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

Jungang Li, Guozhen Lu

TL;DR

The paper addresses establishing global Trudinger–Moser inequalities on complete noncompact Riemannian manifolds under a Ricci lower bound and positive injectivity radius. It presents a streamlined, local-to-global proof that derives uniform local inequalities on geodesic balls and passes to a global bound using a level-set framework and Gromov covering, aided by Green-function representations and Adams' method. The contributions include clarifying the proof of a key theorem and offering an alternative, simpler argument that yields constants independent of the ball location, extending the reach of concentration–compactness techniques to non-Euclidean settings. This work enhances the toolkit for sharp exponential-type inequalities on manifolds and improves understanding of how local bounds aggregate into global results.

Abstract

The main purpose of this short note, on the one hand, to is clarify some part of the proof of Theorem 1.3 in [8] in a simple way, and on the other hand, to give an alternative argument from local inequalities to global ones.

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

TL;DR

The paper addresses establishing global Trudinger–Moser inequalities on complete noncompact Riemannian manifolds under a Ricci lower bound and positive injectivity radius. It presents a streamlined, local-to-global proof that derives uniform local inequalities on geodesic balls and passes to a global bound using a level-set framework and Gromov covering, aided by Green-function representations and Adams' method. The contributions include clarifying the proof of a key theorem and offering an alternative, simpler argument that yields constants independent of the ball location, extending the reach of concentration–compactness techniques to non-Euclidean settings. This work enhances the toolkit for sharp exponential-type inequalities on manifolds and improves understanding of how local bounds aggregate into global results.

Abstract

The main purpose of this short note, on the one hand, to is clarify some part of the proof of Theorem 1.3 in [8] in a simple way, and on the other hand, to give an alternative argument from local inequalities to global ones.
Paper Structure (3 sections, 8 theorems, 24 equations)

This paper contains 3 sections, 8 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Some useful known lemmas
  3. Proof of Theorem \ref{['maintheorem']}

Key Result

Theorem 2.1

Let $(M, g)$ be a complete noncompact Riemannian manifold whose Ricci curvature is lower bounded, i.e. $\textit{Ric} \geq \lambda g$ for some constant $\lambda \in \mathbb{R}$. Moreover, we assume that its injectivity radius has a lower bound, i.e. $\textit{inj} (M,g) \geq i > 0$ for some constant $ Moreover, $\alpha_n$ is sharp.

Theorems & Definitions (11)

  • Theorem 2.1
  • Lemma 2.1: Hebey Hebey, Lemma 1.6
  • Lemma 2.2: Hebey Hebey, Theorem 1.3
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.2: Schauder estimates
  • Theorem 3.3
  • ...and 1 more