Perturbations of the Laplacian in Teichmüller Space
An important consequence of the real analytic structure of Teichmüller space is the analyticity of the Laplacian. In Buser-Courtois , for instance, this was used to show that the spectrum of a compact Riemann surface is determined by a finite part. That article also contains an outline proof of the analytic structure of the Laplacian. In this chapter we give a detailed account. The main theorems are contained in Sections 14.7 and 14.9. The finiteness result follows in Section 14.10. Sections 14.1 - 14.5 provide the necessary material from the perturbation theory of linear operators. These sections are based on Kato’s book  but are written in a self-contained style. Section 14.6 then links the general theory with Teichmüller space, where for convenience we recall the necessary facts from Chapter 6.
KeywordsHolomorphic Function Open Neighborhood Closed Operator Compact Riemann Surface Open Neighborhood Versus
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