Vol. 54, No. 1, 1974

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On deformations of complex compact manifolds with boundary

Garo K. Kiremidjian

Vol. 54 (1974), No. 1, 177–190

Let Mbe a complex Kählr-Einstein maniford of negative scalar curvature. Let M be a relatively compact submanifold of Msuch that dimCM = dimCM= n and the boundary bM is a C submanifold of Mof 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 CΘ. 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.

Mathematical Subject Classification 2000
Primary: 32G05
Received: 5 April 1973
Published: 1 September 1974
Garo K. Kiremidjian