Doubling nodal solutions to the Yamabe equation in $\mathbb{R}^n$ with maximal rank

TL;DR

This paper constructs new solutions to the Yamabe equation in Euclidean space, including the first example of maximal rank solutions in odd dimensions, using a novel doubling method inspired by minimal surface theory.

Contribution

It introduces a new family of entire solutions to the Yamabe equation with maximal rank in odd dimensions, expanding the known solution space.

Findings

Solutions exist for all dimensions n ≥ 3.

First example of maximal rank solutions in odd dimensions.

Construction parallels minimal surface doubling techniques.

Abstract

We construct a new family of entire solutions to the Yamabe equation $$-\Delta u=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u \mbox{ in }\mathcal{D}^{1,2}(\mathbb{R}^n).$$ If $n=3$, our solutions have maximal rank, being the first example in odd dimension. Our construction has analogies with the doubling of the equatorial spheres in the construction of minimal surfaces in $S^3(1)$.