Closed Kaehler--Einstein surfaces are complex ball quotients

Let $$X$$ be a closed Kaehler manifold of real dimension 4 endowed with a Kaehler--Einstein metric of negative curvature. Is it true that $$X$$ is isomorphic, as a Kaehler manifold, to a quotient of a complex ball?

• Which curvature do you want to be negative? – abx Dec 16 '18 at 14:52
• @abx say scalar curvature – geometer Dec 16 '18 at 14:55
• Any surface with ample canonical bundle admits a Kähler--Einstein metric (Aubin-Yau), with constant negative scalar curvature. Those which are quotients of the complex ball form a minuscule part of this huge crowd. – abx Dec 16 '18 at 14:59
• @abx what if we require holomorphic sectional curvature to be negative? What if we require all sectional curvature to be negative? – geometer Dec 16 '18 at 15:30
• Any compact complex surface of general type satisfies $c_1^2\le 3c_2$, with equality just when the surface is a ball quotient, a result of Yau and also of Miyaoka (the Bogomolov-Miyaoka-Yau inequality). For related results on singular ball quotients see Greb et al. arxiv.org/abs/1511.08822 – Ben McKay Dec 16 '18 at 16:54