Action minimizers under topological constraints in the planar equal-mass four-body problem

It is shown that in the planar equal-mass four-body problem, there exist two sets of new action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints on the four bodies: a double isosceles configuration and an isosceles trapezoid configuration. By applying the level estimate method, these minimizers are shown to be collision-free and they can be extended to two new sets of periodic or quasi-periodic orbits.