A simple proof of an isoperimetric inequality for euclidean and hyperbolic cone-surfaces

TL;DR

This paper provides a straightforward proof that isoperimetric inequalities hold for Euclidean and hyperbolic cone-surfaces with negative curvature singularities, extending classical results to a discrete setting.

Contribution

It introduces a simple proof for isoperimetric inequalities on cone-surfaces using discrete conformal deformations, offering a new approach compared to previous complex proofs.

Findings

Isoperimetric inequalities hold for all Euclidean and hyperbolic cone-metrics with negative curvature singularities.

The proof employs discrete conformal deformations to eliminate singularities and increase area.

The approach parallels Weil's method using uniformization and harmonic functions.

Abstract

We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems of Weil and Bol that deal with Riemannian metrics of curvature bounded from above by 0, respectively by -1. A stronger discrete version was proved by A.D.Alexandrov, with a subsequent extension by approximation to metrics of bounded integral curvature. Our proof uses "discrete conformal deformations" of the metric that eliminate the singularities and increase the area. Therefore it resembles Weil's argument, which uses the uniformization theorem and the harmonic minorant of a subharmonic function.