We define an elliptic equation for Hermitian metrics which is related to the Einstein condition. Solutions to this equation are closely related to Kähler-Einstein metrics, and are automatically Kähler-Einstein when a certain non-positive condition is satisfied by c1(M). Given this, a natural flow equation arises taking Hermitian metrics to Kähler metrics. We prove short time existence and regul...

A new asymptotic expansion method is developed to separate the WheelerDeWitt equation into the time-dependent Schrödinger equation for a matter field and the Einstein-Hamilton-Jacobi equation for the gravitational field including the quantum back-reaction of the matter field. In particular, the nonadiabatic basis of the generalized invariant for the matter field Hamiltonian separates the Wheele...

We define an elliptic equation for Hermitian metrics which is related to the Einstein condition. Solutions to this equation are closely related to Kähler-Einstein metrics, and are automatically Kähler-Einstein when a certain non-positive condition is satisfied by c1(M). Given this, a natural flow equation arises taking Hermitian metrics to Kähler metrics. We prove short time existence and regul...

Using the influence functional formalism we show how to derive a generalized Einstein equation in the form of a Langevin equation for the description of the backreaction of quantum fields and their fluctuations on the dynamics of curved spacetimes. We show how a functional expansion on the influence functional gives the cumulants of the stochastic source, and how these cumulants enter in the eq...

A simple conservation law formula for field equations with a scaling symmetry is presented. The formula uses adjoint-symmetries of the given field equation and directly generates all local conservation laws for any conserved quantities having non-zero scaling weight. Applications to several soliton equations, fluid flow and nonlinear wave equations, Yang-Mills equations and the Einstein gravita...

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to Kähler-Einstein metrics, and are automatically Kähler-Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existen...

The Kompaneets equation describes a field of photons exchanging energy by Compton scattering with the free electrons of a homogeneous, isotropic, non-relativistic, thermal plasma. This paper strives to advance our understanding of how this equation captures the phenomenon of Bose-Einstein condensation through the study of a model equation. For this model we prove existence and uniqueness theore...

Using both the Born-Oppenheimer idea and the de Broglie-Bohm interpretation of wavefunction we represent in a different way the semiclassical quantum gravity from the Wheeler-DeWitt equation in an oscillating regime which can preserve completely the unitary quantum evolution of a matter field at the expense of a nonlinear gravitational field equation, but has the same asymptotic limit as the ot...