MINIMAX PERIODIC ORBITS OF CONVEX LAGRANGIAN SYSTEMS ON COMPLETE RIEMANNIAN MANIFOLDS

The paper proves Theorem 1 and Theorem 2—existence of periodic orbits for almost every energy in the subcritical interval (e0(L), cu(L)) under the L-shrinking hypothesis, and for k > cu(L) under the homotopically L-shrinking hypothesis—via a modified minimax principle plus Struwe’s monotonicity trick, localizing the variational problem in a compact K0 using ΦK and controlling periods through a bounded pseudo-gradient flow and Lemma 10 (periods cannot collapse at positive levels) . The candidate solution establishes the same results and overall scheme (minimax + Struwe + localization + compactness of PS sequences, with energy identification from ∂TSk=0) and matches key identities for Sk and energy at critical points . Differences are technical: the paper’s proof avoids completeness issues by building a positively complete truncated flow on sublevels, whereas the candidate invokes Ekeland’s principle on a non-complete ambient manifold without first imposing a lower bound on periods. This is a fixable gap (one can restrict to T ∈ [ε, Tmax] or use the paper’s flow), so we deem both correct, with the model’s proof requiring minor repair. The L-shrinking and homotopically L-shrinking mechanisms used for localization and homotopy preservation in the paper are faithfully mirrored in the candidate solution .