Continuation of multilevel rotating vortex patches to the genuine-hole regime

Establish whether the rotating outer vortex patches with highly concentrated inner components constructed by Baroncini, Cantero, García, Hassainia, and Mateu can be continued to the parameter value at which the inner vorticity level vanishes, thereby reaching configurations with genuine zero-vorticity holes.

Background

The paper contrasts its unit-vorticity characteristic-function patches with the multilevel configurations constructed by Baroncini, Cantero, García, Hassainia, and Mateu. In those configurations, the vorticity on each inner component is nonzero, specifically equal to a value of the form $1-(\pi^2)^{-1}$ in the local regime, so the inner components are not genuine holes.

The cited construction identifies continuation in its parameter to the value at which this inner vorticity becomes zero as an unresolved global problem. Solving it would connect the existing multilevel rotating-patch branch to the genuine-hole regime addressed directly by the present paper.