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Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature

Cornelia Druţu, Urs Lang, Panos Papasoglu, Stephan Stadler

Abstract

We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than $c_2=1/(4π)$ -- the optimal constant for the Euclidean plane -- implies that the underlying space is Gromov hyperbolic, and a linear inequality holds. We establish the first analogous gap theorem in higher dimensions: if a proper CAT(0) space satisfies a Euclidean inequality for 2-spheres with a constant below the sharp threshold $c_3=1/(6\sqrtπ)$, then the space also admits an inequality with an exponent arbitrarily close to 1. As a corollary we obtain a similar result for Lipschitz surfaces of higher genus. Towards our main theorem we prove a (non-sharp) Euclidean isoperimetric inequality for null-homotopies of 2-spheres, apparently missing in the literature. A novelty in our approach is the introduction of minimal tetrahedra, which we demonstrate satisfy a linear inequality.

Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature

Abstract

We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than -- the optimal constant for the Euclidean plane -- implies that the underlying space is Gromov hyperbolic, and a linear inequality holds. We establish the first analogous gap theorem in higher dimensions: if a proper CAT(0) space satisfies a Euclidean inequality for 2-spheres with a constant below the sharp threshold , then the space also admits an inequality with an exponent arbitrarily close to 1. As a corollary we obtain a similar result for Lipschitz surfaces of higher genus. Towards our main theorem we prove a (non-sharp) Euclidean isoperimetric inequality for null-homotopies of 2-spheres, apparently missing in the literature. A novelty in our approach is the introduction of minimal tetrahedra, which we demonstrate satisfy a linear inequality.

Paper Structure

This paper contains 17 sections, 28 theorems, 95 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Strategy of proof
  4. Organization
  5. Acknowledgements
  6. Preliminaries
  7. Basics
  8. Lipschitz maps, area and volume
  9. Plateau problem and minimal discs
  10. Disc decompositions
  11. Nagata dimension
  12. Metric integral currents
  13. The Euclidean isoperimetric inequality for spheres
  14. Minimal triangles and tetrahedra
  15. Quantitative triangulation of discs
  16. ...and 2 more sections

Key Result

Theorem 1

For a proper $\operatorname{CAT}(0)$ space $X$, the following are equivalent:

Figures (4)

  • Figure 1: A polygonal resolution of a disc with two holes.
  • Figure 2: Example of a disc decomposition.
  • Figure 3: Illustration of the decomposition process.
  • Figure 4: Decomposition of $D$.

Theorems & Definitions (43)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 33 more