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This paper characterizes the annealed, topological complexity (both of totalcritical points and of local minima) of the elastic manifold. This classicalmodel of a disordered elastic system captures point configurations withself-interactions in a random medium. We establish the simple-vs.-glassy phasediagram in the model parameters, with these phases separated by a physicalboundary known as the Larkin mass, confirming formulas of Fyodorov and LeDoussal.  One essential, dynamical, step of the proof also applies to a generalsignal-to-noise model of soft spins in an anisotropic well, for which we provea negative-second-moment threshold distinguishing positive from zerocomplexity. A universal near-critical behavior appears within this phaseportrait, namely quadratic near-critical vanishing of the complexity of totalcritical points, and cubic near-critical vanishing of the complexity of localminima.  These two models serve as a paradigm of complexity calculations for Gaussianlandscapes exhibiting few distributional symmetries, i.e. beyond the invariantsetting. The two main inputs for the proof are determinant asymptotics fornon-invariant random matrices from our companion paper [Ben Arous, Bourgade,McKenna 2021], and the atypical convexity and integrability of the limitingvariational problems.

Landscape complexity beyond invariance and the elastic manifold