Hello, I've thought about it a few times and this question interests me. If a brane (M2 in my case) in euclidean signature preserves some supersymmetry, does it automatically solve the equation of motion? In order to be more precise here is what I mean: Take 11D SUGRA with an AdS_4xS^7 background solution. Now embed an M2 brane in this background such that its Lagrangian is simply given by the volume of the brane since the metric has an euclidean signature. Imagine you were able to prove that this brane preserves some supersymmetry, i.e. the Killing spinors on AdS_4 and S^7 satisfy the usual projection equation. Does it automatically follows that the brane satisfies the equation of motion, i.e. has minimal volume? This seams to be true because any susy preserving brane in the literature is also solution to the equation of motion. On top of that I have heard that a proof exists in minkowskian signature by using a Hamiltonian formalism so this might as well be true in euclidean signature. One could wick rotate the metric into an minkowskian one and use the Hamiltonian proof but I find it a bit ugly. Has someone ever seen a proof of this?