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Determinant of the Laplacian on Tori of Constant Positive Curvature with one Conical Point

Part of: Riemann surfaces Partial differential equations on manifolds; differential operators General theory of linear operators

Published online by Cambridge University Press:  04 January 2019

Victor Kalvin  and
Alexey Kokotov
Victor Kalvin
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8 Email: [email protected]@concordia.ca
Alexey Kokotov
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8 Email: [email protected]@concordia.ca
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Abstract

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We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extensions of) the Laplacians on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4\unicode[STIX]{x1D70B}$.

Keywords

determinant of Laplacianmoduli spacespectral zeta-functioncurvature oneconical pointconical singularityRiemann surfacecompact Riemann surface

MSC classification

Primary: 47A10: Spectrum, resolvent
Secondary: 58J52: Determinants and determinant bundles, analytic torsion 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 341 - 347
Copyright
© Canadian Mathematical Society 2018 

References

Chai, C.-L., Lin, C.-S., and Wang, C.-L., Mean field equation, hyperelliptic curves and modular forms: I . Camb. J. Math. 3(2015), no. 1–2, 127–274. https://doi.org/10.4310/CJM.2015.v3.n1.a3.Google Scholar
Clemens, C. H., A scrapbook of complex curve theory. Second ed., Graduate Studies in Mathematics, 55, American Mathematical Society, Providence, RI, 2003.Google Scholar
Eremenko, A., Metrics of positive curvature with conic singularities on the sphere . Proc. Amer. Math. Soc. 132(2004), no. 11, 3349–3355. https://doi.org/10.1090/S0002-9939-04-07439-8.Google Scholar
Kalvin, V., On determinants of Laplacians on compact Riemann surfaces equipped with pullbacks of conical metrics by meromorphic functions . J. Geom. Anal., to appear. https://doi.org/10.1007/s12220-018-0018-2.Google Scholar
Kalvin, V. and Kokotov, A., Metrics of constant positive curvature, Hurwitz spaces and det Δ . Int. Math. Res. Not. IMRN, to appear. https://doi.org/10.1093/imrn/rnx224.Google Scholar
Kitaev, V. and Korotkin, D., On solutions of the Schlesinger equations in terms of theta-functions . Int. Math. Res. Not. IMRN 1998 no. 17, 877–905. https://doi.org/10.1155/S1073792898000543.Google Scholar
Kokotov, A. and Korotkin, D., Tau-functions on Hurwitz spaces . Math. Phys. Anal. Geom. 7(2004), no. 1, 47–96. https://doi.org/10.1023/B:MPAG.0000022835.68838.56.Google Scholar
Kokotov, A. and Korotkin, D., Isomonodromic tau-function of Hurwitz Frobenius manifolds . Int. Math. Res. Not. IMRN 2006, Art. ID 18746. https://doi.org/10.1155/IMRN/2006/18746.Google Scholar
Kokotov, A. and Strachan, I., On the isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one . Math. Res. Lett. 12(2005), no. 5–6, 857–875. https://doi.org/10.4310/MRL.2005.v12.n6.a7.Google Scholar
Polchinski, J., Evaluation of the one loop string path integral . Comm. Math. Phys. 104(1986), no. 1, 37–47.Google Scholar
Ray, D. B. and Singer, I. M., Analytic torsion for complex manifolds . Ann. of Math. 98(1973), 154–177. https://doi.org/10.2307/1970909.Google Scholar
Sarnak, P., Some applications of modular forms. Cambridge Tracts in Mathematics, 99, Cambridge University Press, 1990. https://doi.org/10.1017/CBO9780511895593.Google Scholar
Umehara, M. and Yamada, K., Metrics of constant curvature 1 with three conical singularities on 2-sphere . Illinois J. Math. 44(2000), no. 1, 72–94.Google Scholar