The Isoperimetric Problem for the Curl Operator

TL;DR

This paper solves the isoperimetric problem for the curl operator on compact three-manifolds, establishing optimal bounds for eigenvalues, identifying domains where bounds are attained, and disproving a longstanding conjecture.

Contribution

It provides the first complete solution to the isoperimetric problem for the curl operator, including explicit bounds, optimal domains, and eigenvalue multiplicities, and disproves a previous conjecture.

Findings

All compact three-manifolds have optimal lower bounds for curl eigenvalues.

Bounds are attained by specific domains with known eigenvalue multiplicities.

The paper disproves a longstanding conjecture by Cantarella et al.

Abstract

In the last decades, many mathematicians have studied the curl operator in compact three-manifolds , mainly the structure of its spectrum and some isoperimetric problems associated with it. In this paper, we will see that all the compact three-manifolds (both closed and and with non-empty boundary) have always optimal lower bounds for the absolute value of their non-null eigenvalues of curl. We will also show that these bounds are always attained and compute the optimal domains and the multiplicities of their associated eigenvalues. So, we have solved the isoperimetric problem associated to the curl operator, and, by the way, we have solved and old Cantarella, de Turck, Gluck and Teytel conjecture in the negative.