Singular solutions of the Yamabe problem in the Heisenberg group and their bifurcation

TL;DR

This paper establishes the existence of a singular solution to a critical nonlinear PDE on the Heisenberg group and investigates bifurcation phenomena of non-homogeneous solutions.

Contribution

It introduces a new concept of normal curvature for hypersurfaces and demonstrates bifurcation of solutions in the Heisenberg group setting.

Findings

Existence of a homogeneous singular solution to the critical PDE on the Heisenberg group.

Introduction of a new normal curvature concept for hypersurfaces.

Identification of bifurcation points leading to non-homogeneous solutions.

Abstract

We prove the existence of a homogeneous singular solution of the critical equation $$-\Delta u = u^{\frac{Q+2}{Q-2}}$$ on the Heisenberg group $H^n$, where $Q$ is the \textit{homogeneous dimension}. In order to do this, we introduce a suitable concept of normal curvature for hypersurfaces. Furthermore we study the bifurcation of non-homogeneous solutions from the homogeneous one.