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A note on Poincaré-Sobolev type inequalities on compact manifolds

Romain Gicquaud

TL;DR

This work derives a quantitative Poincaré-Sobolev-type inequality on compact Riemannian manifolds for deviations from a density-weighted average. The key idea is to lift the problem to Euclidean space via a surjective local diffeomorphism from the unit ball and the coarea formula, obtaining explicit dependence of the constant on the density’s L^q-norm. The result fills a gap by showing how the density affects the Poincaré constant without altering the underlying measure or Sobolev norms, with potential applications to coupled elliptic systems on manifolds. The approach combines Euclidean convex-domain estimates with a geometric gluing construction to extend the inequality globally on M.

Abstract

We prove a Poincaré-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical weighted Poincaré inequalities, the density does not enter the measure or the Sobolev norms, but only the reference average. We show that the associated Poincaré constant depends quantitatively on the Lebesgue norm of the density. This framework naturally arises in the analysis of coupled elliptic systems and seems not to have been addressed in the existing literature.

A note on Poincaré-Sobolev type inequalities on compact manifolds

TL;DR

This work derives a quantitative Poincaré-Sobolev-type inequality on compact Riemannian manifolds for deviations from a density-weighted average. The key idea is to lift the problem to Euclidean space via a surjective local diffeomorphism from the unit ball and the coarea formula, obtaining explicit dependence of the constant on the density’s L^q-norm. The result fills a gap by showing how the density affects the Poincaré constant without altering the underlying measure or Sobolev norms, with potential applications to coupled elliptic systems on manifolds. The approach combines Euclidean convex-domain estimates with a geometric gluing construction to extend the inequality globally on M.

Abstract

We prove a Poincaré-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical weighted Poincaré inequalities, the density does not enter the measure or the Sobolev norms, but only the reference average. We show that the associated Poincaré constant depends quantitatively on the Lebesgue norm of the density. This framework naturally arises in the analysis of coupled elliptic systems and seems not to have been addressed in the existing literature.

Paper Structure

This paper contains 4 sections, 4 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. The case of open subsets of TEXT
  3. Local diffeomorphisms onto compact manifolds
  4. Proof of the main result

Key Result

Theorem 1.1

Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, and let $p,q \in (1,\infty)$ satisfy Let $r \in (1,\infty]$, and assume that, if $p < n$, then Then, for all functions $f \in W^{1,p}(M,\mathbb{R})$ and all non-negative densities $\omega \in L^q(M)$ normalized by $\mathbb{E}[\omega] = 1$, the weighted average $\mathbb{E}_\omega[f]$ is well-defined and there exists a constant $C>0$,

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • proof : Proof of Theorem \ref{['thmPoincare']}