Schwarz lemmas via the pluricomplex Green's function
Jaikrishnan Janardhanan




Department of Mathematics, Indian Institute of Technology Madras, Chennai
600036, India


Schwarz lemmas via the pluricomplex Green's function
    Jaikrishnan Janardhanan Jaikrishnan Janardhanan is supported by a DST-INSPIRE fellowship from the 
Department of Science and Technology, India. 
 
mailto:jaikrishnan@iitm.ac.injaikrishnan@iitm.ac.in 
    

===========================================================================================================================================================================================================




  We prove a version of the Schwarz lemma for holomorphic mappings from the unit
  disk into the symmetric product of a Riemann surface.
  Our proof is function-theoretic and  self-contained.  
  The main novelty in our proof is the use of the pluricomplex Green's function.
  We also prove several other Schwarz lemmas using this function.


 Mathematics Subject Classification:
Primary: 32H15, 32H20, 32U35

Key words and phrases: Schwarz lemma, symmetric product,
pluricomplex Green's function





§ INTRODUCTION

The main result of this article is the following: 

  Let X be a Riemann surface and 
  f:→^n(X) be holomorphic. Then 
  
    ℋ^n_ℳ_X(^-1(f(x)), ^
        -1(f(x_0)) ≤ℳ_(x,x_0),   ∀ x,x_0 ∈.


The notation used in the theorem will be explained in Section <ref>.
Briefly, ^n(X) is the n-fold symmetric product of X and
 is the natural map from X^n to ^n(X). The
Möbius pseudodistance associated to X is denoted ℳ_X and 
ℋ_ℳ_X is the Hausdorff pseudodistance induced on
subsets of X by ℳ_X. Given a point
p ∈^n(X), ^-1(p) here denotes the subset of X comprising the
coordinate points of (any element of) ^-1{p}. Note that in our
notation, ^-1{p} is not the same as ^-1(p).


  Note that Theorem <ref> is trivially true whenever X is a compact
  Riemann surface. Also for a domain    
  D ⊂, it is easy to see that either D is Carathéodory hyperbolic 
  (i.e, the Möbius pseudodistance is a distance) or ℳ_D ≡ 0.
  This is not true for Riemann surfaces; see <cit.>.
  We emphasize that Theorem <ref> applies to all Riemann surfaces
  including those for which the Möbius pseudodistance is not a distance
  but yet not identically 0.


The genesis of Theorem <ref> is a result by Nokrane and Ransford <cit.> which is in the setting of
algebroid multifunction taking values in the unit disk. This was later extended
to proper holomorphic correspondences from the unit disk to any bounded planar 
domain by Chandel <cit.>. In our notation, the
result of Nokrane and Ransford is Theorem <ref> with X = while
that of Chandel is the case when X is any bounded planar domain. 

Our motivation for formulating and proving Theorem <ref> comes from an
earlier work  <cit.> in which we investigated the Minkowski
function of a quasi-balanced domain. During the
course of our study, we realized that a special case of 
<cit.> follows easily from simple
observations about the Minkowski function and an extremal function (now
popularly known as
the pluricomplex Green's function) studied by Lempert 
<cit.>, Klimek <cit.> and
Demailly <cit.>. A natural
question to ask is whether these elementary observations have wider
applicability.

The symmetrized bidisk and polydisk have been the subject of intense research
for the past two decades; see, for instance,
<cit.>. 
More recently, the symmetric product of more general objects has also been
studied by several researchers 
<cit.>. The symmetric product of a Riemann surface can be given a
natural complex structure that makes it into a complex manifold. It is, therefore, natural to look for an extension of
the original result of Nokrane and Ransford in the setting of symmetric
products of a Riemann surface and
Theorem <ref> is the desired extension. The proofs of both <cit.> and
<cit.> rely on the holomorphic functional 
calculus (the underlying 
Banach algebra being the space of n × n complex matrices). It is unclear
how these ideas can be generalized to the setting of an arbitrary Riemann
surface.
In contrast, our proof of Theorem <ref> is almost entirely 
self-contained and uses tools solely from complex analysis. Specifically, we 
require only basic facts about plurisubharmonic functions, invariant metrics
and some standard theorems from complex analysis. The central tool in our proof
is the pluricomplex Green's function alluded to in the previous paragraph.

We will also give several applications that illustrate the scope of our
techniques.
A case in point is the situation of equality in (<ref>), which can be 
studied using our techniques in the case when 
X =. This has been studied by
Nokrane and Ransford <cit.> and our analysis is
reminiscent of theirs but simpler. 


















  Let f:→ be a holomorphic function such that 
  
    ℋ^n_ℳ_(π^-1(f(x)), π^-1(f(0)) =
        ℳ_(x,0),
    
  for x ∈ U and U ⊂ a non-empty open subset. Then we can
  find an automorphism of , say g, such that g∘ f
  is the n-th root multi-function, i.e., the map
  
    z ↦π(ζ_1(z),…,ζ_n(z)),
 
  where ζ_1(z),…,ζ_n(z) ∈ are the n-th roots of z.

Here  is the symmetrized polydisk and π is the
map whose coordinates are the elementary symmetric polynomials; see Section 
<ref> for precise definitions including that of g.



As another application, we shall also use our techniques to give a Schwarz lemma
for quasi-balanced domains that extends the well-known Schwarz lemma for balanced domains 
(Result <ref>); see Theorem <ref>. Using this lemma, we
shall then sketch a straightforward proof of a version of Schwarz lemma for the
spectral unit ball originally proved by Bharali <cit.>.




 §.§ Organization
 Section 2 contains a brief treatment of all the
tools required in our proofs. We present our Schwarz lemma for quasi-balanced 
domains in Section <ref>. The proofs of Theorems <ref> and 
<ref> are contained in Section <ref>. Finally, we briefly
sketch the proof of a version of the Schwarz lemma for the spectral unit ball 
in Section <ref>.




 §.§ Notation

We will use  to denote the unit disk in the
complex plane. The space of holomorphic mappings from a complex manifold X
into a complex manifold Y will be denoted (X,Y). We use || for the usual Euclidean norm
in ^n, irrespective of the dimension. All manifolds will be assumed to be
connected. All other notations used will be introduced in Section <ref>.




 §.§ Acknowledgments
 I would like to thank Dr. Pranav Haridas for many
useful discussions. Prof. Aprameyan Parthasarathy and Dr. Vikramjeet Chandel
read parts of this manuscript and made useful suggestions that have greatly
improved the manuscript. Prof. G.P. Balakumar patiently answered  many
questions, both trivial and hard, that were useful in the proofs. I
would also like to thank Prof. Autumn Kent for bringing Stanton's paper 
<cit.> to my attention. I am grateful to the anonymous
referee for her/his careful reading of our paper and the many
useful suggestions that have improved our exposition.  





§ TOOLS





 §.§ The pluricomplex Green's function


In this section, we define and prove basic facts about an extremal function
defined using plurisubharmonic functions. Our treatment is from 
<cit.> where the definition is attributed to Klimek 
<cit.>. The paper by Demailly 
<cit.> contains further properties of this function.


  Let X be a complex manifold. Fix z_0 ∈ X and define the extremal
  function

    λ_X(z,z_0) := sup{ϕ(z): ϕ∈ P_X(z_0)},

where P_X(z_0) is the collection of functions ϕ on X that satisfy:


  
  * ϕ is upper semi-continuous,
  
  * 0 ≤ϕ < 1,
  
  * logϕ is plurisubharmonic on X,
  
  * ϕ(z_0) = 0,
  
  * for any coordinate system z = (z_1,…,z_n) with origin at z_0,
  the quantity ϕ(x)/|z(x)| is bounded above in a neighbourhood
  of z_0.




  In the above definition, functions that are identically -∞ are
  considered to be plurisubharmonic whence the function that is identically 0
  is an element of P_X(z_0). So the collection P_X(z_0) is always
  non-empty.



  The function logλ_X(z,z_0) is known in the literature as the 
  pluricomplex Green's function with a logarithmic pole at z_0. The
  pluricomplex Green's function is well-studied and is at the heart of
  many deep results (see <cit.> and the papers cited in the
  introduction for a small sample). For our
  purposes, the function λ_X—which we will refer to throughout this
  paper as the extremal function—is more convenient to work with. 



  If D ⊂^n is a bounded domain then for each z_0 ∈ D, the function
  |z - z_0| ∈ P_D(z_0). Therefore λ_D(z,z_0)
  > 0   ∀ z ∈ D ∖{ z_0 }.




  Let X and Y  be complex manifolds and let f:X → Y be holomorphic. 
  Then
  
    λ_Y(f(x),f(z_0)) ≤λ_X(x,z_0).




  It suffices to show that if ϕ∈ P_Y(f(z_0)) then ϕ∘ f ∈
  P_X(z_0). Only the final condition in the definition of P_X(z_0)
  needs to be checked. For a coordinate system z = (z_1,…,z_n)
  around z_0 and w = (w_1,…,w_n) around f(z_0), we have
  
    logϕ∘ f(x) - log (|z(x)|)    = logϕ∘ f(x)
        - log |w(f(x))|
       + log|w(f(x))|/|z(x)|.

  The expression on the right hand side is clearly bounded above in a
  neighbourhood of z_0 and
  we are done.


We need a version of Schwarz lemma for subharmonic functions proved by Sibony in
order to compute the extremal function for the unit disk .


  Let u be an upper semi-continuous function on  such that
  
    
  * log u is subharmonic,
    
  * u(z)/|z|^2 is bounded on ^*,
    
  * 0 ≤ u < 1 on .
  
  Then u(z) ≤ |z|^2  ∀ z ∈. If u(z_0) = |z_0|^2 for some
  z_0 ∈, z_0 ≠ 0, then u(z) ≡ |z|^2.



  The extremal function λ_(z,0) = |z|.



  Clearly λ_D(z,0) ≥ |z|. Conversely, if ϕ∈ P_(z,0) then
  ϕ^2 is subharmonic and ϕ^2(z)/|z|^2 is bounded above on ^* by the final condition in the definition of P_(z_0).
  This means that ϕ(z) ≤ |z| by Lemma <ref> and we are done.








 §.§ The Möbius pseudodistance




We now define the Möbius pseudodistance of a complex manifold X and
prove some of its key properties. 


  Let X be a complex manifold. We define the Möbius pseudodistance
  on X to be 
  
    ℳ_X(z_1, z_2) := sup{|f(z_1)|: f ∈(X,), f(z_2) = 0
        }  ∀ z_1, z_2 ∈ X.




  Observe that the above definition is analogous to that of the Carathéodory
  pseudodistance except that we use the Möbius distance of  in the
  definition instead of the Poincaré distance. The proof that the above
  definition actually gives a pseudodistance follows along the same lines as
  that for the Carathéodory pseudodistance. As expected, holomorphic mappings
  are distance decreasing under this pseudodistance and biholomorphisms are
  isometries. It is also clear that if C_X denotes the Carathéodory
  pseudodistance on X then tanh C_X = ℳ_X. See <cit.> for details.



  It follows from Lemma <ref> that
  
    λ_(z,z_0) = ℳ_(z,z_0).



  Let B(a,r) be the ball of radius r centred at point a ∈^n. Then
  
    ℳ(z, a) = |z - a|/r  ∀ z ∈ B(a,r).




  If D_i are disks in the plane then for (z_1, …,z_n)
  ∈ D_1 ×…× D_n, we have
  
    ℳ_D_1 ×…× D_n((z_1,…,z_n),(a_1,…,a_n)) = max_i ℳ_D_i(z_i,a_i).




  Using the Remark <ref> and the fact ℳ_X is distance decreasing
  under the inclusion map, one easily shows that ℳ_X is continuous on
  X× X; see <cit.>.




  We say that the complex manifold X is Carathéodory hyberbolic if
  ℳ_X is a distance.



   Bounded domains are Carathéodory hyperbolic. This follows from
   the observation that if z, w ∈ D, z ≠ w, then some coordinate
   projection is a bounded holomorphic function that separates z and w. 



  Let X be a complex manifold. Then
  
    0 ≤ℳ_X < 1.




  Let z,w ∈ X be such that ℳ_X(z,w) = 1. Then by the very
  definition of ℳ_X, we can find a sequence of holomorphic functions
  f_n : X → such that f_n(w) = 0 and |f_n(z)| → 1. By Montel's
  theorem, (X,) is a normal family. This means that some subsequence
  of f_n must converge in  the compact-open topology to a holomorphic map
  f:X →. But this is absurd as |f_n(z)| → 1.





The next theorem gives the crucial link between the Möbius
pseudodistance of a complex manifold and its extremal
function. This link
is the central tool used in the proof of Theorem <ref>.


  Let X be a complex manifold. Then for a fixed z_0
  ∈ X, the function ℳ_X(,z_0) is plurisubharmonic. In fact,
  logℳ_X(, z_0) is plurisubharmonic. 
 


  From Remark <ref>, ℳ_X is a continuous function. 
  The fact that ℳ_X(,z_0) is plurisubharmonic is now
  straightforward from the fact that ℳ_X(,z_0) is continuous and a 
  supremum of plurisubharmonic functions. The same argument also shows that the
  function logℳ_X(, z_0) is plurisubharmonic. 




  It is now straightforward to prove that for any z_0 ∈ X, the function ℳ(,z_0) ∈ P_X(z_0). Thus, ℳ_X(,z_0) ≤λ_X(,z_0).





























 §.§ The Minkowski function of a quasi-balanced domain


Let p_1,p_2,…,p_n be relatively prime positive integers. We say
that a domain D⊂^n is (p_1,p_2,…,p_n)-balanced
  (quasi-balanced) if

    λ∙ z ∈ D   ∀λ∈, ∀ z ∈ D,

where for z = (z_1, z_2, …, z_n) ∈ D, we define
λ∙ z := (λ^p_1z_1,λ^p_2z_2,
…,λ^p_nz_n).  If p_1 = p_2 = … = p_n = 1 above,
then we say D is a balanced domain 
(balanced domains are also known as complete circular domains in the
literature).

Given a (p_1,p_2,…,p_n)-balanced domain D⊂^n, we
define the Minkowski function _D : ^n → by

    _D(z) := inf{t > 0:  1/t∙ z ∈ D }.

Clearly D = {z ∈^n: _D(z) < 1} and
_D(λ∙ z) = |λ|_D(z).  This function was first studied
by Nikolov <cit.> (see also <cit.>).
It turns out that
_D is plurisubharmonic if and only if D is additionally pseudoconvex; see 
<cit.>.

Section 2.2 of 
<cit.> contains an extensive treatment of the properties
of the Minkowski function of both balanced and quasi-balanced domains.











 §.§ The symmetric product of a Riemann surface



Let X be a Riemann surface. Given (x_1,…, x_n) ∈ X^n, we denote by
⟨ x_1, …, x_n ⟩ the image in the quotient topological
space ^n(X) := X^n/S_n under the
S_n-action on X^n that permutes the entries of (x_1,…, x_n). We will
also abbreviate the element 

    ⟨z_1,…,z_1_μ_1-times, 
      z_2,…,z_2_μ_2-times,…, z_k,…,z_k
      _μ_k-times⟩, μ_1 + … + μ_k = n,

by

    ⟨ z_1;μ_1, …,z_k;μ_k ⟩.


The map

    X^n∋ (x_1,…, x_n) ⟼ ⟨ x_1,…, x_n⟩  ∀ (x_1,…, x_n)∈ X^n

will be denoted by ^n. We shall drop the superscript when
there is no ambiguity. 
It is easy to see that there is a natural complex structure on
^n(X) that makes it a complex manifold of dimension n (see
below). With this
complex structure, the map  is a branched proper holomorphic
mappings whose set of critical points is

    {(z_1,…,z_n) ∈ X^n : z_i = z_j  for some i ≠ j}.



The symmetrized polydisk  is a quasi-balanced domain in
^n with weights (1,2,…,n)
defined using the elementary symmetric polynomials as follows. Let σ_j,
j = 1,…, n, denote the elementary symmetric polynomial of degree j in
n indeterminates. The map π^(n) : ^n→^n is
defined as:

    π^(n)(z_1,…, z_n) := (σ_1(z_1,…, z_n), σ_2(z_1,…, z_n),
      …, σ_n(z_1,…, z_n)),   
     (z_1,…, z_n)∈^n.

Again, we shall drop the superscript when there is no scope for confusion.

The symmetrized polydisk, 𝔾^n, is defined as
𝔾^n := π(^n). It is easy to see that 𝔾^n is
a (1,2,…,n)-balanced domain in ^n, whence 𝔾^n is a
holomorphic embedding of the
n-fold symmetric product of  into ^n. It is also easy to see that the 
Minkowski functional of  is given by

    _(z_1,…,z_n) := max{|λ_1|,…,|λ_n|: π^(n)
      (λ_1,…,λ_n) = (z_1,…,z_n)}.

The above formula implies that  is pseudoconvex. This also follows
by appealing to the fact that the proper image of a pseudoconvex domain is
pseudoconvex. This automatically means that  is a domain of
holomorphy. 


We now give a brief description of the complex structure on the topological
space ^n(X) when X is Riemann surface.
Given subsets V_j⊆ X that are open, let us write:
  
    ⟨ V_1,…, V_n ⟩ := {⟨ x_1,… x_n⟩ : x_j ∈ V_j,
          j = 1, …, n}

  The set ⟨ V_1,…, V_n ⟩ is an open subset of X^n_ by the defining property of the quotient topology. Given a point
  p ∈^n(X), p = ⟨ p_1,… p_n ⟩,
  choose a holomorphic chart (U_j, φ_j) of X at p_j,
  j = 1,…, n, such that
 
    U_j∩ U_k = ∅  if p_j≠ p_k    and     U_j = U_k   if p_j = p_k.

 The above choice of local charts ensures that the map  _p: ⟨
 U_1,…, U_n⟩→^n given by
 
    _p: ⟨ x_1,…, x_n⟩ ⟼ ((φ_1(x_1),…,
        φ_n(x_n)),…,(φ_1(x_1),…,φ_n(x_n)))

is a homeomorphism. This follows from the Fundamental Theorem of Algebra. The
collection of all such charts (⟨ U_1,…, U_n⟩, _p)
produces a holomorphic atlas on ^n(X). The following lemma is easy to
prove and we omit the proof.


  Let X be a Riemann surface and for 1 ≤ k < n, define
  
    V_k := {⟨ z_1, …, z_n ⟩∈^n(X): the set {z_1,…,z_n} has preciselyk elements}

  Then V_k is an analytic subvariety of ^n(X).

    

The book <cit.> contains an exhaustive account of the
various properties of the symmetrized polydisk. The book 
<cit.> is the canonical reference for the symmetric product
in general.





§ A SCHWARZ LEMMA FOR QUASI-BALANCED DOMAINS
 


The following version of Schwarz lemma for balanced domains is well-known. This
version follows easily from the fact that holomorphic maps contract under the
Lempert function and the relationship between the Lempert function and the
Minkowski function of a balanced pseudoconvex domain.

[Proposition 3.1.1 of <cit.>]

  Let D_1 ⊂^m and D_2 ⊂^n be balanced pseudoconvex domains
  with Minkowski functions _1 and _2, respectively. Then given any
  holomorphic map f: D_1 → D_2 with f(0) = 0, we have 
  
    _2(f(z)) ≤_1(z).



We will now prove an analogue of the above result for quasi-balanced domains.



  Let D be a (p_1,…,p_n)-balanced pseudoconvex domain with 
  highest weight p_n. Then 
  
    _D^p_n(z) ≤λ_D(z,0) ≤_D(z).





  First observe that the pseudoconvexity of D ensures that log_D is plurisubharmonic. Fix 0 <  < 1 and consider the set 
  
    K := {w ∈ D: _D(w) = }.

  Note that 0 ∉K. Observe that for any z ∈ D 
  such that 0 < _D(z) <, we can find 0 < t < 1 such that for some z'
  ∈ K, we have t ∙ z' = z. Hence _D^p_n(z) = t^p_n_D^p_n(z'). As
  0 ∉K, we can trivially write the inequality
  
    _D^p_n(w) ≤ C |w|  ∀ w ∈ K,

  for some C > 0, suitably large. It is also easy to see that |z| ≥ t^
  p_n |z'|. Thus, 

  
    _D^p_n(z) = t^p_n_D^p_n(z') ≤ Ct^p_n |z'| ≤ C
         |z|.

  Therefore _D^p_n(z)/|z| is bounded in a
  neighbourhood of 0. Thus, _D^p_n∈ P_D(z,0) whence
  ^p_n(z,0) ≤λ_D(z,0). This inequality is obviously also true if _D(z_0) = 0.
  
  Now for a fixed z with _D(z) ≠ 0, consider the map
  
    ϕ: ∋λ↦λ∙z/_D(z)∈ D.

  We then have 
  
    λ_D(z,0) ≤λ_(_D(z),0) = _D(z).

  On the other hand, if _D(z) = 0, for each n ∈_+, the element n ∙ z ∈ D. We repeat the above argument with the element n ∙
  z instead of z/_D(z). It is clear that λ_D(z,0) ≤ 1/n.
  This proves that λ_D(z,0) ≤_D(z) and we are done.




The above theorem yields the following analogue of Schwarz lemma for
pseudoconvex quasi-balanced domains.
 
  Let D_1 ⊂^n and D_2 ⊂^m be pseudoconvex
  quasi-balanced domains. If f:D_1 → D_2 is holomorphic and f(0) =
  0 then
  
    _D_2^p(f(z),0) ≤_D(z,0)  ∀ z ∈ D,

  where p is highest weight of the quasi-balanced domain D_2



  The above theorem subsumes Result <ref>.
  See <cit.> for a proof of the above Schwarz
  lemma using the Lempert function instead of the extremal function. 



The following is a version of Schwarz lemma that follows from Theorem 
<ref>. This result was proved by 
Ransford–Nokrane <cit.> in a formulation involving
algebroid multi-functions. 


  Let f:→ be holomorphic with f(0) = 0 and
  f(z) = π(λ_1,…,λ_n). Then
  
    max{|λ_1|,…,|λ_n|}≤ |z|^1/n.




  As alluded to in the introduction, the above observation was the impetus for
  this paper.













§ PROOFS OF THE MAIN RESULTS


Our strategy is to establish that the function ℋ^n_ℳ_X
(^-1(z), ^-1(z_0)) (see (<ref>)), is intimately related to
the extremal function of ^n(X) via a function h_1 which we will define
below.

Let X be a Riemann surface  and fix z_0 ∈^n
(X). Define the function h_1:^n(X) → [0,1) by

    h_1(z) := max(max_i∏_jℳ_X(z_i, a_j), max_i∏_j
      ℳ_X(z_j, a_i) ),

where z_0 = ⟨ a_1,…,a_n ⟩ is a fixed point and we have
written z as ⟨ z_1,…,z_n ⟩. We also define the function
h:^n(X)→ [0,1) by

    h(z) := ℋ_ℳ_X(^-1(z), ^-1(z_0)) = ℋ_ℳ_X({z_1,…,z_n}, {a_1,…,a_n}) ,

where ^-1(z) and ^-1(z_0) are defined as in Theorem <ref>. Observe that from the very definitions, 
we have 

    h^n(z) ≤ h_1(z)  ∀ z ∈^n(X).


Our proof of the main theorem hinges on the following
theorem combined with Lemma <ref> and the fact that 
λ_(x,x_0) = ℳ_(x,x_0). We, once again, emphasize 
that the Riemann surface X is arbitrary and in view of this,
Remark <ref> is pertinent in what follows.


   Let V be the set of critical values of the map :X^n →^n(X). For each z_0 ∈^n(X) ∖ V, defining h_1 as in
   (<ref>), we have
  
    h_1 ∈ P_^n(X)(z_0).




  From the very definition, h_1 is continuous, h_1
  (z_0) = 0 and 0 ≤ h_1 < 1. 
  We first show that the function log h_1 is plurisubharmonic on
  ^n(X). Fix z ∈^n(X)∖ V, z = ⟨
  z_1,…,z_n ⟩. Let (U, ψ) be a coordinate chart around z such
  that ψ(z) = 0. We can find an open set B ⊂ U such that: 
  
    
  * The map ψ|_B is a biholomorphism onto a ball B(0,r),
    
  * We can find  an inverse (π_1,…,π_n)
    of  defined on B such that π_i(z) = z_i.
  

  For y ∈ B, we can write
  
    h_1(y) = max(max_i∏_jℳ_X(π_i(y), a_j),
        max_i∏_jℳ_X(π_j(y), a_i) ).

  Now Proposition <ref>, together with basic properties of plurisubharmonic
  functions, shows that log h_1 is plurisubharmonic on ^n(X)
  ∖ V. By Riemann's removable singularities theorem for plurisubharmonic  functions (<cit.>), the function h_1 extends to be
  a plurisubharmonic function on ^n(X). 


  It remains to show that the final condition in the definition of P_X(z_0)
  is satisfied by h_1. Let (U, ψ) be any coordinate chart
  around z_0 such that ψ(z_0) = 0. Choose B and π_i as before. Let D_i
  ⊂ X be open pairwise disjoint coordinate disks that contain a_i. By
  continuity, shrinking B if necessary, we can assume (
  π_1,…, π_n)(B) ⊂
  D_1 ×…× D_n. By the distance decreasing
  property of the Möbius pseudodistance and Remark <ref>, we now have
  
    max_iℳ_D_i(π_i(z), a_i) ≤ℳ_B
        (z,z_0)   ∀ z ∈ B.

  From Remark <ref>, ℳ_B(z,z_0) = |ψ(z)|/r. The
  above equation, combined with the fact that ℳ_X ≤ℳ_
  D_i, therefore shows
  
    max_i ℳ_X(π_i(z), a_i) ≤|ψ(z)|/r ∀ z ∈ B.

  From the very definition of h_1, it is now follows that
  
    h_1(z) ≤|ψ(z)|/r  ∀ z ∈ B.

  The function h_1 satisfies all the conditions required for it to be an
  element of P_X(z_0) and we are done.




  It is not hard to see that λ_X(,z_0) ∈ P_X(z_0) (see 
  <cit.>). Therefore, in the definition of 
  h_1, we might as well have used the function λ_X instead of the
  function ℳ_X and the same proof mutatis mutandis would
   show that the modified function is in P_X(z_0) as well.


The following corollary is immediate from (<ref>) and Theorem 
<ref>.

  For each z_0 ∈^n(X) ∖ V
  
    h^n(z) ≤ h_1(z) ≤λ_^n(X)(z,z_0).



Before we come to the proof of Theorem <ref>, we need one final lemma.


  With the same notation as Theorem <ref>, let 1 ≤ k ≤ n be the
  highest integer such that for some x^0 ∈, writing f(x^0) = ⟨
  x^0_1,…,x^0_n ⟩,  the set {x^0_1,…,x^0_n} has k elements.
  Then: 
  
    
  * Except for x in a discrete set
          E ⊂, f(x) = ⟨ x_1,…,x_n ⟩ also has
          the  property that {x_1,…,x_n} has k elements;

    
  * For each x ∈∖ E, we can find a disk V_x ⊂∖ E centred at x, holomorphic maps f_x,1,…, f_x,k : V_x → X and
           positive integers μ_1, …, μ_k whose sum is n such that 
           
    f(y) = ⟨f_x,1(y);μ_1,f_x,2(y);μ_2,…,
                 f_x,k(y);μ_k ⟩, ∀ y ∈ V_x.

  
 



  Let E ⊂ be the set of all elements each x ∈ with the
  property that f(x) = ⟨ x_1,…,x_n ⟩ is such that {x_1,…,x_n} has fewer than k elements. By Lemma <ref>,
  the collection of
  all points w in ^n(X) with the property that, writing w as
  ⟨ w_1,…, w_n ⟩, the set {w_1,…,w_n} has
  fewer than k elements is an  analytic subvariety of ^n(X) 
  (Lemma <ref>). If E is an indiscrete set, it follows from the 
  principle of analytic continuation that E =, a  contradiction.

  Now let x ∈∖ E and f(x) = ⟨ x_1;μ_1,… x_k;
  μ_k ⟩, μ_1 + … + μ_k =n. Let U_i ⊂ X be pairwise
  disjoint coordinate disks
  centred at x_i. Then by continuity, we can find a disk V_x ⊂∖ E centred at x such that 
  
    f(V_x) ⊂⟨U_1, …, U_1_μ_1-
        times,…, U_k, …, U_k_μ_k-times⟩.

  As the U_i are pairwise disjoint and for each y ∈ V_x and writing f(y) =
  ⟨ y_1,…,y_n ⟩, the cardinality of { y_1,…,y_n}
  is k, it is clear that we can define continuous
  maps f_x,1, …, f_x,k : V → X such that
   
    f(y) = ⟨f_x,1(y);μ_1,f_x,2(y);μ_2,…,
        f_x,k(y);μ_k ⟩, ∀ y ∈ V_x.

  The fact that the maps f_x,1, …, f_x,k
  are holomorphic is a simple consequence of the way the complex structure on
  ^n(X) is defined.





 §.§ Proof of Theorem <ref>
 
Let E and k be as in Lemma <ref>. For x ∈∖ E, we can find a disk V_x and holomorphic maps f_x,1, …, f_x,k defined on V_x satisfying
the conclusion of Lemma <ref>. We now define
f̂ on V_x by

    V_x ∋ y ↦^k(f_x,1(y), …, 
      f_x,k(y)).

The above definition yields a holomorphic map f̂: ∖ E
→^k(X).
By Riemann's removable singularities theorem for subharmonic functions, 
λ_∖ E≡λ_|_∖ E. Fix y ∈∖ E and define the functions ĥ
and ĥ_1 on ^k(X) with respect to the point f̂(y)
and analogous to h and h_1 (see (<ref>) and (<ref>)), 
respectively. It follows from Corollary 
<ref> and Lemma <ref> that

    ĥ^n(f̂(x)) ≤ĥ_1(f̂(x)) ≤ℳ_(x,
      y)  ∀ x ∈∖ E.

It is obvious that 

    h_1(f(x)) ≤ĥ_1(f̂(x)),
 
where h_1 is defined on ^n(X) with respect to the point f(y).
Thus

    h_1(x) ≤ℳ_(x,y)  ∀ x ∈∖ E.

From (<ref>), it is clear that if we view h_1 as a 
function of both x and y, it is continuous on ×. So is
the function ℳ_(x,y). This combined with (<ref>)
delivers the theorem.



 §.§ Proof of Theorem <ref>

  In this proof, we shall tacitly identify ^n() with 
  without explicit mention. 
  Let z_0 := f(0) = ⟨ a_1, …, a_n ⟩ and consider the
  functions h as before defined on ^n() with respect to the
  point f(0). It is harmless to assume that 0 ∉U. 

  Claim: We can find an open disk G ⊂ U
  and a holomorphic function F:G→ such that for some 1 ≤ j_0
  ≤ n, we have
  
    h^n(x) = ℳ_(F(x), a_j_0)   ∀ x ∈ G.


  Proof of claim: 
  We adopt the same notation as Lemma <ref>. Choose x^0 ∈ U ∖ E.
  We have h(x^0) = ℳ_(f_x^0, i_0(x^0),
  a_j_0) for some choice of 1 ≤ i_0, j_0 ≤ k (the choice might not
  be unique). Let i_0,…,i_l and j_0,…,j_l be all the indices such
  that h(x^0) = ℳ_(f_x^0, i_m(x^0), a_j_m)
  where 0 ≤ m ≤ l.  We can find a disk G ⊂ V_x^0∩ U
  centered at x^0
  such that for each x ∈ G, h(f(x)) is one of the functions ℳ_(
  f_x^0, i_m
  (x), a_j_m), 1 ≤ m ≤ l. Define the sets
  
    E_m := {x ∈ G: h(f(x)) = ℳ_(f_
        x^0,i_m(x), a_j_m)},  0 ≤ m ≤ l.

  Each E_m is a closed subset of G and ⋃_m=0^l E_m = G.
  Consequently, one of the sets E_m has non-empty interior and we can 
  rename G to be any disk contained in this E_m and choose F to be the
  corresponding f_i_m.



  With the claim in hand, the proof of the theorem is not hard. Let a_j_0,
  G and F be as in the claim. We may assume that a_j_0∉G. Let
  ϕ∈() be the automorphism that
  interchanges 0 and a_j_0. We have h^n(ϕ∘ f(x)) =
  ℳ_(ϕ∘ F(x),0))^n = |ϕ∘ F(x)|^n  ∀ x ∈
  G. By hypothesis, this means that
  
    |ϕ∘ F(x)|^n = |x|  ∀ x ∈ G.

  But any branch of √() on G satisfies the
  above equation
  as well proving that for some θ, e^iθ(ϕ∘ F) is just
  some branch of the √(). Let Θ be the automorphism of  associated to rotation by e^iθ. 
  Replacing f with Θ∘ϕ∘ f, we
  may assume that f|_G lifts over π to a map into ^n, one of whose
  components is a branch of √().

  Writing f as (f_1,…,f_n), consider the polynomial over ()
  
    P(x,y) := y^n + f_1(x) y^n-1 + … + f_n-1(x)y^n-1 + f_n(x)
 
  From the conclusion of the preceding paragraph,  we can find a n-th root of
  unity ζ  such that P(x^n, x ζ) ≡ 0 on G. Consequently, P
  (x^n, x ζ) ≡ 0 on  by the identity theorem. If η is any
  other n-th root of unity, we see that P(x^n, x ηζ) ≡ 0 on
  . Therefore f(x) = π(ζ_1(x),…,ζ_n(x))
  where √(x) = {ζ_1(x), …, ζ_n(x)}. The theorem
  is proved with g := e^iθϕ.





§ A SCHWARZ LEMMA FOR THE SPECTRAL UNIT BALL


In this section, we sketch a proof of a Schwarz lemma for the spectral
unit ball. This theorem was formulated and proved by Bharali 
<cit.>. But as the ideas fit well with the main themes
of this article, we felt it is worthwhile to sketch a slightly different proof
here.


For n ∈_+, the spectral unit ball _n ⊂^n^2 is the
collection of all matrices A ∈ M_n() (n × n complex matrices) whose
spectrum σ(A) is contained in .  We have the following


  The set _n is an unbounded balanced pseudoconvex domain with Minkowski 
  function given by the spectral radius ρ.



  That _n is balanced and that the spectral radius is the Minkowski
  function is easy to see from the
  definitions. We can define the holomorphic map Ψ_n:M_n() →^n given
  by M ↦π(σ(M)). Observe that Ψ_n^-1() = _n
  which shows that _n is a domain of holomorphy  (from that fact that
   is a domain of holomorphy and <cit.>).
  Pseudoconvexity of _n now
  follows from the characterization of domains of holomorphy (see <cit.>).



  The above proposition shows that ρ|__n is
  plurisubharmonic (see <cit.>). This fact
  is usually proved in the literature using a theorem of 
  Vesentini <cit.>.  




  Given A∈ M_n(), we can write its minimal polynomial 𝐌_A
  as
  
    𝐌_A(t)=∑_∈σ(A)(t-)^m().

  The minimal Blaschke product corresponding to A is defined by
  
    𝐁_A(t):=∏_λ∈σ(A)⊂(t-λ/1 - λt)^m(λ).



Using the minimal Blaschke product corresponding to A, we can construct a
holomorphic map A: _n →_n that takes A to 0. We
define

    A:B ↦∏_λ∈σ(A)(𝕀 - 
      λB)^-m(λ)(B - λ𝕀)^m(λ),

where m(λ) is the multiplicity of the eigenvalue λ in the
minimal polynomial of A. It can be shown that if σ(B) = {λ_1,
…, λ_n} then σ(A(B)) = {𝐁_A
(λ_1), …, 𝐁_A(λ_n)}.
If F:→_n is holomorphic such that F(z) = A and F(w) = B then
A∘ F takes A to 0 and B∘ F takes
B to 0.
The following result is immediate from the Schwarz lemma for
balanced domains (Result <ref>). 

[Bharali, Theorem 1.5 of <cit.>] 
  Let f:→_n be holomorphic. Then for z, w ∈, we have
  
    max   {max_λ∈σ(f(w))∏_μ∈σ(f(z))ℳ_(μ,λ)^m(μ),max_μ∈σ(f(z))∏_
         λ∈σ(f(w))ℳ_(μ,λ)^m(λ)}
       ≤ℳ_(z,w),

  where m(μ) and m(λ) denote the multiplicity of the eigenvalues
  μ and λ in 𝐌_f(z) and 𝐌_f(B),
  respectively.
  




amsalpha 

