The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces

Understand

We give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian $n$-manifolds.

• For some non-compact, finite area 2-manifolds, we prove the existence and regularity of subsets whose isoperimetric ratio is equal to the Cheeger constant.
• To do this, we use results of Hass-Morgan for the isoperimetric problem of these manifolds.
• We also give an example of a finite area 2-manifold where no such subset exists.