Introduction

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

M6b(5'1)\Dific+(6'1)

— a homogeneous space of the Lie group

Diff+(S'1),

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

M6b(51)\Diff+(S'1)

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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