The Miyaoka-Yau inequality for minimal models of general type and the uniformization by the ball

By proving Calabi's conjecture, Yau proved that the first and second Chern classes of a compact manifold with ample canonical bundle encode the symmetries of the Kahler-Einstein metric via a simple inequality ; the so-called Miyaoka- Yau inequality. In the case of equality, such symmetries lead to the uniformization by the ball. Later, by constructing singular KE metrics, Tsuji established this inequality for smooth minimal models of general type. The singularity of these metrics are usually a major obstacle towards uniformization. In a joint project with Greb, Kebekus and Peternell we take a different approach; using Hermitian-Yang-Mills theory and Simpson's groundbreaking work on complex variation of Hodge structures, we prove the MY inequality for minimal models of general type and show that when the equality holds, the canonical models are ball-quotients.