Simultaneous unitarizability of SL<span class="mathjax-formula">$_{\hbox{\textit {n}}}{\mathbb {C}}$</span>-valued maps, and constant mean curvature <span class="mathjax-formula">$k$</span>-noid monodromy

Simultaneous unitarizability of SL nC -valued maps, and constant mean curvature k-noid monodromy

WAYNEROSSMAN ANDNICHOLASSCHMITT

Dedicated to Professor Takeshi Sasaki on his sixtieth birthday Abstract. We give necessary and sufficient local conditions for the simultaneous unitarizability of a set of analytic matrix maps from an analytic 1-manifold into SLnCunder conjugation by a single analytic matrix map.

We apply this result to the monodromy arising from an integrable partial differential equation to construct a family ofk-noids, genus-zero constant mean curvature surfaces with three or more ends in Euclidean, spherical and hyperbolic 3-spaces.

Mathematics Subject Classification (2000):53C42 (primary); 53A35, 49Q10 (secondary).

Introduction

In this paper we find necessary and sufficient conditions for the existence of an SL_n

C

-valued analytic matrix map on an analytic 1-manifold which simultaneously unitarizes a given set of analytic matrix maps via conjugation. We apply these results to construct families of constant mean curvature (CMC) immersions with arbitrarily many ends into ambient 3-dimensional space forms (Theorem 8.5). The ends are conjectured to be asymptotic to half-Delaunay surfaces.

We show that the existence of a global unitarizer is equivalent to the existence of analytic unitarizers defined only on local neighborhoods (Theorem 1.10). In the case of SL₂

C

the necessary and sufficient conditions for global simultaneous unitarization are local diagonalizability, pointwise simultaneous unitarizability, and pairwise infinitesimal irreducibility (Theorem 3.4). This latter condition means that at each point of the 1-manifold, the coefficient of the leading term of the series ex- pansion of the commutator has full rank. For general SL_n

C

, global unitarizability is equivalent to pointwise unitarizability together with a graph condition (Theo- rem 2.7). These results are proven by linearizing the unitarization problem and applying analytic Cholesky decompositions.

The Unitarization Theorem 2.7 is a refinement of the variantr-unitarization Theorem 4.4 (see [22, 6] for the case of SL₂

C

). In Theorem 4.4, the analytic curve Received May 31, 2006; accepted in revised form October 23, 2006.

is the standard unit circle

S

¹, and an analytic simultaneous unitarizerC is found on a radius-r circle for somerless than 1.

While the unitarized loops extend holomorphically to

S

¹, the unitarizerC gen- erally has branch points. The conditions of the Unitarization Theorem 2.7 are the obstructions to extendingCholomorphically to

S

^1.

One application of the Unitarization Theorem 2.7 is to solving the monodromy problem arising in the construction of CMC surfaces via the extended Weierstrass representation. In this construction, using integrable system methods, the problem of closing the surface is solved by unitarizing the monodromy group, whose ele- ments are defined on a loop. The unitarization theorem provides a construction of a global analytic simultaneous unitarizer once it is known that the monodromy group is pointwise simultaneously unitarizable along the loop.

Hence in the second part of the paper we apply the unitarization theorem to the construction of CMC genus-zero surfaces with arbitrary numbers k ≥ 3 of ends which are asymptotic to half-Delaunay surfaces [12], lying in ambient 3- dimensional space forms (see Figures 1.1 and 1.2). We call these surfacesk-noids, and trinoids whenk = 3. Fork = 2 these are the well-known Delaunay surfaces, CMC surfaces of revolution with translational periodicity.

Trinoids in

R

³, either embedded or non-Alexandrov-embedded, were first con- structed by Kapouleas [11] using techniques that glue parts of CMC surfaces to- gether via the study of Jacobi operators, and later there has been further work in this direction [16, 19], but this approach only gives examples that are in some sense

“close” to the boundary of the moduli space of the surfaces.

Figure 1.1. Symmetric CMC 5-noids inR³, one with unduloidal ends and one with nodoidal ends (cutaway view). The images were produced by CMCLab [20].

A construction that gives a broader collection of Alexandrov embedded trinoids [9, 10] andk-noids [8] in

R

³with embedded ends asymptotic to Delaunay unduloids was found by Große-Brauckmann, Kusner and Sullivan, using an isometric corre- spondence between minimal surfaces in the 3-dimensional sphere

S

^{3 and CMC 1} surfaces in

R

³. The family ofk-noids we construct here also includes surfaces with asymptotically Delaunay ends [12].

Constant mean curvature trinoids in

R

³ with embedded ends via loop group techniques are constructed in [22, 5] by methods derived from integrable systems techniques. Developed initially by Dorfmeister, Pedit and Wu [4], this construc- tion employs ther-Unitarization Theorem together with ther-Iwasawa decompo- sition [17], a generalization of the Iwasawa decomposition on the unit circle to a radius-rcircle withr <1.

These trinoids extend to larger classes of non-Alexandrov-embedded trinoids:

In one way, Kilian, Sterling and the second author [13] dressed these trinoids into

“bubbleton” versions, also conformal to thrice-punctured spheres; these bubbleton versions have ends asymptotic to embedded Delaunay unduloids [12], and com- puter graphics suggest that they are not Alexandrov embedded. In another way, Kilian, Kobayashi and the authors [22] extended the class of trinoids to CMC 1 sur- faces in

R

³ whose potentials are perturbations of nonembedded Delaunay nodoid potentials, and hence are not Alexandrov embedded [12]. Also, in [22], trinoids in

S

³and hyperbolic 3-space

H

³were proved to exist, including examples that are not Alexandrov embedded.

In this work, we establish the closing conditions for trinoids and symmetric k-noids via loop group techniques by a more elementary approach using only the 1-unitarization theorem and 1-Iwasawa decomposition.

Figure 1.2.Symmetric CMC 5-noids inS³andH³. Here,S³has been stereographically projected toR³∪ {∞}, andH³is shown in the Poincar´e model.

1.

Simultaneous unitarizability for SL

_n

C

1.1. Preliminaries

An analytic curveis a connected real analytic one-dimensional manifold without boundary. On an analytic curve

C

we denote by

H

C

V

the set of analytic maps

C

→

V

into a space

V

^{, and by}

M

C

V

the set of analytic maps

C

→

V

with possible poles.

M ∈

H

CSL_n

C

^islocally diagonalizable at p ∈

C

if there exists a neighbor- hood

U

⊆

C

^{of pandV} ∈

H

USL_n

C

^{such thatV M V−1}is diagonal.

M₁, . . . ,M_q ∈

H

CSL_n

C

^arelocally simultaneously unitarizable at p ∈

C

^if there exists a neighborhood

U

⊆

C

^ofpandV ∈

H

USL_n

C

^{such thatV M}1V⁻¹, . . . . . . ,V MqV⁻¹∈

H

USUn

C

^.

M₁, . . . ,M_q ∈

H

CSL_n

C

^aresimultaneously unitarizableon

C

if there exists V ∈

H

CSLn

C

^{such thatV M1V−1}, . . . ,V MqV⁻¹∈

H

CSUn

C

^.

For f ∈

M

C

C

^{, define f∗= f, and forM ∈}

M

CM_n×n

C

^{, define M∗= Mt.} For 0 < r < s < ∞, let

A

r,s ⊂

C

denote the open annulus

A

r,s = {λ ∈

C

^{|r <|λ|<s}.}

A subset

G

⊂ M_n×n

C

^isreducibleif there exists a proper non-zero subspace V ⊂

C

^{nsuch thatG V} ⊂V for allG ∈

G

. In the case

G

= {A, B}is a set of two elements, we say A, B∈M_n×n

C

are reducible.

We have the following Schur-type lemma.

Lemma 1.1. If A, B ∈ M_n×n

C

are irreducible and X ∈ M_n×n

C

commutes with A and B, then X is a multiple of the identity matrixI∈M_n×n

C

^.

Proof. Suppose X ∈

C

^{I and let λ ∈}

C

be an eigenvalue of X. Let V = {v ∈

C

^{n|Xv=λv}, soV = {0}. ThenV ⊂}

C

ⁿis a proper subspace becauseX ∈

C

^I.

For anyv∈V, we have X Av= A Xv= λAv, so Av∈ V. HenceAV ⊆ V.

Similarly, B V ⊆V, soAandBwould be reducible.

1.2. Linearizing the simultaneous unitarization problem

We begin with an elementary proof of a specialization of standard results in the theory of holomorphic vector bundles, constructing a global kernel of a suitable bundle map which depends holomorphically (or real analytically) on a parameter.

Lemma 1.2. Let

D

be an analytic1-manifold or2-manifold, N, M ∈

N

^{, and L :}

D

→Hom(

C

^N,

C

^M)a holomorphic map. Supposedim kerL=1on

D

^{away from} a subset S⊂

D

of isolated points. Then

(i) dim kerL≥1on

D

^.

(ii) There exists a holomorphic map X :

D

→

C

^N which is not identically0such that X ∈kerL, that is, for each p∈

D

^{, X}(p)∈kerL(p).

Proof. SinceLhas rank N −1 on

D

^{\S, all theN ×N} minor determinants ofL are holomorphic on

D

and zero on

D

^\S, and hence 0 on

D

. Hence dim kerL≥1 on

D

^.

SinceLhas rankN−1 on

D

\S, then there exists an(N−1)×(N−1)minor determinant ofLwhich is not identically 0 on

D

. Hence by a permutation we may assume without loss of generality that

L = A B

C D

,

where A∈

H

DM_{(N−1)×(N−1)}

C

^{witha:=detA ≡0 on}

D

^,B ∈

H

DM_(N−1)×1

C

^, C ∈

H

DM_{(M−N+1)×(N−1)}

C

^{, andD}∈

H

DM_(M−N+1)×1

C

^.

DefineX =(−a A⁻¹B,a)^t. Thena A⁻¹is holomorphic on

D

because its en- tries are polynomials in the entries of A. HenceXis holomorphic on

D

. Moreover,

X ≡0 becausea ≡0.

It is clear that the upper(N −1)×1 block ofL Xis 0. To show that the lower (M−N+1)×1 block ofL Xis zero, letA_k ∈

H

CM_1×(N−1)

C

^{andBk ∈}

H

C

C

^{be the} respectivek’th rows ofAandB(k ∈ {1, . . . ,N−1}). Then sinceB−A A⁻¹B=0, B_k −A_kA⁻¹B=0, k∈ {1, . . . ,N−1}. (1.1) Fix j ∈ {1, . . . ,M−N +1}and letC_j ∈

H

CM_1×(N−1)

C

^{and Dj ∈}

H

C

C

^{be the} respective j’th rows ofC andD. Because allN ×N minor determinants ofLare 0, thenC_j =

r_kA_k and D_j =

r_kB_k for some holomorphic scalar functions r₁, . . . ,r_N−1. By (1.1),C_jA⁻¹B = D_j. Since this holds for all j ∈ {1, . . . ,M− N +1}, we have D−C A⁻¹B = 0. Hence the lower (M − N +1)×1 block a(D−C A⁻¹B)ofL Xis 0,L X =0, and X∈kerL.

A global simultaneous unitarizer of a suitable set of analytic mapsM₁, . . . ,M_q on an analytic curve is constructed in two steps. First, a global analytic solutionXto the linear system X M_kX⁻¹ = M_k^∗−1,k ∈ {1, . . . ,q}, is found which is Hermitian positive definite. Then the analytic Cholesky decomposition gives X = V^∗V, and V is a simultaneous unitarizer ofM₁, . . . ,M_q.

We continue by defining a linear mapLwhose kernel will containX.

Definition 1.3. ForM₁, . . . ,M_q ∈SL_n

C

^{(n≥1), defineL =}

L

^{(M1, . . . ,Mq)as} the linear mapL: M_n×n

C

^→(Mn×n

C

⁾ⁿ

L(X)=

X M₁−M₁^∗−1X, . . . ,X M_q−M_q^∗−1X . Lis similarly defined forM₁, . . . ,M_q ∈

H

CSL_n

C

^.

The linear mapL(M₁, . . . ,M_q)has the following easily computed properties.

The first of these properties motivates the definition ofL.

Lemma 1.4. With n≥2, let M₁, . . . ,M_q ∈SL_n

C

^{and let L=}

L

(M₁, . . . ,M_q)be as in Definition1.3. Then

(i) If A∈SL_n

C

^{and A∗A∈ker}L, then A simultaneously unitarizes M₁, . . . ,M_q. (ii) If X ∈kerL, then X^∗ ∈kerL.

(iii) Let C ∈GL_n

C

^{and letL =}

L

^{(C M1C−1, . . . ,C MqC−1). Then X ∈kerL if} and only if C^∗X C ∈kerL.

We will require the following further properties ofL.

Lemma 1.5. With n ≥ 2, let M₁, . . . ,M_q ∈ SL_n

C

^{and L =}

L

(M₁, . . . ,M_q). If for some i, j ∈ {1, . . . ,q}, M_i and M_j are irreducible, and if M₁, . . . ,M_q are simultaneously unitarizable, thendim kerL =1.

Proof. LetCbe a simultaneous unitarizer ofM₁, . . . ,M_q, defineP_k =C M_kC⁻¹ ∈ SU_n (k∈ {1, . . . ,n}), and setL=

L

(P₁, . . . ,P_r). SinceP_k = P_k^∗−1, then kerLis the set ofX ∈M_n×n

C

such that [X, P_k]= 0 (k ∈ {1, . . . ,q}). Since M_i andM_j are irreducible, then P_i and P_j are irreducible. By Lemma 1.1, kerL=

C

^{⊗I. By} Lemma 1.4(iii), kerL=

C

^⊗(C∗C), so dim kerL=1.

Lemma 1.6. Let M₁, M₂∈SL_n

C

^{and L =}

L

(M₁, M₂). Suppose M₁and M₂are irreducible and simultaneously unitarizable. Let X ∈ M_n×n

C

^{\ {0}}and suppose

X∈kerL and X^∗ = X . Then X is positive or negative definite.

Proof. First assume the special case thatM₁, M₂ ∈ SU_n. Then X ∈kerLmeans [X, M₁]=0 and [X, M₂]=0. SinceM₁andM₂are irreducible, by Lemma 1.1, X=rI for somer ∈

C

^{∗. SinceX} is Hermitian, thenr ∈

R

^{∗. HenceX} is positive or negative definite.

To show the general case, letC ∈SL_n

C

be a simultaneous unitarizer ofM₁and M₂, and letL =

L

(C M₁C⁻¹,C M₂C⁻¹). ThenX = (C⁻¹)^∗X C⁻¹is Hermitian, and X ∈ kerL by Lemma 1.4(iii). By the special case above, X is positive or negative definite. Hence Xis positive or negative definite.

Lemma 1.7. Let

C

be an analytic curve, and M₁, . . . ,M_q ∈

H

CSL_n

C

^{(q ≥ 2),} and let L =

L

(M₁, . . . ,M_q). Suppose for some subset S ⊂

C

of isolated points, M₁and M₂are irreducible on

C

\S, and M₁, . . . ,M_qare pointwise simultaneously unitarizable on

C

\S. Then there exists X ∈(

H

CM_n×n

C

^{)∩kerL with X∗ = X} and a subset S⊂

C

of isolated points such that X is positive definite on

C

\S. Proof. By Lemma 1.5, dim kerL =1 on

C

\S. By Lemma 1.2, dim kerL ≥1 on

C

and there exists an analytic mapX₁∈

H

CM_n×n

C

^{such that X1∈(kerL)\ {0}.}

If X₁is Hermitian, defineX₂= X₁; otherwise defineX₂=i(X₁−X₁^∗). Then X₂ ≡0,X₂is Hermitian, andX₂∈kerLby Lemma 1.4(ii).

By Lemma 1.6, X₂ is (pointwise) either positive definite or negative definite except at the set of isolated points at which X₂ = 0 or where M₁, . . . ,M_q all commute.

Let v ∈

C

^{n \ {0}}be any non-zero constant vector and define f = v^∗X₂v. Then f ≡ 0 and X = f X₂ is positive definite on

C

away from a set of isolated points.

1.3. Analytic Cholesky decompositions

We prove two analytic versions of the Cholesky decomposition theorem. The first (Proposition 1.8) is for Hermitian positive definite maps on an analytic curve, used in the Unitarization Theorem 1.10. The second (Proposition 4.2), used in ther- unitarization Theorem 4.4, is for meromorphic maps on

S

¹ which are Hermitian positive definite except at a finite set of points.

Proposition 1.8 (Holomorphic Cholesky decomposition). Let

C

be an analytic curve and let X ∈

H

CSL_n

C

be Hermitian positive definite. Then

(i) There exists V ∈

H

CSL_n

C

such that X= V^∗V .

(ii) V is unique up to left multiplication by elements of

H

CSU_n.

Proof. We first prove the following analytic version of the LDU-decomposition: if X ∈

H

CGL_n

C

is Hermitian positive definite, then there exists R, D ∈

H

CGL_n

C

such that X = R^∗D R, where R is upper triangular with diagonal elements≡ 1, andDis diagonal with diagonal elements in

H

C

R

^>0. The proof is by induction on n. The casen=1 is clear, withR=I,D= X.

Now assume the statement is true forn−1 and write X =

X₀ Y₀ Y₀^∗ z

,

with X₀ ∈

H

CGL_n−1

C

^{, Y0 ∈}

H

CM_(n−1)×1

C

^{and z ∈}

H

C

C

^{. Then X0 is Her-} mitian positive definite, so let X₀ = R^∗₀D₀R₀ be the decomposition given by the induction hypothesis. Then D₀ and R₀^∗ are invertible on

C

, so we can define

R, D∈

H

CGL_n

C

^by R=

R₀ S₀ 0 1

, S₀=(R^∗₀D₀)⁻¹Y₀, D= D₀ 0

0 d

, d =z−S₀^∗D₀S₀. Then we have X = R^∗D R. Taking the determinant yields detX = detD = ddetD₀, showing thatdtakes values in

R

^>0. This proves the statement forn.

To show the first part of the theorem, take Hermitian positive definite X ∈

H

CSL_n

C

^{, and let X = R∗D R} be its analytic LDU factorization. With D = diag(d₁, . . . ,d_n), letE =diag(√

d₁, . . . ,√

d_n), choosing positive square roots. Let V = E R, soX =V^∗V. It is clear that detV ≡1, soV ∈

H

CSL_n

C

, proving (i).

To show uniqueness (ii), supposeV, W ∈

H

CSL_n

C

^{withV∗V = W∗W. Let} U =W V⁻¹∈

H

CSL_n

C

^{. ThenU∗=U−1, soU ∈}

H

CSU_n.

Lemma 1.9. If X₁, X₂ ∈SL_n

C

are positive definite, and X₂is a multiple of X₁, then X₁= X₂.

Proof. LetX₂ = c X₁. SinceX₁and X₂ are positive definite, thenc > 0. Taking the determinant,cⁿ=1, socis ann’th root of 1. Hencec=1, soX₁= X₂. 1.4. Simultaneous unitarization

We are now prepared to prove the following unitarization theorem. Conditions equivalent to condition (ii) of this theorem (local simultaneous unitarizability) are found in Sections 2.1–3.

Theorem 1.10. Let

C

be an analytic curve and M₁, . . . ,M_q ∈

H

CSL_n

C

^{(q ≥2).}

Suppose M₁and M₂are irreducible on

C

except at a subset of isolated points. Then the following are equivalent:

(i) M₁, . . . ,M_qare globally simultaneously unitarizable on

C

^. (ii) M₁, . . . ,M_qare locally simultaneously unitarizable at each p∈

C

^.

In this case, any simultaneous unitarizer V is unique up to left multiplication by an element of

H

CSU_n.

Proof. Clearly (i) implies (ii). Conversely, suppose (ii). A global analytic simulta- neous unitarizerV is constructed as follows.

Fix p ∈

C

. By (ii), there exists a neighborhood

U

pof pandW ∈

H

USL_n

C

such that W M_jW⁻¹ ∈

H

UpSU_n, j ∈ {1, . . . ,q}. Define X_p = W^∗W. Then X_p∈

H

UpSL_n

C

^∩kerLandXpis Hermitian positive definite. By Lemma 1.9,X_p is the unique map in a neighborhood of pwhich takes values in SL_n

C

, is in kerL, and is Hermitian positive definite.

Define X :

C

→ SL_n

C

^{by X}(p) = X_p(p). Then for each p ∈

C

^{, X is} analytic at pbecause it coincides with the local analytic map X_p on

U

^{p, again by} Lemma 1.9. This defines the unique mapX ∈

H

CSL_n

C

^∩kerLwhich is Hermitian positive definite on

C

^.

By the Cholesky Decomposition Proposition 1.8, there existsV ∈

H

CSL_n

C

such thatX=V^∗V. Then fork∈{1, . . . ,q},V M_kV⁻¹∈

H

CSU_nby Lemma 1.4(i), soV is the required simultaneous unitarizer.

To show uniqueness, suppose W ∈

H

CSL_n

C

is another simultaneous unita- rizer. ThenW^∗W is a Hermitian positive definite element of

H

CSL_n

C

^{∩kerL. By} Lemma 1.9,W^∗W = X =V^∗V. The uniqueness result follows by the uniqueness result in the Cholesky Decomposition Proposition 1.8.

2.

Local conditions for simultaneous unitarizability

Theorem 1.10 effectively reduces global simultaneous unitarizability to local si- multaneous unitarizability. We will now give more explicit conditions for local simultaneous unitarizability.

2.1. n-graphs

Given two matrix mapsM₁, M₂ ∈

H

CSL_n

C

^withM1diagonal, the local simulta- neous unitarizability ofM₁andM₂at pis equivalent to the equalities of the orders of certain corresponding entries ofM₂andM₂^∗−1at p. These relations are naturally expressed in terms of graphs on{1, . . . ,n}(Definition 2.1).

Definition 2.1. An-graph is a directed graph with vertices V = {1, . . . ,n}. A n-graph is not a multigraph — it may not have two instances of the same edge.

However, it may have an edge connecting a vertex to itself. A n-graph is con- nectedif it is connected as a undirected graph.

Let

C

be an analytic curve, let p∈

C

^{, and let A}, B∈

H

CM_n×n

C

^{. Forµ, ν ∈} {1, . . . ,n}, letA_µνdenote the entry ofAlying in rowµand columnν, and similarly for B.

LetGbe an-graph. AisG-non-zero at pif for every directed edge(µ, ν)∈ V² fromµto ν of G we have ord_pA_µν < ∞. (Since

C

is connected and Ais holomorphic, this order condition is equivalent to the condition A_µν ≡0.)

Let G be a n-graph. A and B areG-compatible at p if for every directed edge(µ, ν)∈V²ofG, we have ord_pA_µν =ord_pB_µν <∞.

Lete₁=(1,0, . . . ,0), . . . ,e_n =(0, . . . ,0,1)∈

C

ⁿ be the standard basis for

C

^{n. Fixing}n, we have the natural correspondence between the nonempty subsets of {1, . . . ,n}and the nonzero subspaces of

C

ⁿgenerated by standard basis elements, where K ⊂ {1, . . . ,n}corresponds to the subspace

E_K =span{e_k|k∈K} ⊆

C

^n.

This correspondence induces a natural one-to-one correspondence between the set of partitions of{1, . . . ,n}all of whose terms are non-empty, and the set of direct sum decompositions of

C

ⁿ relative to its standard basis, all of whose terms are non-zero.

The following lemma expresses in terms ofn-graphs a notion closely related to block-diagonalizability.

Lemma 2.2. Let

C

be an analytic curve, let A∈

H

CM_n×n

C

, and let p∈

C

^{. Then} the following are equivalent:

(i) Everyn-graph H for which A is H -non-zero at p is disconnected.

(ii) There exists a direct decomposition

C

^{n = V1⊕V2}into non-zero summands V₁ and V₂generated by standard basis elements of

C

ⁿ such that AV₁ ⊆ V₁ and

AV2⊆V2near p.

Proof. Suppose (ii) and letG₁G₂ = {1, . . . ,n}be the corresponding partition.

The conditions AV₁ ⊆ V₁ and AV₂ ⊆ V₂ are equivalent to the condition that A_µν ≡ 0 for all (µ, ν) ∈ (G₁ ×G₂)∪(G₂×G₁). Let G be a n-graph for which Ais G-non-zero at p. ThenG does not contain any of the edges in(G₁× G₂)∪(G₂ ×G₁). Hence the two subsetsG₁ andG₂ of the vertex set ofG are disconnected, proving (i).

Conversely, suppose (i) and let G be the maximaln-graph among then- graphsH for whichAisH-non-zero atp. ThenGis disconnected, so letG₁G₂= {1, . . . ,n}be a partition with G₁ andG₂ nonempty such that no edge of G is in (G₁×G₂)∪(G₂×G₁). SinceGis maximal, thenA_µν ≡0 for all(µ, ν)∈(G₁× G₂)∪(G₂×G₁). Let

C

^{n =V1⊕V2}be the direct sum decomposition corresponding to the partitionG₁G₂. Then AV₁⊆V₁andAV₂⊆V₂, proving (ii).

We shall say thatX∈

H

CM_n×n

C

^isinfinitesimally invertible at pif the leading term in its series expansion at pin some local coordinate on

C

^nearphas full rank.

Equivalently, there exists a local meromorphic scalar function f near psuch that f X is holomorphic atpand rank(f X)(p)=n.

Lemma 2.3. Let

C

be an analytic curve, let p ∈

C

^{and let A}, B ∈

H

CM_n×n

C

^. Let X ∈

H

CM_n×n

C

be diagonal with X ≡0, and suppose X A = B X . Then the following are equivalent:

(i) A and B are G-compatible at p for some connectedn-graph G.

(ii) X is infinitesimally invertible at p, and neither of the equivalent conditions of Lemma2.2hold for A at p.

Proof. First suppose (ii) holds. Since X is diagonal, the infinitesimal invertibility of Xis equivalent to

ordp X11= · · · =ord

p Xnn <∞. (2.1)

The components of X A−B X are

0=(X A−B X)_µν= X_µµA_µν−X_ννB_µν. (2.2) LetGbe a connectedn-graph such thatAisG-non-zero at p. Then for each edge (µ, ν)ofG, we have ord_pA_µν<∞. Then (2.1) and (2.2) imply

ordp A_µν =ord

p B_µν<∞. (2.3)

Hence AandBareG-compatible at p.

Conversely, suppose (i), that A and B are G-compatible at pfor some con- nected n-graph G. Then (2.3) holds for each edge (µ, ν) of G. Hence A is G-non-zero at p. By (2.2),

ordp X_µµ=ord

p X_νν <∞.

The connectedness ofGimplies (2.1).

Lemma 2.4. Let

C

be an analytic curve, p ∈

C

, and suppose X ∈

H

CMn×n

C

^is infinitesimally invertible at p. Then

(i) detX has a local analytic n’th root in some neighborhood

U

⊂

C

^{of p.}

(ii) (detX)^−1/nX ∈

H

USL_n

C

^.

(iii) If X is Hermitian positive definite on a punctured neighborhood of p, then the n’th root can be chosen so that(detX)^−1/nX is Hermitian positive definite in a neighborhood of p.

Proof. Lett be a local coordinate on

C

^{near pwitht(p)=0. Since X}is infinitesi- mally invertible at p, we have

X= X_mt^m+O(t^m+1), detX_m =0. (2.4) Taking the determinant of (2.4),

detX=(detX_m)t^nm+O(t^nm+1). (2.5) Since detX_m =0, then ord_p(detX)=nm. Sincenm is divisible byn, then detX has a local analyticn’th root in a neighborhood

U

⊂

C

^of p, proving (i).

Taking ann’th root of (2.5),

(detX)^1/n=(detX_m)^1/nt^m+O(t^m+1), (2.6) so

Y :=(detX)^−1/nX =(detX_m)^−1/nX_m+O(t¹), and henceY ∈

H

USL_n

C

, proving (ii).

To show (iii), suppose

U

is a neighborhood of psuch that X is positive def- inite on

U

\ {p}. Then detX is positive on

U

\ {p}, and hence by its continuity is nonnegative on

U

. By (2.5), detX_m > 0, so we can choose(detX_m)^1/n > 0 in (2.6). Equation (2.4) and a limit argument imply thatm is even. Equation (2.6) then implies that(detX)^1/n is non-negative on

U

^{. ThenY =(detX)1/nXis posi-} tive definite in a punctured neighborhood ofp. HenceY(p)is positive semidefinite, and soY(p)is positive definite since detY(p)=1.

Lemma 2.5. Let

C

be an analytic curve, let p ∈

C

^{and let M}1, M₂ ∈

H

CSL_n

C

^. Suppose M₁and M₂ are irreducible on

C

\ {p}. Suppose M₁is diagonal and no two local analytic eigenvalues of M₁are identically equal. Then the following are equivalent:

(i) M₁and M₂are locally simultaneously unitarizable at p.

(ii) M₁and M₂are pointwise simultaneously unitarizable at each point of a neigh- borhood of p, and M₂ and M₂^∗−1 are G-compatible at p for some connected n-graph G.

Proof. To show (i)⇒(ii), letV ∈

H

USL_n

C

be a local simultaneous unitarizer of M₁andM₂at pand letX =V^∗V. Then

X M_k =M_k^∗−1X, k∈ {1, 2}.

SinceM₁is unitarizable, it has unimodular eigenvalues. SinceM₁is diagonal, then M₁∈

H

CSU_n. HenceM₁= M₁^∗−1, and so [X, M₁]=0. Since no two eigenvalues ofM₁are identically equal,Xis diagonal away from the set of isolated points where two eigenvalues of M₁coincide, so by its continuity,Xis diagonal.

Since M₁ is diagonal and M₁ and M₂ are irreducible in a punctured neigh- borhood of p, then there is no non-zero proper subspaceW ⊂

C

ⁿ generated by standard basis elements of

C

^{nsuch thatM2W ⊂W}in a neighborhood of p. Hence M₂is not block-diagonalizable at pvia a permutation at p. Since X(p) ∈SL_n

C

^, thenXhas full rank at p. By Lemma 2.3,M₂andM₂^∗−1areG-compatible at pfor some connectedn-graphG. This proves (ii).

To show (ii)⇒(i), sinceM₁andM₂are irreducible on

C

\{p}and are pointwise simultaneous unitarizable at each point in a neighborhood ofp, by Lemma 1.7 there exists a neighborhood

U

^{of pand a map X} ∈

H

UM_n×n

C

^{∩kerL for which X is} Hermitian positive definite on

U

\ {p}. By condition (ii) and Lemma 2.3, X is infinitesimally invertible at p. By Lemma 2.4, there exists a neighborhood

V

^of

pand a choice of n’th root of detX such thatY = (detX)^−1/nX ∈

H

VSL_n

C

^is Hermitian positive definite.

By the Cholesky Decomposition Proposition 1.8, there existsV ∈

H

VSLn

C

such thatY = V^∗V. Then by Lemma 1.4(i),V M_kV⁻¹ ∈

H

VSU_n,k ∈ {1,2}, so V is a local simultaneous unitarizer ofM₁andM₂at p.

Lemma 2.6. For q ≥ 2, let M₁, . . . ,M_q ∈ SL_n

C

^{with M1, M2} irreducible and M₁, M₂ ∈ SU_n. If M₁, . . . ,M_q are simultaneously unitarizable by an element of SL_n

C

^{, then M1, . . . ,Mn ∈SUn.}

Proof. LetC ∈ SL_n

C

be a simultaneous unitarizer. Then C^∗C commutes with each ofM₁andM₂. By Lemma 1.1,C^∗C ∈

C

^{I. SinceC∗C} is positive definite, by Lemma 1.9,C^∗C =I. HenceC ∈SU_n. SinceC M_kC⁻¹∈SU_nfork∈ {1, . . . ,q}, thenM_k ∈SU_nfork ∈ {1, . . . ,q}.

Lemma 2.5, Lemma 2.6 and Theorem 1.10 give the following result.

Theorem 2.7 (Unitarization theorem). Let

C

be an analytic curve, let p∈

C

^and let M₁, . . . ,M_q ∈

H

CSL_n

C

^{, q} ≥ 2. Suppose M₁ and M₂ are irreducible on

C

except at a subset of isolated points. Suppose M₁ is locally diagonalizable at each point p∈

C

, and that no two local analytic eigenvalues of M₁are identically equal. Then M₁, . . . ,M_q are globally simultaneously unitarizable if and only if the following conditions hold:

(i) M₁, . . . ,M_qare pointwise simultaneously unitarizable at each point of

C

^. (ii) For each p ∈

C

, let C be a local diagonalizer of M₁ at p, and let P₂ =

C M₂C⁻¹. Then P₂and P₂^∗−1 are G-compatible at p for some connectedn- graph G.

In this case, any simultaneous unitarizer is unique up to left multiplication by an element of

H

CSU_n.

3.

Simultaneous unitarizability for SL

2

C

For the case SL₂

C

^{, then}-graph condition in Theorem 1.10 is particularly simple and can be recast in terms of commutators.

Note that for A, B ∈ M_2×2

C

^{, Aand B}are irreducible if and only if [A, B]

has rank 2.

Definition 3.1. We say thatA, B∈

H

CSL₂

C

^areinfinitesimally irreducibleat p∈

C

^{if [A}, B] ≡0 and the leading order term in the series expansion of [A, B] at p has full rank.

The property infinitesimally irreducible (respectively reducible) is preserved under conjugation by an element of

H

CSL₂

C

^.

Lemma 3.2. If A is diagonal, then A and B are infinitesimally irreducible if and only if the two off-diagonal terms of B have the same finite order.

We give a sufficient condition for local diagonalizability:

Lemma 3.3. Let

C

be an analytic curve, p∈

C

^{, and M} ∈

H

CSU₂be an analytic map. Then M is locally diagonalizable in a neighborhood

U

⊂

C

of p by a map C ∈

H

USU₂.

Proof. It follows from the characteristic equation ofMthat its eigenvaluesµ1, µ2= µ⁻₁¹are analytic in a neighborhood

U

⊂

C

^ofp. It can be shown, for example by an analytic version of the QR-decomposition, that there exist corresponding analytic eigenvector functionsv1, v2 ∈

H

U

C

^{2 such that V} = (v1, v2) ∈

H

USU₂. Then V⁻¹M V =diag(µ1, µ2), soC =V⁻¹is the required diagonalizer.

Theorem 3.4 (Unitarization theorem for SL₂

C

^{). Let}

C

be an analytic curve, M₁, . . . ,M_q ∈

H

CSL₂

C

^{(q ≥}2), and suppose that[M_r, M_s] ≡0for some fixed choice r, s ∈ {1, . . . ,q}. Then M₁, . . . ,M_q are globally simultaneously unitariz- able if and only if the following conditions hold at each p∈

C

^:

(i) M₁, . . . ,M_qare pointwise simultaneously unitarizable at p.

(ii) M_r or M_sis locally diagonalizable at p.

(iii) M_r and M_sare infinitesimally irreducible at p.

In this case, any simultaneous unitarizer V is unique up to left multiplication by an element of

H

CSU₂.

Remark 3.5. Examples exist which show the independence of the three conditions (i)–(iii) of Theorem 3.4.

Proof. Renumber so thatr = 1 and s = 2. Suppose M₁, . . . ,M_q are globally simultaneously unitarizable on

C

. Then (i) clearly holds, and condition (ii) holds by Lemma 3.3. To show condition (iii), letV be a local unitarizer ofM₁and M₂, and let P_k = V M_kV⁻¹, k ∈ {1,2}. By Lemma 3.3, there exists a local unitary diagonalizerC of P₁. LetQ_k =C P_kC⁻¹,k ∈ {1, 2}. ThenQ₁is diagonal. Since Q₂is unitary, its off-diagonal terms have the same order. Since [M₁, M₂] ≡0, then [Q₁, Q₂] ≡ 0, so Q₂is not identically diagonal. Hence the off-diagonal entries of Q₂are not both identically zero. By Lemma 3.2,Q₁andQ₂are infinitesimally irreducible, and henceM₁andM₂are infinitesimally irreducible.

Conversely, assume conditions (i)–(iii) and assume M₁ is locally diagonaliz- able at p. We show that the conditions of Theorem 2.7 are satisfied. SinceM₁ is locally diagonalizable, and [M₁, M₂] ≡0, thenM₁does not have identically equal eigenvalues. Let C be a local diagonalizer of M₁ at p, and let P_k = C M_kC⁻¹, k ∈ {1,2}. By (iii), the two off-diagonal terms of P₂ and those of P₂^∗−1 all have the same finite order. Hence P₁andP₂are irreducible in a punctured neighborhood of p, soM₁andM₂are irreducible in a punctured neighborhood of p. LetGbe the connected2-graph with single edge(1, 2). ThenP₂andP₂^∗−1areG-compatible.

The existence and uniqueness of the global simultaneous unitarizer follows by The- orem 2.7.

4.

Simultaneous

r

-unitarization

Ther-Unitarization Theorem 4.4 for SL_n

C

-valued loops is a variant of the Unita- rization Theorem 1.10 on the standard unit circle

S

¹ = {λ ∈

C

| |λ| = 1}. This variant has been proven for the case of SL₂

C

[22], where it finds application to the construction of non-simply-connected CMC surfaces.

In ther-Unitarization Theorem, a holomorphic map V is constructed on an annulus

A

^r,1which simultaneously unitarizes the given set of loopsM₁, . . . ,M_qin the sense that theV M_kV⁻¹extend holomorphically to

S

¹and are unitary there. In this case, the unitarizing loopV does not in general extend holomorphically to

S

^1, but has zeros and poles there.

We note that a unitary map cannot have poles. This follows from the fact that SU_n and U_n are compact:

Proposition 4.1. Let

C

be an analytic curve. Then

M

CU_n=

H

CU_nand

M

CSU_n=

H

CSU_n.

We now extend the Cholesky decomposition theorem to the case of an analytic map on

S

¹which is Hermitian positive definite except at a finite subset.

Proposition 4.2 (Meromorphic Cholesky decomposition). Let

C

=

S

¹⊂

C

^and let X ∈

M

CM_n×n

C

be Hermitian positive definite except at a finite subset of points S⊂

C

^{. Then}

(i) There exists V ∈

M

CM_n×n

C

such that X =V^∗V . (ii) V is unique up to left multiplication by elements of

H

CU_n.

Proof. We will apply the LDU-decomposition stated at the beginning of the proof of Proposition 1.8, with holomorphicity replaced by meromorphicity.

Letρ:

C

→

C

be a double cover and letτ :

C

→

C

be the deck transformation induced by a single counterclockwise traversal of

C

^{. Let} ρ^∗ and τ^∗ denote the respective pullbacks. Write D = diag(d₁, . . . ,d_n)and fork ∈ {1, . . . ,n}, define b_k to be either of the global square roots ofd_k on

C

^{. DefineB= diag(b1, . . . ,bn)} andV = B(ρ^∗R), soρ^∗X =V^∗V. Withλthe standard unimodular parameter on

S

^{1, define}

c_k =1 ifτ^∗b_k =b_k, and c_k =√

λifτ^∗b_k = −b_k,

and define C = diag(c₁, . . . ,c_n) ∈

H

CU_n. Then τ^∗(CV) = CV. Let V ∈

M

CM_n×n

C

be the unique map satisfyingρ^∗V = UV. Then X = V^∗V, prov- ing (i).

To show uniqueness (ii), suppose V, W ∈

M

CM_n×n

C

^{withV∗V = W∗W.} LetU = W V⁻¹ ∈

M

CM_n×n

C

^{. ThenU∗ =U−1, soU} takes values in U_n on

S

¹ away from its poles. By Proposition 4.1,U ∈

H

CU_n.