摘要：

Lying at the center of mathematical physics for several decades, the Einstein constraint equations have already been studied intensively since an effective version of the conformal method first appeared in the early 1970s. However, only until the Yamabe problem was solved, the existence and uniqueness of solutions to the constraints were known but limited to the constant mean curvature (CMC) case. This was in the early 90s. Analogous results for the near-CMC case began to appear thereafter. In the last twenty years, there has been some limited progress toward the understanding of solutions to the constraints in the far-from-CMC case. Although it was initially conceivable that these far-from-CMC results would lead to a new picture for the non-CMC case that would mirror the good properties of the CMC and near-CMC cases, remarkable examples of bifurcations, of non-existence, and of non-uniqueness of solutions have been discovered. In this talk, I will present a simple approach to constructing solutions to the constraints in certain far-from-CMC regimes. This new approach implies that some known existence results on Yamabe positive manifolds with arbitrary mean curvature can be thought of as perturbations of the CMC case. This is based on joint work with Romain Gicquaud.