Holomorphic extensions of Laplacians and their determinants |
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Authors: | Young-Heon Kim |
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Affiliation: | Department of Mathematics, Northwestern University, Evanston, IL 60208, USA |
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Abstract: | The Teichmüller 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 {Δμ,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 {Δμ,ν} and show how this family realizes the holomorphic extension of det′(Δ) as its determinant. |
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Keywords: | 32G15 58J52 |
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