  The gradient flow of the Yang-Mills action acts pointwise on closed loops of
gauge fields. We construct a topologically nontrivial loop of SU(2) gauge
fields on S4 that is locally stable under the flow. The stable loop is written
explicitly as a path between two gauge fields equivalent under a topologically
nontrivial SU(2) gauge transformation. Local stability is demonstrated by
calculating the flow equations to leading order in perturbations of the loop.
The stable loop might play a role in physics as a classical winding mode of the
lambda model, a 2-d quantum field theory that was proposed as a mechanism for
generating spacetime quantum field theory. We also present evidence for
2-manifolds of SU(2) and SU(3) gauge fields that are stable under the
Yang-Mills flow. These might provide 2-d instanton corrections in the lambda
model. For Isidore M. Singer in celebration of his eighty-fifth birthday.
