Normalized Ricci flow on Riemann surfaces and determinants of Laplacian

Details Normalized Ricci flow on Riemann surfaces and determinants of Laplacian Normalized Ricci flow on Riemann surfaces and determinants of Laplacian

2004-05-02

arxiv.org/abs/math/0405010v2

Mathematics

Spectral Theory

a reference to paper math.DG/9904048 by W.Mueller and K.Wendland where the main theorem of this paper was proved a few years e

Scientific paper

10.1007/s11005-005-1451-2

In this note we give a simple proof of the fact that the determinant of Laplace operator in smooth metric over compact Riemann surfaces of arbitrary genus $g$ monotonously grows under the normalized Ricci flow. Together with results of Hamilton that under the action of the normalized Ricci flow the smooth metric tends asymptotically to metric of constant curvature for $g\geq 1$, this leads to a simple proof of Osgood-Phillips-Sarnak theorem stating that that within the class of smooth metrics with fixed conformal class and fixed volume the determinant of Laplace operator is maximal on metric of constant curvatute.

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