Global dynamics above the ground state for the energy-critical Schrodinger equation with radial data

Consider the focusing energy critical Schrodinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with the paper by Schlag and the first author on the subcritical equation, the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference from the subcritical case is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.