Optimal Trudinger-Moser inequalities on complete noncompact Riemannian manifolds: Revisit of the argument from the local inequalities to global ones

The main purpose of this short note, on the one hand, to is rigorize some part of the proof of Theorem 1.3 in [11] in a simple way, and on the other hand, to give an alternative argument from local inequalities to global ones.

💡 Research Summary

The paper addresses two intertwined problems concerning optimal Trudinger‑Moser inequalities on complete non‑compact Riemannian manifolds. The manifolds under consideration are assumed to have a positive injectivity radius, a Ricci curvature lower bound (\operatorname{Ric}\ge\lambda g), and an upper bound on sectional curvature. Under these geometric hypotheses the authors provide a rigorous justification of Theorem 1.3 (referred to as Theorem 1.1 in the present note) from a previous work