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> It can be flat, concave or convex. Convex is be ruled out:
> 
> https://www.quantamagazine.org/black-holes-prove-that-anti-de-sitter-space-time-is-unstable-20200511/
> 
> "The three simplest solutions to Einstein’s vacuum equations are the most symmetric ones — those in which the curvature of space-time is the same everywhere. In Minkowski space-time, where the cosmological constant is zero, the universe is perfectly flat. In de Sitter space-time, where the cosmological constant has a positive value, the universe is shaped like a sphere. And when the cosmological constant is negative, you get AdS space-time, which has a saddle shape. In the early days of cosmology, scientists wondered which one of these three space-times describes our universe.
> 
> ***
> 
> "Here’s the proof of instability: Moschidis showed that when he adds even a minuscule amount of matter to an AdS space-time, a black hole (or black holes) will inexorably form. However, AdS space-time has, by definition, uniform curvature everywhere, which means it cannot harbor space-contorting objects like black holes. “If you perturb AdS space-time and wait a sufficient time,” Moschidis said, “you’ll end up with a different geometry — one that contains black holes — and it’s no longer AdS. That’s what we mean by unstable.'”
> 
> Comment: Getting rid of convex, saddle-shaped space time, simplifies the issue of getting real quantum gravity understood.