I've been looking into anti-matter and a relatively realistic way to contain it through gravity in my story. I'm probably incorrect in the theoretical physics of the idea but, I'm imagining that the fabric of space/time is the barrier between the matter and anti-matter in the universe. Now if you could somehow stretch space/time like a piece of cloth to its limit and then somehow make a tear in it, antimatter would flow from the other side. I'm wondering if this idea has merit and if I'm missing anything that's needed to describe the experiment.
This comment too long, so I'm posting it in the answers area. This is not an answer to the main question itself, but hopefully it will be enlightening.
Note: I do not believe a true answer is possible with the Science-Based tag in play.
In your question you talk about stretching space and time until you create a tear. Obviously how the tear itself is made is stuff of Science Fiction and not going to stand up to the Science-Based tag, but let's allow that one aspect to stand.
How the Tear Came About: Build Something Denser than a Black Hole (Sci-Fi)
Black holes create such an stretch space-time to such a limit that our current mathematics start to treat many aspects as "infinity". Let's apply Science-Fiction for a moment and assume that you made something "more massive per unit area than a black hole", taking those numbers beyond infinity and ripping a hole in space-time.
At this point physics and mathematics cannot identify what would occur. We can, after all, only create hypothesis about the observable universe, but this would be - by definition - a view into something outside the known universe. The fact that it obeys our physical laws at all is frankly astounding.
Here's where we start to (loosely) use the Science-Based tag. Let's talk about antimatter/gravity interaction (reference the link for the next three cases).
Caveat 1: Once we start talking about a hole in space-time, I'm going to assume that there aren't unforeseen consequences like gravity or time stopping/warping everywhere. Those would be interesting topics to write about, but it sounds like you're going for "business as usual" everywhere else. So let's assume the hole is only locally significant.
Caveat 2: Whatever exists beyond the tear is pretty much by definition not from our universe, so frankly I'm surprised that it manifests as something that follows the same laws of physics as our own universe (it doesn't have to). But maybe there's a meta-universal reason for it we just aren't aware of. So I'm assuming that the other universe has the same laws of physics... it's just full of anti-matter.
Case 1: The Leading Antimatter/Gravity Theory Holds*
Currently the standard theory of antimatter/gravity interaction is that antimatter reacts to gravity in the same was as regular matter. We haven't been able to demonstrate it yet, but that's the leading theory.
If that is the case, on our side of the superdense region you created we see it as a black hole. On the other side they would see about the same thing. The black hole would then be sucking in both matter and anti-matter. Assuming (as it sounds like) the other universe is just wall-to-wall antimatter and that the hole on our side opened in space (where there is very little matter), the black hole would eat antimatter quickly until it lost so much mass that it stopped being a black hole.
At that point either the hole to the other universe closes or it remains open. It's worth noting that if the hole remains open, the door between this universe and the other is only as large as the black hole that made the tear... anywhere between 0.1mm - 400au.
Case 2: Anti-Matter is Repelled by Gravity
In this case the other universe would have a hole suddenly pop up that repels that universe's standard makeup (antimatter), so presumably none would be hitting the black hole to wear it down. The result would be no "leakage" between universes.
Case 3: Anti-Matter is Attracted by Gravity, but At a Different Rate than Matter
At the end of the day I think this basically devolves back to Case 1. As long as there is attraction and a sufficient supply of anti-matter the black hole would eventually cause the black hole to evaporate.
This is more or less the range of possibilities available for your story. If you created a very large black hole (eg. 400AU) on Earth or anywhere in the vicinity, you would wipe us all out in an instant. On the other hand, if you created a pinprick you would get very little anti-matter out of it, so that may not be entirely what you want either.
Another, far more concerning, area of concern is what happens when the black hole evaporates as a result of the antimatter. Per Wikipedia "a 1-second-lived black hole has a mass of 2.28 × 10^5 kg, equivalent to an energy of 2.05 × 1022 J that could be released by 5 × 10^6 megatons of TNT" - though we could halve that if we assume half the energy were released into the other universe (we see 2.5 x 10^6 megatons of TNT worth of explosion). Let's put that in perspective. The largest theoretical nuclear weapon can deliver only 1,300 megatons of explosive - still multiple orders of magnitude less than the destruction of this black hole.
Worse, a 228,000kg black hole would only have an event horizon radius of 3.386 x 10^-22m (1.333 x 10^-20 inches), which maybe doesn't play out very well when you consider how little anti-matter is going to come through a hole that size in realistic time... especially when it needs to come out of a tear a VERY GREAT DISTANCE from Earth.
Even allowing for some hand-waving to enable a tear in the universe using a black hole, I just can't justify a science-based tag. So you're going to have to find a different way to rip the fabric of the universe. You probably were going to anyway, but hopefully this thought-experiment demonstrates why this particular route would have no good science-based ending.
Anti-matter is currently stored using electric fields on time scales of minutes. In order to store it with gravity, you need to invent a device that can generate gravity (or negative gravity, if that is a thing?) Here is an article about theories for a gravity generating machine, though it is pretty short on the facts.
Asking for science-based on this question is a bit much for a topic that is on the farthest edges of the event horizon of theoretical physics. Also, your description, "the fabric of space/time is the barrier between the matter and anti-matter in the universe" sounds like it is grossly in error conceptually. It sounds like you are describing sleep as the boundary between our world and the Fey world...i.e. it sounds like you are describing fantasy.
Anti-particles don't leak into our universe from their own anti-matter universe...well if they do we don't know about it. They exist in our universe, and are created by certain high energy interactions. The big cosmological question surrounding anti-matter is: why is the universe mostly composed of matter instead of equal parts matter and anti-matter? This field of inquiry is called Baryogenesis.
Such an exotic technology as anti-matter storage using gravitational fields is beyond the boundaries of science-based on this site, so, like Separatrix says, stick to sci-fi and let it happen.
Using reasonably known physics, you could use gravity to create an antimatter trap, but not the way you are describing.
Robert L Forward wrote about this in his book "Future Magic", describing various ways to manipulate gravity. Newtonian manipulation involves doing things like squeezing 4 million ton asteroids into spheres about 30 cm in diameter to create a 1 g field on the surface of the sphere. IF you did this multiple times you could set up a ring of spheres where the gravity cancels out in the centre of the sphere. Some sort of auxiliary system would be needed to assist in stabilizing the antimatter, because even a slight perturbation or irregularity in the gravitational fields would cause the antimatter to start moving towards one of the spheres at an ever increasing rate.
imagine each planet in the picture is only 30 cm in diameter and the antimatter in suspended in the space in the centre
Forward also suggested more elegant ways to manipulate gravity. Imagine a smoke ring. Now imagine the particles of smoke are replaced by neutronium, and the ring is whirling around at close to the speed of light. The frame dragging would create an area in the centre where gravity would grip the object and fling it through the centre of the ring at high rates of acceleration, but the object being accelerated would seem to be in free fall. An object approaching the central ring could also be decelerated if moving against the direction of the whirling ring. In the picture below, the smoke ring would be turning in such a direction that an object moving from the bottom of the frame would be accelerated towards the top, while an object approaching the ring from the top would be decelerated before it reached the bottom.
Stay away from the neutronium cloud at the bottom!
Once again, a series of these "smoke rings" could be used to keep antimatter contained. If the rings were in a vacuum and arranged so the antimatter was either passed back and forth like a ping pong ball, or in a large ring so the antimatter was moving through the centres like a car in a racetrack. then the antimatter wold be much less likely to interact with normal matter.
I think I’ve answered this before… yes, in this answer. Your question asks about using gravity, but that’s totally impractical. For the amount of power needed to manipulate gravity in any form, instead of powering that machinery you can just use electric and magnetic forces directly for the containment!
I see another Forward fan answered already, but I want to point out the “trap chip” and other questions that are more general on containing antimatter.
As for your unusual physics, the closest thing that sounds like this in the real universe is this paper (boy, that took a while to find again just recalling the general idea!) “Antimatter Production at a Potential Boundry” from 2001.
It’s very interesting to read as background if you will be featuring antimatter in your story, regardless. Check it out!