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Holomorphic Extensions of Laplacians and Their Determinants

Young-Heon Kim
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Abstract

The Teichmueller space Teich(S) of a surface S in genus g>1 is a totally real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det'(Δ) on Teich(S) has a unique holomorphic extension to QF(S). To realize this holomorphic extension as the determinant of differential operators on S, we introduce a holomorphic family {Δ_{μ,ν}} of elliptic second order differential operators on S whose parameter space is the space of pairs of Beltrami differentials on S and which naturally extends the Laplace operators of hyperbolic metrics on S. We study the determinant of this family {Δ_{μ,ν}} and show how this family realizes the holomorphic extension of det'(Δ) as its determinant.

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Young-Heon Kim. 2005-05-25. Holomorphic Extensions of Laplacians and Their Determinants. https://arxiv.org/abs/math/0505530

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