An isoperimetric inequality for surfaces formed from spherical polygons

TL;DR

This paper presents a new proof of an isoperimetric inequality for certain convex surfaces formed from spherical polygons, with implications for eigenvalue bounds on these surfaces.

Contribution

It provides a novel proof of an isoperimetric inequality for surfaces with Gaussian curvature one, formed from spherical polygons, extending previous results.

Findings

Inequality applies to surfaces formed from spherical polygons with non-smooth points.

Implication for lower bounds on the first Dirichlet eigenvalue of regions.

Extension to bounds on the first Dirichlet-Neumann eigenvalue via approximation.

Abstract

We give a new proof of an isoperimetric inequality for a family of closed surfaces, which have Gaussian curvature identically equal to one wherever the surface is smooth. These surfaces are formed from a convex, spherical polygon, with each vertex of the polygon leading to a non-smooth point on the surface. For example, the surface formed from a spherical lune is a surface of revolution, with two non-smooth tips. Combined with a straightforward approximation argument, this inequality was first proved by B\'erard, Besson, and Gallot, where they provide a generalization of the L\'evy-Gromov isoperimetric inequality. The inequality implies an isoperimetric inequality for geodesically convex subsets of the sphere, and, using a Faber-Krahn theorem, it also implies a lower bound on the first Dirichlet eigenvalue of a region of a given area on the closed surfaces. Via approximation, we convert this into a lower bound on the first Dirichlet-Neumann eigenvalue of domains contained in geodesically convex subsets of the sphere.