Isoperimetric inequalities in the hyperbolic plane

Abstract

Isoperimetric problem is a mathematical problem where we enclose a given area with the shortest possible curve. The isoperimetric problem in the Euclidean plane states ``Among all piecewise smooth simple closed planar curves enclosing a fixed area $A>0$, the circle enclosing area $A$ is the unique perimeter minimizer". The history of the Isoperimetric Problem is fascinating. Many mathematicians, starting from the ancient Greek mathematician Zenodorous (c. 200 - c. 140 BC), contributed to the development of the proof. The proof of this classical theorem was settled completely by the year 1882. Inspired by the legend of Queen Dido mentioned in the Latin epic poem Aeneid (c. 29 - c. 19 BC), where she wanted to enclose the largest piece of land using an ox hide thong, the isoperimetric problem was later considered in other settings and proven for non-Euclidean surfaces. The theorem was proved for curved surfaces such as the unit sphere (by Bernstein) in 1905 and by Fiala and the hyperbolic space (by Schmidt in 1940) among several other Riemannian manifolds with nonzero curvature. In this thesis, we present a proof of The Isoperimetric problem and the isoperimetric inequality for the hyperbolic plane $\mathbb{H}^2$. The isoperimetric inequality states that for any piecewise smooth simple closed curve $\mathcal{C}$ in $\mathbb{H}^2$ with arc-length $\ell$ and enclosing area $A>0$ we have $\ell^2 \geq 4 \pi A+ A^2$ and equality holds if and only if $\mathcal{C}$ is a circle in $\mathbb{H}^2$ of radius $\displaystyle \sinh^{-1}\left(\dfrac{\sqrt{A(4 \pi+ A)}}{2 \pi}\right)$.