Planar doubling nodal solutions to the Yamabe equation with maximal rank

TL;DR

This paper constructs two new families of nodal solutions to the Yamabe equation concentrating along planar circles, with one family being a novel twisted variant and solutions in dimension 3 achieving maximal rank.

Contribution

It introduces a new twisted family of solutions not invariant under Kelvin transformations and analyzes their interaction, including a crossing phenomenon.

Findings

Two families of solutions concentrating on planar circles

One family is a new twisted variant

Solutions in dimension 3 attain maximal rank

Abstract

This article constructs two families of nodal solutions to the Yamabe equation, each concentrating along two planar circles. One family is conformally equivalent to the one previously obtained by Medina--Musso. The second family is a twisted variant of the first; it is new and is derived from ansatzes that are not Kelvin invariant, in contrast to a standard assumption in earlier works. In addition, in dimension 3, these solutions attain maximal rank. By means of a continuous family of conformal transformations, we then analyze the interaction of the two circles, which display a crossing phenomenon reminiscent, in some sense, of leap-frogging behavior in vortex dynamics.