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Let M′ be a complex
Kählr-Einstein maniford of negative scalar curvature. Let M be a relatively
compact submanifold of M′ such that dimCM = dimCM′ = n and the
boundary bM is a C∞ submanifold of M′ of real dimension 2n − 1. It is further
assumed that the following condition holds: There exists a constant c > 0 such
that for all φ ∈ C0,q(M,Θ), q = 1,2, ((2□′−Δ)φ,φ) ≧ o(φ,φ) where Θ is
the holomorphic tangent bundle of M′,Cpq(M,Θ) is the space of all C∞Θ.
valued (p,q) forms extendible to a neighborhood of M,□′ (resp., Δ) is the
complex (resp., the real) Laplacian on Cp,q(M,Θ) and (,) is the L2-inner
product.
The main result of this paper is that there exists a universal family of
deformations of M whose parameter space is, in general, a Banach analytic set. In
the case when M is a compact Riemann surface with boundary it is shown that real
analytic families of complex structures on M can be described in terms of an open set
in Rm where m is the dimension of the reduced Teichmüller space. The proof of this
fact is independent of the theory of quasiconformal mappings and Schwarzian
derivatives.
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