A simple proof of the isoperimetric theorem for the hyperbolic plane

TL;DR

This paper provides a concise, accessible proof of the hyperbolic plane's isoperimetric theorem, highlighting the maximal area of triangles with fixed side lengths and clarifying its relation to the isoperimetric problem.

Contribution

It offers a simplified, pedagogical proof of a key hyperbolic isoperimetric result, making it accessible for students and clarifying its connection to the broader problem.

Findings

Maximal area triangles occur when =+.

The proof is similar to existing work but simplified for clarity.

The result is important for understanding hyperbolic geometry.

Abstract

In this pedagogical note we present a short proof of the following main result of arxiv.org/abs/0911.5319, and clarify its relation to the isoperimetric problem. On the hyperbolic plane consider triangles ABC with fixed lengths of AB and AC. The maximal area of these triangles is attained for the triangle ABC such that \angle A=\angle B+\angle C. The proof is essentially the same as in the above-cited paper. However, since the result is beautiful and important, a short proof cleared of unnecessary details could be interesting for a reader. The note is accessible for students familiar with elementary hyperbolic geometry. (This note is based on referee reports to the above paper, see www.mccme.ru/mmks)