Desingularization of Clifford Torus and Nonradial Solutions to Yamabe Problem with Maximal Rank

TL;DR

This paper constructs nonradial solutions to the critical Yamabe problem by desingularizing Clifford tori, providing the first example with maximal rank in four dimensions and advancing understanding of solution structures.

Contribution

It introduces a novel desingularization method for Clifford tori to produce nonradial solutions with maximal rank for the Yamabe problem.

Findings

Established existence of nondegenerate nonradial solutions

First example of maximal rank solutions in four dimensions

Advances understanding of solution multiplicity in Yamabe problem

Abstract

Through desingularization of Clifford torus, we prove the existence of a sequence of nondegenerate (in the sense of Duyckaerts-Kenig-Merle nodal nonradial solutions to the critical Yamabe problem $$-\Delta u=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u,\qquad u\in {\mathcal{D}}^{1,2}(\mathcal{R}^n). $$ The case $n=4$ is the first example in the literature of a solution with {\em maximal rank} ${\mathcal N}=2n+1+\frac{n(n-1)}{2}$.