Unstable harmonic maps into real hypersurfaces of a complex Hopf manifold

Abstract

Let $\phi: M\rightarrow N$ be a pseudohermitian immersion ([6]) of a compact strictly pseudoconvex $CR$ manifold $M$ into a totally umbilical real hypersurface $N$, of nonzero mean curvature of $(\Vert H\Vert\neq 0)$, of a complex Hopf manifold $\bm{C}H^{n}$, tangent to the Lee field $B_{0}$ of $\bm{C}H^{n}$. If $B_{0}$ is orthogonal to the $CR$ structure of $N$ and $E(\phi) \gt \mathop{V\!ol}(M)/[(1+\Vert H\Vert^{2})\Vert H\Vert^{2}]$ then $\phi$ is an unstable harmonic map.