Cheng and Yau’s Work on the Monge-ampère Equation and Affine Geometry

S. T. Yau has done extremely deep and powerful work in differential geometry and partial differential equations. His resolution of the Calabi conjecture on the existence of KählerEinstein metrics, by solving a complex Monge-Ampère equation on Kähler manifolds, is of fundamental importance in both mathematics and physics. We would like to recall in this article the contributions of S. Y. Cheng and S. T. Yau to the real Monge-Ampère equation and its applications to affine geometry. Many definitions and details are omitted here. We refer the reader to the surveys by Loftin [34] and Trudinger and Wang [51], as well as other references cited below, for more extensive discussions of the topics covered here.