Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature

Debora Impera, Stefano Pigola, Michele Rimoldi, Giona Veronelli

TL;DR

This work develops a robust framework for global geometric inequalities on noncompact manifolds whose curvature is asymptotically nonnegative. By combining an extension principle that propagates exterior isoperimetric control to the whole manifold with a localized Brendle ABP method, the authors obtain isoperimetric and $L^1$ Sobolev inequalities under mild infinity-geometry assumptions. They further derive Michael-Simon and Log-Sobolev inequalities for submanifolds in ambient spaces with nonnegative curvature outside a compact set, with constants governed by end volume growth AVR. The results apply to manifolds with ends and ALE-type geometries, providing sharp constant behavior tied to the geometry at infinity and offering tools for geometric analysis on noncompact spaces.

Abstract

We establish the validity of the isoperimetric inequality (or equivalently, an $L^1$ Euclidean-type Sobolev inequality) on manifolds with asymptotically non-negative sectional curvature. Unlike previous results in the literature, our approach does not require the negative part of the curvature to be globally small. Furthermore, we derive a Michael-Simon inequality on manifolds whose curvature is non-negative outside a compact set. The proofs employ the ABP method for isoperimetry, initially introduced by Cabré in the Euclidean setting and subsequently extended and skillfully adapted by Brendle to the challenging context of non-negatively curved manifolds. Notably, we show that this technique can be localized to appropriate regions of the manifold. Additional key elements of the argument include the geometric structure at infinity of asymptotically non-negatively curved manifolds, their spectral properties - which ensure the non-negativity of a Bakry-Émery Ricci tensor on a conformal deformation of each end - and a result that deduces the validity of the isoperimetric inequality on the entire manifold, provided it holds outside a compact set.

Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature

TL;DR

This work develops a robust framework for global geometric inequalities on noncompact manifolds whose curvature is asymptotically nonnegative. By combining an extension principle that propagates exterior isoperimetric control to the whole manifold with a localized Brendle ABP method, the authors obtain isoperimetric and Sobolev inequalities under mild infinity-geometry assumptions. They further derive Michael-Simon and Log-Sobolev inequalities for submanifolds in ambient spaces with nonnegative curvature outside a compact set, with constants governed by end volume growth AVR. The results apply to manifolds with ends and ALE-type geometries, providing sharp constant behavior tied to the geometry at infinity and offering tools for geometric analysis on noncompact spaces.

Abstract

We establish the validity of the isoperimetric inequality (or equivalently, an Euclidean-type Sobolev inequality) on manifolds with asymptotically non-negative sectional curvature. Unlike previous results in the literature, our approach does not require the negative part of the curvature to be globally small. Furthermore, we derive a Michael-Simon inequality on manifolds whose curvature is non-negative outside a compact set. The proofs employ the ABP method for isoperimetry, initially introduced by Cabré in the Euclidean setting and subsequently extended and skillfully adapted by Brendle to the challenging context of non-negatively curved manifolds. Notably, we show that this technique can be localized to appropriate regions of the manifold. Additional key elements of the argument include the geometric structure at infinity of asymptotically non-negatively curved manifolds, their spectral properties - which ensure the non-negativity of a Bakry-Émery Ricci tensor on a conformal deformation of each end - and a result that deduces the validity of the isoperimetric inequality on the entire manifold, provided it holds outside a compact set.

Paper Structure

This paper contains 7 sections, 20 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Extending isoperimetry inside compact sets
  3. Geometric preliminaries
  4. Asymptotically non-negative curvature
  5. Submanifolds in ambient manifolds with nonnegative sectional curvature outside a compact set
  6. Michael-Simon inequality
  7. Log-Sobolev inequality

Key Result

Theorem 1.1

Let $(M^n, g)$ be a Riemannian manifold. Assume that $M$ has infinite volume and that the $L^1$-Sobolev inequality e:L1Sob holds for all $h\in C_c^\infty(M \setminus F)$ for some compact $F\subset M$. Then, the same Sobolev inequality, possibly with a different constant, holds for all $h\in C^\infty

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • proof : Proof (of Theorem \ref{['th:extension']})
  • Remark 2.2
  • Corollary 2.3
  • proof
  • ...and 31 more