First, there are only three type of leaves: enter image description here If we assume lamination is compact ,all leaves is only can be a simple closed geodesic. I want to ask about how could union of uncountably many leaves be a minimal compact lamination,actually all leaves is complete simple geodesic,but all leaf in a compact minimal lamination in a punctured torus can only be a simple closed geodesic cause leaf of compact lamination can not goes up to cusp ,so it can only be closed simple geodesic ,so we take closure of any leaf ,it can only be itself, that contrict to minimal.
cause i dont know that question above, i can not understand why we cut a punctured torus along a compact minimal lamination which is not a simple closed geodesic ,we can gent a ideal bigons which is patch two side of two ideal triangels.