Extremals of determinants of Laplacians

The Ricci Flow on Noncommutative Two-Tori

“…This means that gaining a good understanding of how to extend the Ricci flow to 2-dimensional noncommutative spaces, such as the noncommutative tori, may also yield useful suggestions on how to extend to noncommutative geometry the Yamabe flow and the Yamabe problem in higher dimensions. Also, by working with the determinant of the Laplacian and its evolution equation under the Ricci flow, one would, by similar techniques, be able to extend to noncommutative 2-tori the results of [40] and obtain an analog of the Osgood-Phillips-Sarnak theorem in this setting, [48].…”

Extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of genus two Riemann surfaces

“…The extremal properties of the function f (2.1) are well-known (see [23,27] Both points i and e πi/3 are orbifold points (with cone angle π and 2π/3, respectively) of the moduli space which can be obtained by an appropriate identification of the boundary of the fundamental domain of the group : there are non-trivial subgroups of the modular group leaving these two points invariant. Introduce the standard generators of :…”

“…The proof of the vanishing of the gradient of det at these two points is based on the existence of subgroups of the modular group leaving these points invariant [23] (i.e., the corresponding elliptic surfaces have non-trivial automorphisms groups); thus the gradient of any modular invariant functional, not only det , vanishes at these two points.…”

Analysis of the horizontal Laplacian for the Hopf fibration

“…The sharp form of Corollary 3 has eluded us and it would be interesting to see if techniques used to prove Onofri's inequality, such as spherically symmetric rearrangements, see [4], or regularized determinants, see [13], can be adapted to obtain a sharp inequality for ∆ H .…”