Asymptotics of polybalanced metrics under relative stability constraints

Under the assumption of asymptotic relative Chow-stability for polarized algebraic manifolds $(M, L)$, a series of weighted balanced metrics $\omega_m$, $m \gg 1$, called polybalanced metrics, are obtained from complete linear systems $|L^m|$ on $M$. Then the asymptotic behavior of the weights as $m \to \infty$ will be studied.Comment: to appear in Osaka J. Mat

Coherent-feedback control strategy to suppress spontaneous switching in ultra-low power optical bistability

An optical resonator with intracavity Kerr nonlinearity can exhibit dispersive bistability suitable for all-optical switching. With nanophotonic elements it may be possible to achieve attojoule switching energies, which would be very attractive for ultra-low power operation but potentially problematic because of quantum fluctuation-induced spontaneous switching. In this manuscript I derive a quantum-optical model of two Kerr-nonlinear ring resonators connected in a coherent feedback loop, and show via numerical simulation that a properly designed `controller' cavity can significantly reduce the spontaneous switching rate of a bistable `plant' cavity in a completely embedded and autonomous manner.Comment: 8 pages, 4 color figure

Chow-stability and Hilbert-stability in Mumford's Geometric Invariant Theory

In this note, we shall show that the Chow-stability and the Hilbert-stability in GIT asymptotically coincide.Comment: I slightly changed the title of the paper. Moreover, a proof using an equivariant version of Serre's Conjecture is given. This new version appeared in Osaka J. Math. 45 (2008

A stronger concept of K-stability

In this paper, by introducing a wider class of one-parameter group actions for test configurations, we have a stronger form of the definition of K-stability. This allows us to obtain some key step of my preceding work in proving that constant scalar curvature polarization implies K-stability for polarized algebraic manifolds