  We present a detailed proof of a previously announced result (C.M. Newman and
D.L. Stein, Phys. Rev. Lett. v. 84, pp. 3966--3969 (2000)) supporting the
absence of multiple (incongruent) ground state pairs for 2D Edwards-Anderson
spin glasses (with zero external field and, e.g., Gaussian couplings): if two
ground state pairs (chosen from metastates with, e.g., periodic boundary
conditions) on the infinite square lattice are distinct, then the dual bonds
where they differ form a single doubly-infinite, positive-density domain wall.
It is an open problem to prove that such a situation cannot occur (or else to
show --- much less likely in our opinion --- that it indeed does happen) in
these models. Our proof involves an analysis of how (infinite-volume) ground
states change as (finitely many) couplings vary, which leads us to a notion of
zero-temperature excitation metastates, that may be of independent interest.
