Einstein metrics and Mostow rigidity

Abstract. Using the new diffeomorphism invariants of Seiberg and Wit-ten, a uniqueness theorem is proved for Einstein metrics on compact quo-tients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality. A smooth Riemannian manifold (M, g) is said [1] to be Einstein if its Ricci curvature is a constant multiple of g. Any irreducible locally-symmetric space is Einstein, and, in light of Mostow rigidity [5], it is natural to ask whether, up to diffeomorphisms and rescalings, the stan-dard metric is the only Einstein metric on any compact quotient of an irreducible symmetric space of non-compact type and dimension> 2. For example, any Einstein 3-manifold has constant curvature, so the answer is certainly affirmative in dimension 3. In dimension ≥ 4, however, so-lutions to Einstein’s equations can be quite non-trivial. Nonetheless, the following 4-dimensional result was recently proved by means of an entropy comparison theorem [2]: Theorem 1 (Besson-Courtois-Gallot). Let M4 be a smooth compact quotient of hyperbolic 4-space H4 = SO(4, 1)/SO(4), and let g0 be its stan-dard metric of constant sectional curvature. Then every Einstein metric g on M is of the form g = λϕ∗g0, where ϕ: M → M is a diffeomorphism and λ> 0 is a constant. In this note, we will prove the analogous result for the remaining 4-dimensional cases: Theorem 2. Let M4 be a smooth compact quotient of complex-hyperbolic 2-space CH2 = SU(2, 1)/U(2). Let g0 be its standard complex-hyperbolic metric. Then every Einstein metric g on M is of the form g = λϕ∗g0, where ϕ:M →M is a diffeomorphism and λ> 0 is a constant. In contrast to Theorem 1, the proof of this result is based on the new 4-manifold invariants [4] recently introduced by Witten [6]