CHAPTER 1
Curvature Properties and Chern Forms
1. The universal Teichmiiller space
1.1. Teichmiiller theory. Here we present, in a succinct form, necessary
facts from Teichmiiller theory (for more details, see monographs [Ahl87, Leh87,
Nag88] and the exposition in [Teo04]).
1.1.1. Main definitions. Let B = {z G C : \z\ l } b e the open unit disk and
let B* = {z G C : \z\ 1} be its exterior. Denote by L°°(B*) and L°°(B) the
complex Banach spaces of bounded Beltrami differentials on B* and ID respectively,
and let L°°(B*)i be the open unit ball in L°°(B*). Two classical models of the
universal Teichmiiller space T(l) are the following.
Model A. Extend every /x G L°°(B*)i to B by the reflection
(i.i) ^
) = / x
( i ) i 2 '
zeB
and consider the unique quasiconformal (q.c.) mapping w^ : C C, which fixes
1, i and 1 (i.e., is normalized) and satisfies the Beltrami equation
Here and in what follows subscripts z and z always stand for the partial derivatives
^ and £=, unless it is explicitly stated otherwise. Due to the reflection symme-
try (1.1) the q.c. mapping w^ satisfies
(1-2)
-^=WA~)
and fixes domains D, B*, and the unit circle 5
1
. For /i, v G L°°(B*)i, set /x ~ v
if Wplsi =
vJi/ls1-
The universal Teichmiiller space T(l) is defined as a set of
equivalence classes of normalized q.c. mappings w^,
T(l) = L°°(B*)i/~ .
Model B. Extend every \x G L°°(B*)i to be zero outside B*, and consider the
unique q.c. mapping w^ which satisfies the Beltrami equation
w* = Vwz,
and is normalized by the conditions /(0) = 0, f'(0) = 1 and /"(0) = 0. Here / =
w^li® is holomorphic on B and prime stands for the derivative. For /i, v G L°°(B*)i,
set /i rsj y if w^liB) = wu\^. The universal Teichmiiller space T(l) is defined as a set
of equivalence classes of normalized q.c. mappings w^,
T(l) = L°°(B*)i/ - .
Since u^l^i =
w^ls1
if
a n
d only if w^l® = U^IB, these two definitions of the
universal Teichmiiller space are equivalent. The set T(l) is a topological space with
7
Previous Page Next Page