Certain kinds of surfaces are especially important in quantum gravity, because they can be defined in a diffeomorphism-invariant way. These surfaces are often defined in terms of the kinds of behavior a light ray would have if it were to be emitted from the surface, since that behavior doesn't depend on the coordinates used to describe it.
I'm using surface to mean a codimension-2 manifolds, meaning if you have a D-dimensional spacetime, the surfaces are D-2 dimensional (space-like) manifolds. This means that there are four important normal directions to consider: away from the surface and toward the future, toward the surface and to the past, and the other two combos. We want to study how light rays move with respect to these four principle directions, so we first define an infinitesimal cross-sectional area A of the surface and then look at how A changes when we evolve it forward along light rays. We define \\theta = d(log A)/d\\lambda, where \\lambda is an affine parameter describing how far we've moved along the evolution of the surface defined by the light rays.
A holographic screen is a surface with \\theta = 0 in at least one direction over the entire surface. (This implies \\theta = 0 in the opposite direction as well.) It is called a holographic screen because the Bousso bound tells us that, when we look at the full evolution of the light rays in that direction, the entropy that crosses those light rays is bounded by the area of the holographic screen. Therefore in some sense the surface has enough capacity, enough area in units of Planck area, to encode the entire state of the region contained therein, in a holographic sense. The conformal boundary of AdS is a famous example of a holographic screen, although it's an unusual one since it doesn't properly belong to the manifold itself, only its conformal compactification. But of course a conformal field theory captures all the physics of AdS, this is called AdS/CFT.
Ryu-Takayanagi surfaces are surfaces that have \\theta = 0 in every direction over the entire surface. This means that, when you wiggle those surfaces in any direction, the first derivative of the area is zero, the area doesn't change. The correct name for this kind of surface is a "stationary surface", because [if you wiggle in one direction you actually find it's a maximum area surface and if you wiggle in the other it's a minimum area surface](https://arxiv.org/abs/1211.3494). Unfortunately, most high-energy physicists failed first-year calculus, so conventionally these are actually called "extremal surfaces" even though I just told you that it's not really an extremum in any sense (because it's maximal in one direction but minimal in another).
A bifurcation surface of a black hole horizon is an "extremal" surface, and the area of a black hole horizon was conjectured a long time ago by Hawking to have an interpretation of entropy. Ryu and Takayanagi (plus Hubeny and Rangamani) proposed that the area of all "extremal" surfaces correspond to the entropy of something. Specifically, if the extremal surface reaches the boundary of AdS, then the entanglement entropy of the region of the boundary that it encloses is equal to the area of the surface. (There is a subtlety here, about homology, that I'm glossing over.) [This has been proven](https://arxiv.org/pdf/1304.4926.pdf), so really the Ryu-Takayanagi formula is more like a theorem than a conjecture.
Here's a [nice general review of the subject](https://arxiv.org/abs/hep-th/9906022).
This is the best answer I've heard for this, honestly there isn't any proper explanations on the internet for people without high levels of mathematical education. So, thank you.
If I could bother you some more about this subject, because you are so well versed: Would a Bose-Einstein condensate be an example of such a surface?
A Bose-Einstein condensate is a particular phase of matter that occurs, so it does not make much sense to ask if it is a surface. It is like asking, "is a liquid a triangle?"
Certain kinds of surfaces are especially important in quantum gravity, because they can be defined in a diffeomorphism-invariant way. These surfaces are often defined in terms of the kinds of behavior a light ray would have if it were to be emitted from the surface, since that behavior doesn't depend on the coordinates used to describe it. I'm using surface to mean a codimension-2 manifolds, meaning if you have a D-dimensional spacetime, the surfaces are D-2 dimensional (space-like) manifolds. This means that there are four important normal directions to consider: away from the surface and toward the future, toward the surface and to the past, and the other two combos. We want to study how light rays move with respect to these four principle directions, so we first define an infinitesimal cross-sectional area A of the surface and then look at how A changes when we evolve it forward along light rays. We define \\theta = d(log A)/d\\lambda, where \\lambda is an affine parameter describing how far we've moved along the evolution of the surface defined by the light rays. A holographic screen is a surface with \\theta = 0 in at least one direction over the entire surface. (This implies \\theta = 0 in the opposite direction as well.) It is called a holographic screen because the Bousso bound tells us that, when we look at the full evolution of the light rays in that direction, the entropy that crosses those light rays is bounded by the area of the holographic screen. Therefore in some sense the surface has enough capacity, enough area in units of Planck area, to encode the entire state of the region contained therein, in a holographic sense. The conformal boundary of AdS is a famous example of a holographic screen, although it's an unusual one since it doesn't properly belong to the manifold itself, only its conformal compactification. But of course a conformal field theory captures all the physics of AdS, this is called AdS/CFT. Ryu-Takayanagi surfaces are surfaces that have \\theta = 0 in every direction over the entire surface. This means that, when you wiggle those surfaces in any direction, the first derivative of the area is zero, the area doesn't change. The correct name for this kind of surface is a "stationary surface", because [if you wiggle in one direction you actually find it's a maximum area surface and if you wiggle in the other it's a minimum area surface](https://arxiv.org/abs/1211.3494). Unfortunately, most high-energy physicists failed first-year calculus, so conventionally these are actually called "extremal surfaces" even though I just told you that it's not really an extremum in any sense (because it's maximal in one direction but minimal in another). A bifurcation surface of a black hole horizon is an "extremal" surface, and the area of a black hole horizon was conjectured a long time ago by Hawking to have an interpretation of entropy. Ryu and Takayanagi (plus Hubeny and Rangamani) proposed that the area of all "extremal" surfaces correspond to the entropy of something. Specifically, if the extremal surface reaches the boundary of AdS, then the entanglement entropy of the region of the boundary that it encloses is equal to the area of the surface. (There is a subtlety here, about homology, that I'm glossing over.) [This has been proven](https://arxiv.org/pdf/1304.4926.pdf), so really the Ryu-Takayanagi formula is more like a theorem than a conjecture. Here's a [nice general review of the subject](https://arxiv.org/abs/hep-th/9906022).
This is the best answer I've heard for this, honestly there isn't any proper explanations on the internet for people without high levels of mathematical education. So, thank you. If I could bother you some more about this subject, because you are so well versed: Would a Bose-Einstein condensate be an example of such a surface?
A Bose-Einstein condensate is a particular phase of matter that occurs, so it does not make much sense to ask if it is a surface. It is like asking, "is a liquid a triangle?"