On Complete Calabi-Yau Metrics and a Free-Boundary Monge-Ampere Equation

Calabi-Yau metrics are central objects in K\"ahler geometry. They are special solutions to the vacuum Einstein's equations and also serve as canonical metrics on K\"ahler manifolds. The existence problem for Calabi-Yau metrics on compact manifolds was answered by Yau in his solution of the Calabi conjecture. The situation in the non-compact setting is much more delicate, and many questions related to the existence and uniqueness of non-compact Calabi-Yau metrics remain unanswered. A major difficulty lies in the lack of suitable model metrics that model the asymptotics of the Calabi-Yau metric at spatial infinity. In this talk, I will give an introduction to this subject and discuss some joint work with T. Collins and S.-T. Yau, on a new relationship between complete Calabi-Yau metrics and a free-boundary problem for a real Monge-Ampere equation.