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Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

Gioacchino Antonelli, Marco Pozzetta

TL;DR

This work analyzes the isoperimetric problem on Alexandrov spaces with nonnegative curvature under a non-collapsing condition. It establishes an equivalence between linear volume growth, asymptotic cylindrical geometry, and uniform isoperimetric bounds, showing that such spaces are tubular neighborhoods of a line or ray and proving existence of large-volume isoperimetric sets. The authors develop a robust non-smooth variational framework, leveraging pmGH convergence, Busemann functions, and RCD theory to extend results to RCD(0,N) spaces and Euclidean convex bodies, with a complete treatment in dimension two. The findings provide new rigidity and structure theorems at infinity and offer a comprehensive description of isoperimetric regions in noncompact, noncollapsed settings with wide applicability to smooth and non-smooth geometries alike.

Abstract

In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces with a non-collapsing condition, i.e., such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces. In this setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov--Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray. Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and we characterize the geometric rigidity at the level of the isoperimetric profile. Specializing our study to the $2$-dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary. The proofs exploit a variational approach, and in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

TL;DR

This work analyzes the isoperimetric problem on Alexandrov spaces with nonnegative curvature under a non-collapsing condition. It establishes an equivalence between linear volume growth, asymptotic cylindrical geometry, and uniform isoperimetric bounds, showing that such spaces are tubular neighborhoods of a line or ray and proving existence of large-volume isoperimetric sets. The authors develop a robust non-smooth variational framework, leveraging pmGH convergence, Busemann functions, and RCD theory to extend results to RCD(0,N) spaces and Euclidean convex bodies, with a complete treatment in dimension two. The findings provide new rigidity and structure theorems at infinity and offer a comprehensive description of isoperimetric regions in noncompact, noncollapsed settings with wide applicability to smooth and non-smooth geometries alike.

Abstract

In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces with a non-collapsing condition, i.e., such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces. In this setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov--Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray. Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and we characterize the geometric rigidity at the level of the isoperimetric profile. Specializing our study to the -dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary. The proofs exploit a variational approach, and in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.
Paper Structure (18 sections, 48 theorems, 207 equations)

This paper contains 18 sections, 48 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and general facts about $\mathrm{CBB}(0)$ metric spaces
  4. Pointed measured Gromov--Hausdorff convergence
  5. Tangent and asymptotic cones of $\mathop{\mathrm{CBB}}\nolimits(0)$ metric spaces
  6. $\mathrm{BV}$ and Sobolev functions on metric measure spaces
  7. BV functions and sets of finite perimeter in metric measure spaces
  8. Sobolev functions in metric measure spaces
  9. Geometric Analysis and isoperimetric sets on RCD spaces
  10. Cylindrical splitting
  11. Isoperimetric sets and profile on $\mathop{\mathrm{RCD}}\nolimits$ spaces
  12. On the asymptotic geometry of $\mathrm{CBB}(0)$ spaces
  13. Structure of asymptotically cylindrical $\mathrm{CBB}(0)$ spaces
  14. Volume non-collapsedness of $2$-dimensional $\mathrm{CBB}(0)$ spaces
  15. Isoperimetric problem
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $N\geq 2$, and let $(X,\mathsf{d})$ be a noncompact $N$-dimensional $\mathop{\mathrm{CBB}}\nolimits(0)$ metric space such that $\mathcal{H}^N(B_1(x))\geq v_0$ for some $v_0>0$, and for every $x\in X$. Let $o \in X$. Then the following are equivalent. If any of the previous items holds, then there exists a unique compact $\mathop{\mathrm{CBB}}\nolimits(0)$ space $K$ such that any pGH limit of

Theorems & Definitions (103)

  • Theorem 1.1: Isoperimetry and structural rigidity, cf. \ref{['thm:raggiosselimiteainfinito']}, \ref{['cor:Equivalenze']}, \ref{['prop:ProfiloCostiffSPlitCilindro']}
  • Theorem 1.2: Existence and structural rigidity, cf. \ref{['thm:AsintoticaProfilo']}
  • Theorem 1.3: $2$-dimensional $\mathop{\mathrm{CBB}}\nolimits(0)$ spaces, cf. \ref{['thm:Noncollapsed']}, \ref{['thm:Existence2D']}, \ref{['thm:IsopBoundaryConnected']}
  • Theorem 1.4: cf. \ref{['thm:ConvergenzaBusemann']}
  • Definition 1
  • Lemma 1
  • Definition 2: pGH and pmGH convergence
  • Definition 3: $L^1$-strong and $L^1_{\mathrm{loc}}$ convergence
  • Definition 4: Hausdorff convergence
  • Remark 1: Uniform convergence of curves and functions along pGH limits
  • ...and 93 more