INITIAL PERTURBATION OF THE MEAN CURVATURE FLOW FOR CLOSED LIMIT SHRINKER

The paper proves a precise global genericity theorem (Theorem 3.1) for RMCF approaching a non-spherical closed shrinker Σ, constructing local invariant manifolds for a truncated flow and, crucially, using a positivity/Harnack route (Li–Yau type inequalities for the time-dependent linearized operator) to drive orbits into an invariant cone so that the first eigenfunction dominates and the trajectory stays close to the unstable manifold before exiting with strictly lower entropy; see the theorem statement and proof scaffolding, including the local invariant manifold theorem (Theorem 2.1), the variational equation, Harnack/averaging estimates, cone-entry, and the quantitative entropy drop via second variation in the φ1 direction . By contrast, the model’s solution outlines a similar local picture but (i) assumes a stable foliation and an open-dense complement of a center-stable manifold without engaging the truncated-flow/maximal-regularity framework used in the paper, and (ii) replaces the paper’s irreversible-flow handling (positivity + Harnack + invariant cones) with an unsubstantiated surjectivity/submersion claim from initial data to the unstable component at a fixed future time. Most critically, its entropy-drop step appeals only to monotonicity and nonstationarity, which does not by itself ensure λ(M̃T) < λ(Σ) in the strong, quantitative sense proved in the paper via the F-unstable φ1-direction calculation . Hence the paper’s argument is correct and complete for the stated result, while the model’s proof is incomplete on key global and entropy steps.