The Calabi Conjecture

Abstract: In this essay we aim to explore the Geometric aspects of the Calabi Conjecture and highlight the techniques of nonlinear Elliptic PDE theory used by S.T. Yau [SY] in obtaining a solution to the problem. Yau proves the existence of a Geometric structure using differential equations, giving importance to the idea that deep insights into geometry can be obtained by studying solutions of such equations. Yau's proof of the existence of a specific class of metrics have found a natural interpretation in recent developments in Theoretical Physics most notably in the formulation of String Theory. We will also attempt to explore the importance of a special case of Yau's solution known as Calabi-Yau Manifolds in the context of holonomy.