Fractional perimeters on the sphere

TL;DR

This paper investigates the properties of fractional perimeters on the sphere, establishing an isoperimetric inequality with spherical caps as equality cases and analyzing their limits as the fractional parameter varies.

Contribution

It proves the spherical fractional isoperimetric inequality, characterizes equality cases, and describes the limits of fractional perimeters as the parameter approaches 1 and negative infinity.

Findings

Spherical fractional isoperimetric inequality established

Equality cases are spherical caps

Limits of fractional perimeters relate to surface area and volume

Abstract

This note treats several problems for the fractional perimeter or $s$-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as $s \nearrow 1$ is proven. It is shown that their limit as $s \searrow -\infty$ can be expressed in terms of the volume.