= Solution
\b[The universal cover of maximally extended AdS is not globally hyperbolic.] Its timelike <conformal boundary> can send signals into the interior after any chosen initial slice, so bulk initial data alone do not determine evolution. In the conformal strip, a null ray can run from $\rho\to\pi/2$ to the center in global time $\pi/2$; it need not meet a sufficiently earlier constant-time slice within the interior spacetime.
The compact-diamond test makes the obstruction explicit. Choose two central events with times $-2$ and $2$ in units $L=1$. Events at $T=0$, $\rho\to\pi/2$ lie between them causally, because their radial coordinate separation is less than2. They form a sequence in the causal diamond with no limit in the AdS interior. The diamond is therefore noncompact. Boundary conditions, such as reflecting conditions on the conformal boundary, can supply a well-posed initial-boundary-value problem; they do not turn the boundary-free interior into a globally hyperbolic spacetime in the usual definition. The periodic-time hyperboloid is also non-globally-hyperbolic, already because it has closed timelike curves.
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