  This paper characterizes the annealed, topological complexity (both of total
critical points and of local minima) of the elastic manifold. This classical
model of a disordered elastic system captures point configurations with
self-interactions in a random medium. We establish the simple-vs.-glassy phase
diagram in the model parameters, with these phases separated by a physical
boundary known as the Larkin mass, confirming formulas of Fyodorov and Le
Doussal.
  One essential, dynamical, step of the proof also applies to a general
signal-to-noise model of soft spins in an anisotropic well, for which we prove
a negative-second-moment threshold distinguishing positive from zero
complexity. A universal near-critical behavior appears within this phase
portrait, namely quadratic near-critical vanishing of the complexity of total
critical points, and cubic near-critical vanishing of the complexity of local
minima.
  These two models serve as a paradigm of complexity calculations for Gaussian
landscapes exhibiting few distributional symmetries, i.e. beyond the invariant
setting. The two main inputs for the proof are determinant asymptotics for
non-invariant random matrices from our companion paper [Ben Arous, Bourgade,
McKenna 2021], and the atypical convexity and integrability of the limiting
variational problems.
