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Sharp Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below

Carlo Morpurgo, Liuyu Qin

TL;DR

The paper proves sharp Sobolev and Moser–Trudinger inequalities on complete noncompact Riemannian manifolds with a lower Ricci bound and positive injectivity radius. It develops a four-step framework: (1) a local $I_1^α$ inequality via a weak pointwise bound in $C^{0,α}$ harmonic coordinates, (2) a first-order isoperimetric profile expansion for small volumes, (3) translation of the profile data into small-volume Sobolev and MT_k inequalities, and (4) a global extension from small to all volumes. The main achievements are the validity of the $I_p^α$ Sobolev inequality with best constant $K(n,p)^α$ for $α∈(0,1)$ and $p∈[1,n)$, and the sharp MT_k inequalities for $k=1,2$ in the same geometric setting. These results extend sharp functional-inequality theory to noncompact manifolds under weaker curvature assumptions, providing a robust mechanism to pass from local isoperimetric information to global analytic inequalities.

Abstract

We establish Sobolev and Moser-Trudinger inequalities with best constants on noncompact Riemannan manifolds with Ricci curvature bounded below, and positive injectivity radius.

Sharp Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below

TL;DR

The paper proves sharp Sobolev and Moser–Trudinger inequalities on complete noncompact Riemannian manifolds with a lower Ricci bound and positive injectivity radius. It develops a four-step framework: (1) a local inequality via a weak pointwise bound in harmonic coordinates, (2) a first-order isoperimetric profile expansion for small volumes, (3) translation of the profile data into small-volume Sobolev and MT_k inequalities, and (4) a global extension from small to all volumes. The main achievements are the validity of the Sobolev inequality with best constant for and , and the sharp MT_k inequalities for in the same geometric setting. These results extend sharp functional-inequality theory to noncompact manifolds under weaker curvature assumptions, providing a robust mechanism to pass from local isoperimetric information to global analytic inequalities.

Abstract

We establish Sobolev and Moser-Trudinger inequalities with best constants on noncompact Riemannan manifolds with Ricci curvature bounded below, and positive injectivity radius.
Paper Structure (7 sections, 9 theorems, 179 equations, 1 table)

This paper contains 7 sections, 9 theorems, 179 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A local $I_1^\alpha$ inequality
  4. From local $I_1^\alpha$ to profile expansion
  5. From profile expansion to small volumes inequalities
  6. From small volumes to global inequalities
  7. Proofs of Theorems \ref{['main1']} and \ref{['main2']}

Key Result

Theorem 1

On a complete, smooth Riemannian $n-$dimensional manifold $(M,g)$, suppose that for some $K\in\mathbb{R}.$ Then, the $I_p^\alpha$ Sobolev inequality holds, for any $\alpha\in (0,1)$ and $p\in[1,n)$.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • Remark 6
  • ...and 8 more