Is there any physics theory that depicts our universe as $2+1$ dimensional?
I heard that black holes seem to suggest that the world might be $2+1$ dimensional, so I am curious whether such theory exists?
Just for curiosity.
I personally haven't heard of one, but then again I haven't heard much.
First thing, are you sure that it's two you want? Or four? Because by relativity et al, our universe is 4D. Go into string theory, it becomes 10/11/26 D.
I think you may have misinterpreted the rubber-sheet explanation of gravity in general relativity as a theory. The 2D rubber sheet is only a way for us to imagine gravity acting, since it bends space. We can easily imagine a 2D sheet bending into the third dimension, as we live in a 3D world (3 macroscopic spatial dimensions, ti be exact). What's actually happening; a 3D bit of space bending into the fourth dimension, is not that easy to imagine. Possibly impossible to imagine. So the 2D-rubber-sheet-and-rock is just a visual aid.
On a side note, the reason I doubt that there is any such theory is this:
You've probably come across the holographic principle: see wikipedia page for details.
The idea is that because the entropy of a black hole is proportional to the area of the event horizon, this means all the information about the black hole is present on the event horizon, and this has dimension 2+1D. However I don't think this should be taken to mean that our universe is 2+1 dimensional.
Having said that, there are ideas from the more fringe areas of physics that at a very small scale/high energy the universe may be 1+1 dimensional. For example causal dynamical triangulation seems to show 2D behaviour at very small scales.
There is an idea from string theory called the AdS/CFT correspondence that physics in an n dimensional gravity theory can be encoded by a n-1 dimensional bounding surface. However this doesn't mean our world is 2+1D, but rather that our physics can be represented by a theory in 4+1D.