By a surface of minimal genus I mean in it's homology class: A surface $S_0$ embedded in a smooth manifold $M$ such that any other surface $S$ with $[S]=[S_0]\in H_2(M)$, we have $g(S)\geq g(S_0)$.
So my question is: If I have a surface $S$ of minimal genus in a smooth 3-manifold $N$ which is a submanifold of a smooth 4-dimensional manifold $M$, is $S$ also minimal in $M$?
I am vaguely aware of "the minimal surface problem" in 4-manifolds, and how the existence of a taut foliaton (with leaves that are minimal sufaces) in $N$ could be used to bound the genus (or Euler characteristic) of surfaces in $M$, but I think since I'm talking about the same surface in both $N$ and $M$, this might just be a topological question, rather than some complicated Seiberg-Witten thing.