The universal Teichmiiller space T(l) is the simplest Teichmiiller space that
bridges spaces of univalent functions and general Teichmiiller spaces. Introduced
by Bers [Ber65, Ber72, Ber73], the universal Teichmiiller space is an infinite-
dimensional complex manifold modeled on a Banach space. It contains Teichmiiller
spaces of Riemann surfaces as complex submanifolds. The universal Teichmiiller spa-
ce T(l) also came to the forefront with the advent of string theory. Its complex sub-
manifold an infinite-dimensional complex Frechet manifold
M6b(S'1)\ Diff+(51),
plays an important role in one of the approaches to non-perturbative bosonic closed
string field theory based on Kahler geometry [BR87a, BR87b]. The manifold
a homogeneous space of the Lie group
also has
an interpretation as a coadjoint orbit of the Bott-Virasoro group, and as such carries
a natural right-invariant Kahler metric [Kir87, KY87].
The complex geometry of the finite-dimensional Teichmiiller spaces Teichmiil-
ler spaces T(r) of cofinite Fuchsian groups, has been extensively studied in the con-
text of Ahlfors-Bers deformation theory of complex structures on Riemann surfaces.
In particular, A. Weil defined a natural Hermitian metric on T(T) by the Petersson
inner product on the tangent spaces. Called Weil-Petersson metric, it was shown
to be a Kahler metric by Weil and Ahlfors. In his seminal paper [Ahl62] Ahlfors
has studied the curvature properties of the Weil-Petersson metric. In particular, he
proved that the Bers coordinates on T(T) are geodesic at the origin, and computed
the Riemann curvature tensor of the Weil-Petersson metric in terms of multiple
principal value integrals. Using these formulas, Ahlfors proved that T(T) has nega-
tive Ricci, holomorphic sectional, and scalar curvatures. Further results have been
obtained by Royden [Roy75]. Wolpert re-examined Ahlfors' approach in [W0I86].
He developed a different method for computing Riemann and Ricci curvature ten-
sors, and obtained explicit formulas in terms of the resolvent kernel of the Laplace
operator of the hyperbolic metric on the corresponding Riemann surface.
Curvature properties of the infinite-dimensional manifold M6b(Sfl)\Diff+(6'1)
have been studied by Kirillov and Yuriev [KY87], and by Bowick and Rajeev
[BR87a, BR87b]. In particular, they computed the Riemann curvature tensor of
the right-invariant Kahler metric and proved that
M6b(S'1)\ Diff+(Sfl)
is a Kahler -
Einstein manifold.
Since both the finite-dimensional Teichmiiller spaces T(T) and the homoge-
neous space
are complex submanifolds of T(l), it is natural to
investigate whether the latter space carries a "universal" Kahler metric which can
be pulled back to the submanifolds. The immediate difficulty is that the universal
Teichmiiller space T(l) is a complex Banach manifold, so that its tangent spaces do
not carry Hermitian metric. Nag and Verjovsky [NV90] were the first to address
this problem. They have shown that the Kahler metric on M6b(Sfl)\Diff+(S'1)
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