Continuity of certain differentials on finitely augmented Teichmüller spaces and variational formulas of Schiffer-Spencer's type (Q1084558)

The author extends some of his earlier work [see J. Math. Kyoto Univ. 25, 597-617 (1985; Zbl 0597.30056)] on the continuity properties of holomorphic and harmonic differentials on Riemann surfaces. The Riemann surfaces are permitted to have nodes hence the surfaces R vary over the augmented Teichmüller spaces [see the reviewer, Ann. Math., II. Ser. 105, 29-44 (1977; Zbl 0347.32010) for the relevant topologies]. Given a one parameter family \(\{R_ t| t>0\}\) of nonsingular Riemann surfaces of finite conformal type, there is often a marking preserving deformation \(f_ t: R_ t\to R_ 0\) which collapses curves into nodes. The family \(R_ t\) may be chosen so that each \(f_ t\) is conformal away from a neighborhood of the nodes. Under these conditions, the author finds that, when normalizations are respected, the Green's function \(g_ t\) converge uniformly to a Green's function \(g_ 0\) as \(t\to 0\). Here we assume that \(g_ 0\) exists. If \[ \phi (q;R)=idq(\cdot,q;R) - *dg(\cdot,q;R) \] then \(\phi (f_ t^{-1}(q);R)\to \phi (q;R)\) in the Dirichlet norm \(\iint \omega \wedge *\omega.\) Variational formulas of Schiffer-Spencer type are also presented.