Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 109
Take a holomorphic section α : D → U0 and a holomorphic vertical vector field
v along α, i.e.foreveryz ∈ D, v(z) is a vector in T α(z)U0 tangent to the fiber over
z (and nonvanishing). We need to prove that log�v(z)� Poin(Dz) is a subharmonic
function on D. By another change of coordinates, we may assume that α(z)=(z,0)
and v(z)= ∂
∂w | (z,0).
For every z, let
g(z,·) : Dz → [0,+∞]
be the Green function of Dz with pole at 0. That is, g(z,·) is harmonic on Dz \{0},
zero on ∂Dz, and around w = 0 it has the development
g(z,w)=log 1
+ λ (z)+O(|w|).
|w|
The constant λ (z) (Robin constant) is related to the Poincaré metricofDz: more
precisely, we have
�
�
λ (z)=−log�
∂
�∂w
| �
�
�
(z,0)
� Poin(Dz)
(indeed, recall that the Green function gives the radial part of a uniformisation of
Dz). Hence, we are reduced to show that z ↦→ λ (z) is superharmonic.
Fix z0 ∈ D. By real analyticity of ∂U0, the function g is (outside the poles)
also real analytic, and thus extensible (in a real analytic way) beyond ∂U0. This
means that if z is sufficiently close to z0, theng(z,·) is actually defined on Dz0 ,
and harmonic on Dz0 \{0}. Of course, g(z,·) does not need to vanish on ∂Dz0 .
The difference g(z,·) − g(z0,·) is harmonic on Dz0 (the poles annihilate), equal to
λ (z) − λ (z0) at 0, and equal to g(z,·) on ∂Dz0 . By Green formula:
and consequently:
λ (z) − λ (z0)=− 1
�
g(z,w)
2π ∂Dz0 ∂g
∂n (z0,w)ds
∂ 2 �
λ 1 ∂
(z0)=−
∂z∂ ¯z 2π ∂Dz0 2g ∂g
(z0,w)
∂z∂ ¯z ∂n (z0,w)ds.
We now compute the z-laplacian of g(·,w0) when w0 is a point of the boundary
∂Dz0 .
The function −g is, around (z0,w0), a defining function for U0. By pseudoconvexity,
the Levi form of g at (z0,w0) is therefore nonpositive on the complex tangent
space T C
(z0,w0) (∂U0), i.e.ontheKernelof∂g at (z0,w0) [Ran, II.2]. By developing,
and using also the fact that g is w-harmonic, we obtain
∂ 2g ∂z∂ ¯z (z0,w0)
⎧
⎪⎨
∂
≤ 2Re
⎪⎩
2g ∂w∂ ¯z (z0,w0) · ∂g
∂z (z0,w0)
∂g
∂w (z0,w0)
⎫
⎪⎬
.
⎪⎭