Does an event horizon imply a singularity? I surprisingly did not find anyone addressing this directly.
The weak cosmic censorship states, very loosely that singularities imply event horizons. I am wondering whether the converse is true.
I.e. is it possible for an event horizon to exist, but it harbors no singularity inside? This does not look obvious to me since the definition of a BH never mentions any singularity whatsoever.
Since there seems to be some confusion on the notion of event horizon, I define it thus,
Let $(M,g)$ be a spacetime that is asymptotically flat at null infinity.
Define the BH region to be $$\mathcal{B} := M \setminus[M \cap J^-(\mathcal{I^+})].$$. Then the (future) event horizon is defined to be the boundary $$\mathcal{H}^+ := \dot{\mathcal{B}} = M \setminus[M \cap \dot{J}^-(\mathcal{I^+})].$$
In a sentence, the event horizon is the boundary of the BH region, which is defined to be the region of spacetime that is causally disconnected from (future) null infinity.
 A: No. You can make an event horizon in flat space by constantly accelerating forever in some direction. 
The crucial thing about event horizons is that they are boundaries beyond which things cannot affect an observer. While black holes have very conspicuous event horizons standard FRLW model accelerating expanding universes have cosmological event horizons. Note that there is no singularity in such models except for the big bang itself far in the past. 
A: The weak cosmic censorship hypothesis is basically that if there are any singularities, then they are hidden from "us" by event horizons. Whether this is actually true of the universe isn't known. 
Both event horizons and singularities are mathematical phenomena that arise in abstract mathematical systems such as our models of gravitation and space-time. My understanding is that most people studying this stuff still think those models are more-or-less correct in the sense that there are real phenomena in the universe that behave enough like the models of event horizons and singularities that we can safely use those words to describe them. But these are not settled questions.
Can we show abstract configurations of our models of space, time, and gravity, such that there's an even horizon and no singularity? Certainly! 


*

*Conventional models of the formation of a black hole have the appearance of the event horizon as a separate and "earlier" event than the formation of the singularity.

*The "edge of the observable universe" is technically an event horizon, albeit of a different kind. 

*Physicist over the decades have expressed all sorts of different exotic space-time configurations; by all means go looking for one that meets the specific idea of an "event horizon" and a "singularity" that you're interested in.


Are there or will there ever be closed gravitational event horizons in the universe that neither contain nor imply the future existence of singularities? It seems kinda unlikely, but I'm not taking bets. 
A: It has been my understanding that from our point of view, time stops at the event horizon.  If that is true for all points inside the horizon, then we would say that the collapse which caused the event horizon has stopped, and the singularity never forms.
