Ricci flow on K\"ahler-Einstein manifolds

In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the K\"ahler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow is the gradient like flow of these functionals. We successfully find such functionals in case of Kaehler manifolds. On K\"ahler-Einstein manifold with positive scalar curvature, if the initial metric has positive bisectional curvature, we prove that these functionals have a uniform lower bound, via the effective use of Tian's inequality. Consequently, we prove the following theorem: Let $M$ be a K\"ahler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler Ricci flow will converge exponentially fast to a K\"ahler-Einstein metric with constant bisectional curvature. Such a result holds for K\"ahler-Einstein orbifolds.Comment: 49 pages. This is a revised version. Sections 4 and 5 are simplified and streamline

Oxygen Isotope Effect on the Spin State Transition in (Pr0.7_{0.7}Sm0.3_{0.3})0.7_{0.7}Ca0.3_{0.3}CoO3{_3}

Oxygen isotope substitution is performed in the perovskite cobalt oxide (Pr$_{0.7}$Sm$_{0.3}$)$_{0.7}$Ca$_{0.3}$CoO${_3}$ which shows a sharp spin state transition from the intermediate spin (IS) state to the low spin (LS) state at a certain temperature. The transition temperature of the spin state up-shifts with the substitution of $^{16}O$ by $^{18}$O from the resistivity and magnetic susceptibility measurements. The up-shift value is 6.8 K and an oxygen isotope exponent ($\alpha_S$) is about -0.8. The large oxygen isotope effect indicates strong electron-phonon coupling in this material. The substitution of $^{16}$O by $^{18}$O leads to a decrease in the frequency of phonon and an increase in the effective mass of electron ($m$$^\ast$), so that the bandwidth W is decreased and the energy difference between the different spin states is increased. This is the reason why the $T_s$ is shifted to high temperature with oxygen isotopic exchange.Comment: 4 pages, 3 figure

Transonic Shocks In Multidimensional Divergent Nozzles

We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity(non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Frechet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.Comment: 54 page

Novel dynamical effects and glassy response in strongly correlated electronic system

We find an unconventional nucleation of low temperature paramagnetic metal (PMM) phase with monoclinic structure from the matrix of high-temperature antiferromagnetic insulator (AFI) phase with tetragonal structure in strongly correlated electronic system $BaCo_{0.9}Ni_{0.1}S_{1.97}$. Such unconventional nucleation leads to a decease in resistivity by several orders with relaxation at a fixed temperature without external perturbation. The novel dynamical process could arise from the competition of strain fields, Coulomb interactions, magnetic correlations and disorders. Such competition may frustrate the nucleation, giving rise to a slow, nonexponential relaxation and "physical aging" behavior.Comment: 5 pages, 4 figure