Kähler-Einstein metrics, Bergman metrics, and higher alpha-invariants

Abstract

The question of the existence of Kähler-Einstein metrics on a Kähler manifold M has been a subject of study for decades. The Kähler manifolds on which this question may be studied divide naturally into three types. For two of these types the question was long ago settled by Yau and Aubin. For the third type, Fano manifolds, the question is (despite great recent progress) open for many individual manifolds. In the first part of this thesis we define algebraic invariants B_{m,k}(M) of a Fano manifold M, which codify certain properties of M's Bergman metrics. We prove a criterion (Theorem 1.1.1) in terms of these invariants B_{m,k}(M) for the existence of a Kähler-Einstein metric on M. The proof of Theorem 1.1.1 relies on Székelyhidi's deep recent partial C^0-estimate, and on a new family of estimates for Fano manifolds. We furthermore introduce a very general hypothesis on Bergman metrics, Conjecture 6.1.2, offering some partial results (Section 6.3) in evidence. Modulo this conjecture, we prove a variation of Theorem 1.1.1, which gives a criterion for the existence of a Kähler-Einstein metric on M in terms of the well-known alpha-invariants, \alpha_{m,k}(M). This result extends a theorem of Tian. The second part of this thesis concerns Riemannian manifolds more generally. We give a characterization (Theorem 1.2.1) of conformal classes realizing a compact manifold's Yamabe invariant. This characterization is the analogue of an observation of Nadirashvili for metrics realizing the maximal first eigenvalue, and of Fraser and Schoen for metrics realizing the maximal first Steklov eigenvalue.