Majorization by Hemispheres & Quadratic Isoperimetric Constants

TL;DR

The paper proves that in certain metric spaces, Lipschitz curves can be extended to Lipschitz maps on the hemisphere, establishing a quadratic isoperimetric inequality that influences minimal disc regularity in Finsler geometry.

Contribution

It demonstrates the extension of Lipschitz curves to hemispheres in spaces with conical geodesic bicombings, establishing a quadratic isoperimetric inequality with specific constant.

Findings

Lipschitz curves extend to hemispheres in these spaces.

Spaces satisfy a quadratic isoperimetric inequality with constant 1/2π.

Implications for regularity of minimal discs in Finsler manifolds.

Abstract

Let $X$ be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed $L$-Lipschitz curve $\gamma:S^1\rightarrow X$ may be extended to an $L$-Lipschitz map defined on the hemisphere $f:H^2\rightarrow X$. This implies that $X$ satisfies a quadratic isoperimetric inequality (for curves) with constant $\frac{1}{2\pi}$. We discuss how this fact controls the regularity of minimal discs in Finsler manifolds when applied to the work of Alexander Lytchak and Stefan Wenger.