A Degenerate Isoperimetric Problem in the Plane

TL;DR

This paper investigates conditions for the existence of length-minimizing curves with a degenerate metric in the plane that enclose a fixed Euclidean area, revealing scenarios where minimizers do not exist and connecting these curves to traveling wave solutions.

Contribution

It extends previous work by allowing a broader class of degenerate metrics with smooth, positive definite Hessians at singularities, and links isoperimetric curves to Hamiltonian system solutions.

Findings

Established sufficient conditions for the existence of minimizers.

Provided examples where minimizers do not exist.

Connected isoperimetric curves to traveling wave solutions.

Abstract

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in [ABCDS] where the metric ds^2 near the singularities equals a quadratic polynomial times the standard metric. Here we allow the conformal factor to be any smooth non-negative potential vanishing at isolated points provided the Hessian at these points is positive definite. These isoperimetric curves, appropriately parametrized, arise as traveling wave solutions to a bi-stable Hamiltonian system.