In algebraic geometry, the existence of canonical metrics as proposed by Calabi allows one to give equally canonical representatives of characteristic classes by differential forms. Due to Yau's initial efforts at disproving the Calabi conjecture by showing that it would lead to contradictions in such contexts, he was able to draw striking corollaries to his primary theorem. In particular, the Calabi conjecture implies the Miyaoka–Yau inequality on Chern numbers of surfaces, as well as homotopical characterizations of the complex structures of the complex projective plane and of quotients of the two-dimensional complex unit ball.