Phase Transition Of K(?)hler Metrics Via Moment Maps

In geometry,an important topic is to consider the limits of Einstein metrics.When we have explicit constructions of K(?)hler-Einstein metrics on some manifolds,we can see the behavior of those metrics more clearly when taking limits.And if we allow the metrics become pseudo when changing the parameters,then the phase transition phenomenon will happen.In literature,many Kahler-Einstein metrics which have explicit formulas are toric.This enable us to use tools in symplectic geometry like moment maps and Hessian geometry to study those phase transition phenomena.In this paper,we will study those phase transition phenomena of Kahler-Einstein metrics via the images of moment maps and the Hessian geometry on them.We discuss all the possible phase transition phenomena in those metrics with explicit construction in which we find a phase transition of a new type.Also we find more general convexity of the moment maps and more ways to realize the flop between two resolved conifold.We also give examples of phase transitions on Kahler metrics of other types like translation Kahler-Ricci Soliton.