Kahler-Einstein Metrics and the Complex Monge-Ampere Equation

The HADES seminar on Tuesday, April 11th will be at 3:30 pm in Room 740.

Speaker: Garrett Brown

Abstract: A central question in geometric analysis is as follows: given a smooth manifold, can one find a Riemannian metric with “special” curvature properties? A classic example of this is the uniformization theorem, which states that any smooth 2-manifold has a metric of constant curvature, and the Gauss-Bonnet theorem relates the sign of the curvature to the genus of the surface.

In complex geometry, one can consider the possible higher dimensional generalizations of the uniformization theorem. One candidate is the following: given a complex manifold, does it have a metric which is Kahler-Einstein, that is, the complex structure is parallel with respect to the metric, and the metric is proportional to the Ricci curvature? This question was answered in the affirmative by Aubin and Yau in the negative first Chern class case, and by Yau in the zero first Chern class case via the more general Calabi conjecture in the late 70s (the positive case was resolved in 2015, requiring a deeper analysis). The crucial step is establishing a priori estimates for a fully nonlinear elliptic equation.

I will do my best to explain the ideas from geometry that are beyond a basic acquaintance with Riemannian geometry.

Leave a Reply

Your email address will not be published. Required fields are marked *