ∼ by(1) If p , p
6∈ Z , p ∼ p if there exists open sets Ω , Ω such that p i ∈ Ω i and aholomorphic diffeomorphism h : Ω → Ω such that f = f ◦ h on Ω .(2) If one of p , p is a zero of Φ, then for any pair of neighbourhoods Ω i containing p i one can find smaller neighbourhoods Ω ′ i ⊂ Ω i containing p i such that for each q ∈ Ω ′ \{ p } there exists q ∈ Ω ′ \{ p } such that q ∼ q , and for each q ∈ Ω ′ \{ p } there is a q ∈ Ω ′ \{ p } such that q ∼ q .If p ∼ p then f (Ω ′ ) = f (Ω ′ ) and f ( p ) = f ( p ) are apparent from the definition. Recall Z = { p ∈ Σ : Φ( p ) = 0 } . Proposition 3.2. ∼ is an equivalence relation.Proof. Reflexivity and symmetry are obvious. As for transitivity, this is clear if p , p , p are all not zeros of Φ. If at least one is a zero, we consider two cases:(i) p , p are zeros, or(ii) p is a zero while p , p are notThe other cases are trivial. Case (i) can be seen from the definitions: take Ω , Ω containing p , p respectively such that for all p ′ ∈ Ω \{ p } there exists p ′ ∈ Ω \{ p } with p ′ ∼ p ′ .Within Ω we find an open set Ω ′ , and then an open set Ω ′ containing p with the sameproperty. SetΩ ′ = { p ′ ∈ Ω \{ p } : there exists p ′ ∈ Ω ′ such that p ′ ∼ p ′ } ∪ { p } . We can find an open disk centered at p inside Ω ′ by applying the definition of ∼ to theopen sets Ω , Ω ′ . It is also clear that Ω ′ is open away from p , and hence it is open. It isnow simple to check that Ω ′ and Ω ′ satisfy the definition of ∼ .The second case requires more work. Select Φ-disks U , U of radius R > p and p respectively such that there are no points q i with q i ∼ p i and no zeros of Φ. Let U ′ , U ′ be Φ-disks centered at the same points with half the radius. Using ∼ , we can findopen sets Ω i ⊂ U ′ i containing p i such that f (Ω ) ⊂ f (Ω ) and every point in q ∈ Ω \{ p } isequivalent to a point in Ω \{ p } . We shrink Ω to turn it into an open disk in the Φ-metriccentered at p with radius δ < R/ p ′ i ∈ Ω i be such that p ′ ∼ p ′ . Viewing Ω in natural coordinates, let γ be the straightline from p ′ to p . We have a holomorphic map h taking a neighbourhood of p ′ to one of p ′ that leaves f invariant. We analytically continue along γ as much as we can. Either h ( γ )hits a zero of Φ or we can continue up until the endpoint. The Φ-length of any segment of h ( γ ) is at most δ , and we infer h ( γ ) is contained in B R/ δ ( p ) ⊂ U . Thus, h ( γ ( t )) cannever be a zero for any time t , and we can continue to the endpoint. From the proof ofProposition 2.6, p = γ (1) is equivalent to the endpoint h ( γ (1)).To finish the proof, we need to argue h ( γ (1)) = p . Let q = h ( γ (1)). We do know p ∼ q . We claim we could have chosen R small enough to ensure no point other thanpossibly p is equivalent to p . Indeed, if this is not possible, then we get a sequence ofpoints ( q n ) ∞ n =1 converging to p such that p ∼ q n for all n . Using transitivity of ∼ forpoints in Σ \ Z , we can then construct a sequence of points q ′ n converging to p that are allequivalent to p . This directly contradicts Lemma 2.7 and completes the proof. (cid:3) We use Proposition 3.2 to prove another useful property of ∼ . Lemma 3.3.
Suppose p , p
6∈ Z . Then there is no sequence ( q n ) ∞ n =1 such that q n ∼ p forall n and q n → p as n → ∞ .Proof. Again going by way of contradiction, assume such a sequence q n exists. Firstly, byLemma 2.7, we cannot have p ∼ p . Using the definition of ∼ , we see that in any pair ofneighbourhoods Ω i of p i , we can find points p ′ i ∈ Ω i such that p ′ ∼ p ′ .Let δ, ǫ > τ = ǫ + 2 δ . We choose δ, ǫ to be small enough to ensure(i) there is no point equivalent to p in B τ ( p ) \{ p } ,(ii) there is no point equivalent to p in B δ ( p ) \{ p } , and(iii) there are no zeros of the Hopf differential in either ball.Choose p ′ ∈ B ǫ ( p ) that is equivalent to a point p ′ ∈ B δ ( p ). In natural coordinates, let γ be the straight line path from p ′ to p . γ has length at most δ , and hence the image of anysegment of γ along an analytic continuation of h lies in B τ ( p ). Thus, we can continue h along γ as much as we like, and we extend to the boundary point p . The endpoint h ( γ (1))is then equivalent to p . Since p p , h ( p ) = p .Set q ′ = h ( p ). Replace δ, ǫ, τ with smaller numbers δ ′ , ǫ ′ , τ ′ satisfying the same relationsas above and q B τ ′ ( p ). By repeating the previous procedure we secure another point q ′ ∼ p that is closer to p . Continuing in this way, we can build a sequence ( q ′ n ) ∞ n =1 converging to p such that q ′ n ∼ p for all n .We now find our contradiction. Given that both such sequences exist, f cannot be anembedding around p nor p and has rank 1 at both points. Choose normal coordinates on M centered at f ( p ) = f ( p ), and a conformal coordinate centered at p in which f takesthe form f k = p k + r k as in the previous subsection. Since f is not regular at p , there is at least one k such thatdeg p k = m > m = ∞ . Choosing a conformal coordinate at p , f takes the form f k = ˜ p k + ˜ r k with ˜ p k a spherical harmonic and ˜ r k decaying faster. The images of p k and ˜ p k in R intersecton open sets, so ˜ p k is clearly non-zero. Thus, the set of points near p on which f k is equalto f k ( p ) is some collection of arcs intersecting at that point. However, since deg p k > (cid:3) The Hausdorff condition.
The main result of this subsection is Proposition 3.4,which implies the topological quotient of Σ by ∼ is Hausdorff. We say p ∼ ′ p if for everypair of neighbourhoods U i containing p i , there exists p ′ i ∈ U i with p ′ ∼ p ′ . Proposition 3.4.
Suppose p ∼ ′ p . Then p ∼ p . Proposition 3.4 is our “holomorphic unique continuation property.” Combined with [Sam78,Theorem 3], Proposition 3.4 implies the following result of independent interest.
Proposition 3.5.
Let D ⊂ R be the unit disk. Suppose u , u : D → M are harmonicmaps maps such that, for all open sets D ⊂ D containing , there is an open subset D ⊂ D (possibly not containing ) such that u ( D ) ⊂ u ( D ) . Moreover, assume thatfor any subsets D ′ i ⊂ D i on which u i is regular such that u ( D ′ ) ⊂ u ( D ′ ) , the map u − | u ( D ′ ) ◦ u | D ′ is holomorphic. Then there exists an open subset D ′ ⊂ D containing such that u ( D ′ ) ⊂ u ( D ) . FACTORIZATION THEOREM FOR HARMONIC MAPS 11
Turning toward the proof of Proposition 3.4, if p and p are both not zeros of Φ, thenone can follow the argument from the proof of Proposition 3.2, almost word-for-word, upuntil the last paragraph. We just need to note that Lemma 3.3 shows we can choose aΦ-disk surrounding p that is small enough that it contains no point equivalent to p . .Going forward, we assume at least one of the two points is a zero of Φ. The main stepin the proof is the next lemma. Lemma 3.6.
There exists δ, τ > such that every p ′ ∈ B δ ( p ) \{ p } is equivalent to a point p ′ ∈ B τ ( p ) \{ p } .Proof. Let δ, ǫ > τ = ǫ + 3 δ . We choose δ, ǫ to be small enough such that B δ ( p ) ∩ B τ ( p ) = ∅ and that in B δ ( p ) \{ p } and B τ ( p ) \{ p } ,(i) we have no points equivalent to the centers, and(ii) there are no zeros of Φ.We take open sets p ′ ∈ B δ ( p ), p ′ ∈ B ǫ ( p ) with p ′ ∼ p ′ , and let h be the associatedholomorphic diffeomorphism. Let q ∈ B δ ( p ), q = p , and let γ be a path from a point p ′ to q . We choose γ to be either the straight line from p ′ to q , or a slight perturbation of thatline to make sure the path does not touch p . Regardless, we can arrange so the Φ-lengthis bounded above by 5 δ/ h along γ as much as we can. Since the starting point lies in B ǫ ( p ), we see the image under h of any segment lies in B τ ( p ). If we can continue h along γ to the endpoint, and the endpoint of h ( γ ) is not p , then we have q = γ (1) ∼ h ( γ (1)).The only way we could not extend is if some segment of h ( γ ) touches p . Notice that,regardless, we have a point q ∈ B δ ( p ) that satisfies q ∼ ′ p (here we are relabelling q to bethe endpoint of a bad segment if that happens). We rule this out with the lemma below. Lemma 3.7.
In the setting above, we can choose our Φ -disks to be small enough so thatno point q ∈ B δ ( p ) \{ p } satisfies q ∼ ′ p .Proof. We first show that given such a point q , we have q ∼ ′ p . Let U , U , U be opensets containing p , p , q respectively. Let δ , δ , δ > p ′ ∈ B δ ( p ), p ′ ∈ B δ ( p )with p ′ ∼ p ′ , as well as p ′′ ∈ B δ ( p ), q ′ ∈ B δ ( q ) with p ′′ ∼ q ′ . We choose the δ j ’s so that B δ +3 δ ( q ) contains no zeros of the Hopf differential, and B δ i ( p i ) can only have zeros at p i .We also choose the δ i ’s so that all balls mentioned above are contained in U , U , U anddisjoint. Let h be the holomorphic map relating p ′′ to q ′ . We analytically continue h alonga path γ from p ′′ to p ′ with length at most 5 δ / p . Then theimage path lies in B δ +3 δ ( q ) and so we can continue to the endpoint. The endpoint h ( γ (1))is equivalent to p ′ . If the endpoint is not q , then h ( γ (1)) ∼ p ′ ∼ p ′ , and this proves theclaim. If the endpoint h ( γ (1)) is q itself, then q ∼ p ′ ∼ p ′ , and we can find q ′′ very close to q that is equivalent to a point very close to p ′ (in particular, contained in B δ ( p )).Therefore, we see that if the lemma is false, we can construct a sequence ( q n ) ∞ n =1 con-verging to p such that q n ∼ ′ p for all n . Fix a q n , along with a δ ′ > B δ ′ ( q n )contains no zeros and no points equivalent to q n and B δ ′ ( p ) has no zeros other than pos-sibly p . We find q ′ n ∈ B δ ′ ( q n ) and p ′ ∈ B δ ′ ( p ) such that q ′ n ∼ p ′ . There is another point q N ∈ B δ ′ ( p ) such that q N ∼ ′ p . Connect p ′ to q N via a path of length at most 5 δ ′ / p . Analytically continue the associated map h along this path. The imagelies in B δ ′ ( q n ), so we can always continue. The endpoint h ( γ (1)) ∈ B δ ′ ( q n ) is equivalentto q N . We claim we can choose q N with the property that h ( γ (1)) = q n . To this end, if h ( γ (1)) = q N , we take the straight line path σ from q N to q N +1 . According to [Str84, The-orem 8.1], if p is a zero of Φ of order n , then geodesics in the Φ metric are either straightlines or the concatenation of two radial lines enclosing an angle of at least 2 π/ ( n + 2). Bypigeonholing, we can pass to a subsequence where every q n lies in a closed sector of angle π/ ( n + 2) around the origin. This guarantees that the straight line path from any q j to q k isa geodesic in the Φ-metric and has length at most δ ′ . As Φ( q n ) = 0, the image h ( σ ) is thena straight line contained in B δ ′ ( q n ) with initial point q n , so it certainly cannot terminateat q n . We prove the claim by replacing q N with q N +1 and taking the concatenation of ouroriginal path with the straight line σ . We now just want to show q N ∼ q n , and we will havea contradiction. Toward this, it is enough to show q N ∼ ′ q n , since Φ does not vanish atthese points.This last step is similar to the beginning of our proof, and so we only sketch the argument.Recall that we have p ∼ ′ q n and p ∼ ′ q N . Find smalls balls containing q n , p , and q N .Then within the ball containing p we have two points p ′ and p ′′ , with p ′ equivalent to apoint near q n and p ′′ equivalent to a point near q N . Connect p ′ and p ′′ via a small arc thatdoes not touch p . We can arrange for the arc to stay in a ball around q n in which it canalways be continued. We thus get a point near q n that is equivalent to a point near q N . Wemay need to wiggle the path so the point is not q n . As discussed above, we are done. (cid:3) Returning to the proof of Lemma 3.6, we see that we can always extend our chosensegments, and moreoever each q ∈ B δ ( p ) \{ p } has an equivalent point in B τ ( p ) \{ p } . (cid:3) With Lemma 3.6 in hand, we are now ready to complete the proof of Proposition 3.4. LetΩ ′ be the set of points in B τ ( p ) \{ p } that have an equivalent point in B δ ( p ) \{ p } . LetΩ = Ω ′ ∪ { p } . By repeating the previous argument, we can find a very small ball B α ( p )such that every point in B α ( p ) \{ p } is equivalent to a point in B δ ( p ) \{ p } . This showsthat p is an interior point of Ω . Away from p , Ω is open by elementary considerations.It is now simple to conclude p ∼ p by using the open sets B δ ( p ) and Ω .3.3. Ramification at branch points.
We now investigate the local behaviour of the map f near zeros of the Hopf differential. This leads us to define a notion of ramification forbranch points. Our definition is slightly different from the one given in Section 1. Lemma 3.8.
Suppose p is a branch point of f , and hence a zero of Φ of some order n .Let h : Ω → Ω be a holomorphic diffeomorphism with f ◦ h = f , and suppose Ω , Ω areboth contained in a ball B ǫ ( p ) , where ǫ > is chosen so that there are no other zeros andno other point is equivalent to p in B ǫ ( p ) . Then, in the natural coordinates for Φ , h is arational rotation of angle πj/ ( n + 2) Proof.
Select p i ∈ Ω i with h ( p ) = h ( p ). Let γ : [0 , → B ǫ be a straight line path startingat p that terminates at the point p . We analytically continue h in a simply connectedneighbourhood of γ , as far as we can. Either there is an interior point q in the straightline that is mapped via h to p , or we can continue along the whole curve and extend to theboundary point p . In the first case, Proposition 3.4 guarantees q ∼ h ( q ) = p , which by ourchoice of ǫ means q = p , contradicting the definition of q . In the second case, Proposition3.4 yields p ∼ h ( p ) and we deduce h ( p ) = p .We now prove h is a rotation. Work in the interior of the extension of Ω in which h hasbeen continued. If we write the Hopf differential in local coordinates as Φ = φ ( z ) dz , then φ ( z ) = φ ( h ( z ))( h ′ ( z )) . FACTORIZATION THEOREM FOR HARMONIC MAPS 13
In the natural coordinate for the Hopf differential this becomes z n = ( h ( z )) n ( h ′ ( z )) . Since we’re in a simply connected region that doesn’t touch zero we can choose a branch ofthe square root. h then satisfies z n/ = ( h ( z )) n/ h ′ ( z ) = ∂∂z ( h ( z )) n/ n/ . Integrate to get z n/ = ( h ( z )) n/ + c for some complex constant c . Since h ( p ) = p , taking z → γ forces c = 0. Thisimplies z n +2 = ( h ( z )) n +2 and the result is now clear. (cid:3) Definition 3.9.
A non-minimal harmonic map g with Hopf differential Φ is holomorphicallyramified of order r − r is the largest integer such that there exists a Φ-disk Ω centeredat p and a holomorphic degree r branched cover ψ : Ω → D with one branch point at p onto a disk D with ψ ( p ) = 0 and such that ψ ( p ) = ψ ( p ) implies f ( p ) = f ( p ).A map is called unramified if r = 1. Clearly, a map can only ramify non-trivially at abranch point. Lemma 3.10.
A non-minimal harmonic map g with Hopf differential Φ is ramified oforder r > at p if and only if for all ǫ > , there exists p , p ∈ B ǫ ( p ) \{ p } such that p ∼ p and p = p , where p ∼ p in the sense that there is a holomorphic map h takinga neighbourhood of p to one of p that leaves g invariant. Remark 3.11.
A similar statement holds for minimal maps. See [GOR73, Lemma 3.12].
Proof. If g is ramified we take a Φ-disk Ω of p and a map ψ : Ω → D as in the definition.Select two points p i = p such that ψ ( p ) = ψ ( p ) as well as neighbourhoods Ω i on which ψ isinjective and share the same image under ψ . Setting ψ i = ψ | Ω i , the map ψ − ◦ ψ : Ω → Ω is a holomorphic diffeomorphism that leaves g invariant and hence p ∼ p . Conversely,pick ǫ > B ǫ ( p ) and so we have a coordinate z such that Φ = z n dz . There exists p , p ∈ B ǫ ( p ) with p ∼ p but p = p . Lemma 3.8shows there are small disks surrounding p , p that are related by a rotation h of the form z e πijn +2 z such that g = g ◦ h . By the Aronszajn theorem, g is invariant under this rotation in all of V . Dividing by the gcd, we see g is invariant under a rotation of the form z e πij r z, where j and r are coprime. It follows that g ◦ α = g in B ǫ ( p ), where α is the rotation z e πi/r z . In these coordinates, we define a holomorphic branched cover ψ : B ǫ ( p ) → D by ψ ( z ) = z r , and note that ψ ( p ) = ψ ( p ) implies g ( p ) = g ( p ). (cid:3) Lemma 3.12.
Let p be a branch point of f of order m − at which f is ramified of order r − . Then there is a Φ -disk Ω of p such that f admits a factorization f | Ω = f ◦ ψ , where (i) ψ : Ω → D is a holomorphic map onto a disk {| ζ | < δ } such that ψ | Ω \{ p } is an r -sheeted covering map,(ii) f is harmonic with respect to the flat metric on D and the given metric on M , and(iii) f : D → M is unramified with a single branch point of order s − at the origin,where s = m/r Proof.
Define f by f ( ψ ( z )) = f ( z ). (i) is given and we begin with (ii). Harmonicity is a localmatter, and at any point away from zero we can choose a neighbourhood surrounding thatpoint where ψ − exists and we have the factorization f = f ◦ ψ − in that neighbourhood.Since ψ − is conformal, f is harmonic off 0. Near 0, we compute f ζ in coordinates to realise C bounds. Via Schauder theory we promote to C (or even C ∞ ) bounds. This impliesthat the tension field is continuous and therefore vanishes everywhere. As for (iii), we canwrite each component f k in certain coordinates as f k = p k + r k , where p k is a spherical harmonic and r k decays faster than p k . In this form, it is easy tocheck the branching orders of f and f .It remains to show that f is unramified. Toward this, let Θ be the Hopf differential of f and note the image of a Φ-disk under ψ is a Θ-disk. Indeed, if Φ = φ ( z ) dz , Θ( ζ ) = θ ( ζ ) dζ in local coordinates, then φ ( z ) = θ ( z r ) (cid:16) ∂z r ∂z (cid:17) = θ ( z r ) z r − r . We rearrange to see θ ( ζ ) = θ ( z r ) = z n − r +2 r − . and the fact that the image is a Θ-disk is derived from direct computation. If f is ramified,we can build another holomorphic branched covering map ψ ′ as in Lemma 3.10. Since both ψ and ψ ′ have finite fibers, the composition ψ ′ ◦ ψ yields a branched cover of degree greaterthan r , which is impossible. This finishes the proof. (cid:3) Remark 3.13.
Our computations show that the ramification order is constrained by r | m , r | ( n +2), and 2 r ≤ n +2. The last condition is superfluous, since we always have 2 m ≤ n +2. Lemma 3.14.
For i = 1 , , let p i be branch points of f of order m i − (we are allowing m i = 1 ), ramified of order r i − . Then p ∼ p if and only if(i) f ( p ) = f ( p ) ,(ii) m /r = m /r , and(iii) if s is the common value m i /r i , there exist maps ψ i : U i → D , f i : D → M , ψ i ( p i ) = 0 , such that ψ i | U i \{ p } is an r i -sheeted holomorphic covering map, f factorsas f | U i = f ◦ ψ i , and f is a harmonic map for the flat metric on the disk with abranch point of order s − .Proof. If m = 0 this is trivial, so we assume m >
0. Suppose the conditions hold. Givenany two open sets Ω i containing p i , we can radially shrink our Φ-disks to have U i ⊂ Ω i (the argument from Lemma 3.8 shows any two points with ψ i ( q ) = ψ i ( q ) have the sameΦ-distance to p i ). For p ′ ∈ U \{ p } let ψ ′ be the restriction to a neighbourhood U ′ of p ′ onwhich ψ is injective. Let ψ ′ be the restriction onto some neighbourhood V ′ such that ψ maps U ′ injectively onto ψ ( U ′ ). Set p ′ = ψ ′− ◦ ψ ′ ( p ′ ) and h = ψ ′− ψ . h is holomorphicand leaves f invariant. The result follows. FACTORIZATION THEOREM FOR HARMONIC MAPS 15
Conversely, assume p ∼ p . (i) was already discussed. We first want to show that we canchoose Φ-disks U i that satisfy condition (2) in the definition of ∼ . We take ψ i : U i → D i and f i : D i → M as in Lemma 3.12. If p ′ ∈ U is equivalent to p ′ ∈ U , then combining ourreasoning from Proposition 3.2 with Proposition 3.4 shows d ( p , p ′ ) = d ( p , p ′ ). We’ve runthis type of argument a few times at this point, but we feel a duty to elaborate. Pick subdisks U ′ i ⊂ U i that satisfy condition (2) and balls B δ ( p ), B ǫ ( p ) contained in the subdisks, suchthat in B δ ( p ) and B ǫ +2 δ ( p ) there are no points equivalent to p , p respectively and noother possible zeros of Φ. Find p ′ ∈ B δ ( p ) \{ p } and p ′ ∈ B ǫ ( p ) \{ p } with p ′ ∼ p ′ . Takethe straight line path γ from p ′ to p and analytically continue h along γ as much as wecan. The image of any segment of this path under h is also a straight line contained in B ǫ +2 δ ( p ). We now have two possibilities:(i) the path h ( γ ) runs into p before we have finished extending, or(ii) we can extend h to the boundary point γ (1) = p In the first scenario, we obtain d ( p , p ′ ) ≤ d ( p , p ′ ). In the latter, Proposition 3.4 ensures h ( γ (1)) ∼ p ∼ p , so that h ( γ (1)) = p . Regardless of the situation, we have d ( p ′ , p ) ≤ d ( p ′ , p ) . To reverse the argument for the other inequality, we go via a straight line from p ′ to p .For any segment γ ′ along which we can continue, the length of h ( γ ′ ) is now bounded aboveby d ( p ′ , p ) < δ . Thus, we can continue along the whole curve so long as we don’t hit p .In the same way as above we get the opposite inequality. This is the desired result.Using the definition of ∼ , we can now assume the Φ-disks U i are such that f ( U ) = f ( U )and that for all p ′ ∈ V , p ′ = p , there is p ′ ∈ V , p ′ = p , such that p ′ ∼ p ′ , and viceversa. We construct a holomorphic diffeomorphism G : D → D such that f ◦ G = f . Let w ∈ D \{ } . We take a small neighbourhood of w and a lift to an open set via ψ such that the restriction of ψ is injective. Let w ′ be the given preimage under ψ . There isthen a point w ′ ∈ V related by a holomorphic map such that f agrees in neighbourhoodssurrounding w ′ and w ′ . Set w = G ( w ) = ψ ( w ′ ). We claim there can be no other pointwith this property. If there was such a w ′ , then we would have w ∼ w ′ with respect tothe corresponding equivalence relation for f . However, we know the map f is unramified,and by Lemma 3.10 we can choose our disks small enough that there are no two distinctpoints in D with this property. The association w w ′ defines our map G . If we set G (0) = 0, then we see G is a diffeomorphism from D \{ } → D \{ } , because we can invertthe construction. The map G is holomorphic off { } . Since it is bounded near 0, it extendsto a holomorphic diffeomorphism on all of D .From Lemma 3.12 the branching order of f i is m i /r i −
1, and since G is a diffeomorphism,it is clear that these branching orders agree. Defining D = D and f to be the commonmap f ◦ G = f , ψ = ψ , ψ = G − ◦ ψ , (iii) can be verified easily. (cid:3) Constructing the Riemann surface.
Preparations aside, we build the coveringspace. Our work here is drawn from Propositions 3.19 and 3.24 in [GOR73]. Let Σ denotethe space of equivalence classes of Σ with respect to ∼ , equipped with the quotient topology.We denote by π : Σ → Σ the projection map. Proposition 3.15. Σ is an orientable surface. Proof.
For each p ∈ M let U be a neighbourhood of p with no other point equivalent to p and as in Lemma 3.12, so that we have a map ψ : U → D , a harmonic map f : D → M ,and a factorization f = f ◦ ψ . Let U = { [ q ] : q ∈ U } . To prove such a set is open, we showany π − ( U ) ⊂ Σ is open. If p ∈ π − ( U ), then there is p ∈ V such that p ∼ p . Thenwe can find neighbourhoods Ω i containing p i with Ω ⊂ U and such that for each p ′ ∈ Ω there exists p ′ ∈ Ω with p ′ ∼ p ′ . This implies the U define an open cover of Σ .On each U we have a map ψ : U → D given by ψ ([ q ]) = ψ ( q ). We will see that thesemaps define charts. If q , q ∈ U are such that q ∼ q , then ψ ( q ), ψ ( q ) are equivalentwith respect to f and hence we can choose U so that ψ ( q ) = ψ ( q ), since f is unramified.This proves ψ is well-defined.For injectivity, suppose [ p ] , [ p ] ∈ U are such that ψ ([ p ]) = ψ ([ p ]). Choosing repre-sentatives p , p , either p = p = p or neither of them is equal to p . In the second case,since ψ is a holomorphic covering map on U \{ p } we can use it to build a holomorphicdiffeomorphism from a neighbourhood of p to a neighbourhood of p . Since f = f ◦ ψ on U , this map leaves f invariant.As for continuity and openness, the argument is the same as the one found in [GOR73,page 779]. The Hausdorff condition is immediate from Proposition 3.4. Σ is orientablebecause π respects the orientation of Σ. (cid:3) There exists a continuous map f : Σ → M such that f = f ◦ π , defined by f ([ p ]) = f ( p ). Proposition 3.16.
There exists a complex structure on Σ so that π : Σ → Σ is holo-morphic and the map f is harmonic with respect to the conformal metric µ obtained viauniformization.Proof. We use the collection of charts specified in Lemma 3.14. Let ( U , ψ ) and ( U , ψ )be two charts for Σ arising from open sets U , U centered at points p , p . We have maps ψ i : U i → D i , ψ i : U i → D i , π : Σ → Σ , and harmonic maps f i : D i → M such that f = f i ◦ ψ i , ψ i = ψ i ◦ π . We show the map ψ ◦ ψ − : ψ ( U ∩ U ) ⊂ D → ψ ( U ∩ U ) ⊂ D is holomorphic.By the removeable singularities theorem, it suffices to check holomorphy away from thecopies of 0 in D i . Let [ q ] ∈ U ∩ U be so that ψ i ([ q ]) = 0, and choose a neighbourhood U around [ q ] and U ′ ⊂ π − ( U ) such that(i) 0 ψ i ( U ),(ii) the map π | U ′ : U ′ → U is injective, and so we can define an inverse π − : U → U ′ ,and(iii) the holomorphic map ψ i is injective in U ′ , so that we can define a holomorphicinverse ψ − i : ψ i ( U ′ ) → U ′ .Note that ψ i ( U ′ ) = ψ i ( U ). Clearly, the map ψ ◦ ψ − is holomorphic in ψ ( U ). Meanwhile,since we can invert π , we obtain ψ ◦ ψ − = ( ψ ◦ π − ) ◦ ( ψ ◦ π − ) − = ψ ◦ ψ − . It follows that the map in question is holomorphic near [ q ], and hence everywhere.In holomorphic local coordinates, the map π is of the form z z or z z n , so it issurely holomorphic. From conformal invariance of the harmonic map equation, f = f i ◦ ψ i FACTORIZATION THEOREM FOR HARMONIC MAPS 17 is harmonic away from images of branch points of π . The argument of Lemma 3.12 shows f is globally harmonic. (cid:3) This completes the proof of Theorem 1.1 for holomorphic diffeomorphisms.4.
Klein Surfaces
We explain the adjustments required to prove Theorem 1.1 for anti-holomorphic diffeo-morphisms h : Ω → Ω .4.1. Preparations.
We begin with a review of Klein surfaces. More details on the theoryof Klein surfaces can be found in the book [AG71]. Set C + = { z ∈ C : Im z ≥ } to be the closed upper half plane. Definition 4.1.
Let Ω ⊂ C + be open. A function f : Ω → C is (anti-)holomorphic if thereis an open set U ⊂ C containing Ω such that f extends to an (anti-)holomorphic functionfrom U → C . Definition 4.2.
A map between open subsets of C is dianalytic if its restriction to anycomponent is holomorphic or anti-holomorphic. Definition 4.3.
Let X be a topological surface, possibly with boundary. A dianalytic atlason X is a collection of pairs U = { ( U α , ϕ α ) } where(i) U α is an open subset of X , V α is an open subset of C + , and ϕ α : U α → V α is ahomeomorphism.(ii) If U α ∩ U β = ∅ , the map ϕ α ◦ ϕ − β : ϕ β ( U α ∩ U β ) → ϕ α ( U α ∩ U β )is dianalytic.A Klein surface is a pair X = ( X, U ).Closely related is the notion of a Real Riemann surface. Definition 4.4.
A Real Riemann surface is the data (
X, τ ) of a Riemann surface X andan anti-holomorphic involution τ : X → X .Given a Real Riemann surface ( X, τ ), the quotient
X/τ has the structure of a Kleinsurface, and as a matter of fact every Klein surface X arises in this fashion (see Chapter1 in [AG71]). The associated Real Riemann surface is called the analytic double, and it isunique up to isomorphism in the category of Real Riemann surfaces. The boundary of theKlein surface corresponds to the fixed-point set of the involution. Definition 4.5.
A harmonic (minimal) map on a Klein surface is a continuous map thatlifts to a harmonic (minimal) map on the analytic double with respect to the conformalmetric obtained via uniformization.To prove Theorem 1.1 for anti-holomorphic maps, as previously done we define an equiva-lence relation ∼ and build a dianalytic atlas on the topological quotient Σ = Σ / ∼ . Beforewe get into details, we make an important reduction: we apply the holomorphic case ofTheorem 1.1 to Σ and acquire a new Riemann surface Σ ′ , as well as maps π : Σ → Σ ′ , f ′ : Σ ′ → M . The key property of the pair (Σ ′ , f ′ ) is that equivalences classes underDefinition 3.2 are singletons.We define a relation ∼ on Σ by taking Definition 3.1, but this time insisting the mapsinvolved are merely conformal rather than holomorphic. Lemma 4.6.
Given p ∈ Σ , there is at most one other point q ∈ Σ ′ such that p ∼ q .Proof. Suppose p, q , q are distinct points and p ∼ q and p ∼ q . If all points are notin Z , then we have anti-holomorphic maps h , h relating to q , q to p . The composition h ◦ h − is then a holomorphic map relating q to q , which means they are equivalentfor Definition 3.2, and this is impossible. If at least one of them is a zero, then we canfind disjoint neighbourhoods Ω containing p and Ω i containing q i such that every point inΩ \{ q } is equivalent to a point in Ω \{ p } , and every point in Ω \{ p } is equivalent to a pointin Ω \{ q } . This brings us to the non-zero case. (cid:3) By the previous lemma, transitivity for ∼ holds vacuously. Accordingly, the proof of thelemma below is trivial. Lemma 4.7. ∼ is an equivalence relation. Proof of the main theorem.
Referencing our earlier work, we prove Theorem 1.1for anti-holomorphic h . Henceforth we abuse notation and set Σ = Σ ′ , f = f ′ .The first thing to note is that h is an orientation-reversing isometry for the Φ-metric.Indeed, if Φ does not vanish on an open subset U ⊂ Ω and z is a natural coordinate for Φ,then the function w = ι ◦ z ◦ h − defines a holomorphic coordinate on h ( U ), where ι is the complex conjugation operator onthe disk. In this coordinate, w ( h ( z )) = z , and it can be easily checked that df p (cid:16) ∂∂z (cid:17) = df h ( p ) (cid:16) ∂∂w (cid:17) ∈ T f ( z ) M ⊗ C . We infer h f w , f w i = 1and furthermore h f w , f w i = h f w , f w i = 1 . As in Lemma 2.5, we find that w is a natural coordinate for Φ. The result follows.Moreover, we can analytically continue h exactly as we did in Proposition 2.6. Movingtoward the main proof, we follow the proof of Lemma 3.6, word-for-word, and note thatLemma 3.7 is immediate from Lemma 4.6. The proof of the analogue of Proposition 3.4follows. As for ramification, we do see new behaviour. Lemma 4.8.
Suppose p is a zero of Φ of order n ≥ . Let h : Ω → Ω be an anti-holomorphic diffeomorphism with f ◦ h = f , and so that Ω , Ω are both contained in aball B ǫ ( p ) , where ǫ > is chosen so that there are no other zeros and no other point isequivalent to p in B ǫ ( p ) . Then, in the natural coordinates for Φ , h ( z ) = e πijn +2 z on its domain. FACTORIZATION THEOREM FOR HARMONIC MAPS 19
Proof.
We follow the proof of Lemma 3.8, except now we have a map h that satisfies z n = ( h ( z )) (cid:16) ∂h∂z (cid:17) . As in the proof of Lemma 3.8, h is defined in a simply connected open set whose distanceto zero can be taken to be arbitrarily small. We observe that h is holomorphic, and take abranch of the square root and integrate to derive h ( z ) = e − πijn +2 z for some j = 0 , , . . . , n + 1. We conjugate to finish the proof. (cid:3) The lemma implies that in a neighbourhood of a ramification point p , f is invariant underthe map ψ ( z ) = e πijn +2 z. This is an anti-holomorphic involution that fixes every point on the line L = { re πijn +2 : − < r < } and acts by reflection across this line on all other points. Lemma 4.9.
Let p and ψ be as above. If ψ ( q ) = q , then q has no equivalent points withrespect to ∼ .Proof. ψ is two-to-one in a neighbourhood of q . Suppose there exists q ′ ∈ Σ with q ∼ q ′ .Then q ′ B ǫ ( p ). Using the definition of ∼ , we can find a small disk B ǫ ′ ( q ) and points p , p ∈ B ǫ ′ ( q ) with p ∼ p , but we can also find a point q ′′ near q ′ such that p ∼ q ′′ . Thiscontradicts Lemma 4.6. (cid:3) We deduce the following.
Lemma 4.10.
Every q ∈ B ǫ ( p ) \L is equivalent to ψ ( q ) and only ψ ( q ) . We say f anti-holomorphically ramifies near p if f is invariant under an anti-holomorphicinvolution in a neighbourhood of p . In contrast to the holomorphic definition, f can anti-holomorphically ramify near rank 1 singularities. If f does ramify at p , we form the quotient K = B ǫ ( p ) /ψ by identifying points z and ψ ( z ). This has the structure of a Klein surface with boundary,the boundary being identified with L . Lemma 4.11. p ∈ Σ satisfies [ p ] = { p } if and only if f ramifies at p .Proof. We need only to show that every point at which f is unramified admits an equivalentpoint. Looking toward a contradiction, suppose there exists p ∈ Σ with [ p ] = { p } and atwhich f does not ramify and choose ǫ > B ǫ ( p ) andthat there are no zeros of Φ in B ǫ ( p ).We claim [ q ] = { q } for every q ∈ B ǫ ( p ). If not, there is a q ∈ B ǫ ( p ) that admits an equiv-alent point q ′ = q . Let h be the anti-holomorphic diffeomorphism relating a neighbourhoodof q to one of q ′ . In coordinates, analytically continue h along a straight line γ from q to p .It follows from our assumption { p } = [ p ] that no segment h ( γ ′ ) for γ ′ ⊂ γ can touch p , forotherwise we get a point equivalent to p . Thus, we can continue to the endpoint, and theendpoint of h ( γ ) is p itself. This implies d ( p, q ) = d ( p, q ′ ) , which contradicts our choice of ǫ >
0, and therefore settles the claim.With the claim in hand, we define a map τ : Σ → Σas follows. If [ q ] = { q } , set τ ( q ) = q . If [ q ] = { q, q ′ } , we put τ ( q ) = q ′ . If f is unramified at q and [ q ] = { p, q } , then τ is an anti-holomorphic diffeomorphism near p. If [ q ] = { q } , thenour claim above shows it is the identity map in a neighbourhood of q . If f ramifies at q , τ acts like the map ψ considered above. In any event, τ is real analytic. Since we know theset { q : | [ q ] | = 2 } is non-empty, τ is globally anti-holomorphic and moreover cannot fix thepoint p . This gives a contradiction. (cid:3) We now come to the main goal. Simply take the anti-holomorphic map τ defined in theproof above. Checking on a topological base for Σ, it is clear that τ is a continuous andopen mapping. As τ = 1, it is an anti-holomorphic diffeomorphism of Σ. The quotientΣ = Σ /τ = Σ / ∼ is the sought Klein surface. Remark 4.12.
We can read off an atlas as follows. If p is not a ramification point, ∼ iden-tifies a small neighbourhood of p with no ramification points to some other neighbourhood.The coordinate chart near p then gives the chart on Σ . Transition maps can be holomor-phic or anti-holomorphic. If p is a ramification point, the quotient gives us a space K asabove, with two different choices for coordinates: natural coordinates for Φ, or the complexconjugation of those coordinates. Both holomorphic and anti-holomorphic transition mapsexist. We omit the technical details.With regard to Theorem 1.1, we are left to discuss the projection π : Σ → Σ andthe harmonic map f . The remark gives coordinate expressions for π in which we see it isdianalytic. Σ is actually the analytic double of Σ , and f clearly descends to a continuousmap f on Σ that is harmonic by definition. This finishes the proof of Theorem 1.1.4.3. Minimal Klein surfaces.
For completeness, we extend the work of Gulliver-Osserman-Royden on minimal maps to the anti-holomorphic case. To the author’s knowledge, theresult of this subsection is new.We begin with a minimal map f : (Σ , µ ) → ( M, ν ) and anti-holomorphic h : Ω → Ω such that f ◦ h = f . As in our approach for non-minimal maps, we first apply [GOR73,Proposition 3.24] to assume Σ has no points that are holomorphically related. We then de-fine ∼ exactly as in subsection 1.1, but allow the diffeomorphisms involved to be conformal.The application of their result assures that Lemma 4.6 goes through for ∼ . The proof ofProposition 3.14 in [GOR73] applies to the map f , which proves the relation ∼ is Hausdorff.For ramification, the distinction is that Φ = 0, so we cannot apply the usual methods.At the same time, all singular points are good branch points. Recall from subsection 1.1that near a branch point p of order m we can find a neighbourhood of p with a holomorphiccoordinate z and coordinates ( x , . . . , x n ) around f ( p ) so that f is given by x = Re z m , x = Im z m , x k = η k ( z ) , k ≥ , where η k ( z ) ∈ o ( | z | m ). If we have distinct p , p in this neighbourhood with p ∼ p , thenthe anti-holomorphic map h that relates the two must satisfy( h ( z )) m = z m . FACTORIZATION THEOREM FOR HARMONIC MAPS 21
Consequently, h is of the form h ( z ) = e πijm z for some j = 0 , , . . . , m −
1. Up until Lemma 4.11, one can run through subsection 4.2almost word-for-word. The only difference is that we use coordinate disks rather thannatural coordinates for a holomorphic differential. The analogue of Lemma 4.11 can beworked out without difficulty.
Lemma 4.13.
In this setting, p ∈ Σ satisfies [ p ] = { p } if and only if f ramifies at p .Proof. Even if f is minimal, analytic continuation is possible. Given a curve γ starting inΩ , we can analytically continue h along γ as long as γ and h ( γ ) stay sufficiently far awayfrom the set { p ∈ Σ : [ p ] intersects the branch set of f } . To do so, we first can assume f is a diffeomorphism on Ω i and injective on Ω i . If q is thefirst point at which γ strikes ∂ Ω , then h ( q ) is well-defined. We choose disks U and U around q and h ( q ) respectively such that f | U i is a diffeomorphism. We then invoke theunique continuation property of Gulliver-Osserman-Royden to find a smaller disk U ′ ⊂ U such that f ( U ′ ) ⊂ U . Setting U ′ = f | − U ( f ( U ′ )), the map f | − U ′ ◦ f | U ′ : U ′ → U ′ is a conformal diffeomorphism that continues h , and is therefore anti-holomorphic. Thisestablishes the continuation result. We also note that [GOR73, Proposition 3.14] impliesthat if γ is a curve along which we have continued h , then p ∼ h ( p ) for all p in the imageof γ .We suppose there is a point p at which f is unramified and such that [ p ] = { p } . Choose acoordinate disk Ω around p in which no two points are equivalent. We show that under thisassumption we must have [ q ] = { q } for all q ∈ Ω. If not, then there is a q ∈ Ω and a q ′ Ωsuch that q ∼ q ′ , and an anti-holomorphic diffeomorphism h relating a neighbourhood of q to one of q ′ . We analytically continue h along a simple curve from q to p that does nottouch any point that is equivalent to a branch point of f . It is easy to build such a curve,since the branch set is discrete, and equivalence classes can have only two points. Usingthe reasoning from Lemma 4.11, we can continue along all of γ and h ( γ (1)) = p . Now, notethat by assumption there is no pair p , p ∈ B ǫ ( p ) with p ∈ γ ([0 , p ∼ p . Taking γ to the endpoint gives that h ( γ ( t )) lies outside B ǫ ( p ) for t ∈ [0 ,
1] sufficiently close to 1.This contradicts h ( p ) = p , and hence yields [ q ] = { q } for all q ∈ Ω. We can now concludethe proof exactly as we did in Lemma 4.11. (cid:3)
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