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Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian

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Abstract

In this letter, we give a simple proof of the fact that the determinant of Laplace operator in a smooth metric over compact Riemann surfaces of an arbitrary genus g monotonously grows under the normalized Ricci flow. Together with results of Hamilton that under the action of the normalized Ricci flow a smooth metric tends asymptotically to the metric of constant curvature, this leads to a simple proof of the Osgood–Phillips–Sarnak theorem stating that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of the Laplace operator is maximal on the metric of constant curvatute.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke West, Montreal, Quebec, Canada, H4B 1R6

    A. Kokotov & D. Korotkin

Authors
  1. A. Kokotov
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  2. D. Korotkin
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Correspondence to A. Kokotov.

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Mathematical Subject Classifications (2000). 58J52, 53C44.

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Kokotov, A., Korotkin, D. Normalized Ricci Flow on Riemann Surfaces and Determinant of Laplacian. Lett Math Phys 71, 241–242 (2005). https://doi.org/10.1007/s11005-005-1451-2

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  • DOI: https://doi.org/10.1007/s11005-005-1451-2

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