Nonlinear gluing of extremal trajectories

Determine whether an extremal trajectory of the nonlinear classical equation on an entire manifold is obtained by gluing extremal trajectories on submanifolds, and determine the number of such extremal trajectories.

Background

The paper establishes compatibility between renormalization and gluing for the three-dimensional quartic model when the background field is chosen as a solution of the free linear equation. Remark 5 considers instead decomposition around a solution of the interacting classical equation, which permits the treatment of linear terms without the same formal difficulties.

The author states that, in this nonlinear setting, it is unclear whether the extremal trajectory on the full manifold is compatible with gluing along submanifolds and whether the resulting extremal trajectory is unique or how many such trajectories exist. These unresolved issues prevent the gluing theorem proved for the free background from being directly extended to the nonlinear decomposition.