You give the subject a single command, which it obeys to the best of its ability at its earliest opportunity. You may select from the following options.
Approach: On its turn, the subject moves toward you as quickly and directly as possible for 1 round. The creature may do nothing but move during its turn, and it provokes attacks of opportunity for this movement as normal.
If someone is commanded to approach what kind of limitations do they have?
What if they're on a ledge 30' above me?
Does "as quickly" mean they have to run if there's a straight path available?
Can they acrobatics to negate attacks of opportunity?
While using acrobatics to avoid AoOs, you move at half speed. This would conflict with "as quickly as possible". However, by increasing the DC of the check by 10, you could do this at full speed.
So my answer would be "Yes, you can use acrobatics to avoid AoOs if under the 'approach' command, but you must use the full speed option by adding 10 to the DC."
If they're on a ledge, then they'd begin making climb checks (unless they've got a feather fall item that can be used).
In general, I do not believe that command, as a first-level spell, is intended to be lethal, so I would rule that the subject will not take any obviously suicidal action (such as trying to run across a chasm or jump off a high ledge).
I agree with SlimGauge; the subject is required to advance "as quickly as possible," which would normally preclude taking ordinary acrobatics checks. I don't think that running would be necessary if they were only 10' away at the beginning of their turn, but if they were further away than they could reach without running, they would need to run.
The spell specifically says it provokes attacks of opportunity for this movement as normal.
Yes, it provokes as normal. Using acrobatics is a normal way to move, which does not provoke when the check is successful. All of which is normal.
Just because they can choose a mode of movement that normally provokes doesn't mean they are prevented form choosing a mode of movement that doesn't, as long as they move at full speed.