Explicit harmonic self-maps of complex projective spaces (Q6143860)

Harmonic maps between manifolds \(f:(M,g)\rightarrow (N,h)\) have been well-studied (beginning with Eels-Sampson) when \(N\) has non-positive sectional curvature. When \((N,h)\) has positive sectional curvature, the problem becomes tricky. In this paper, the authors consider the case when \(M=N=\mathbb{P}^n\) equipped with the Fubini-Study metric. They impose an \(SU(p+1)\times SU(n-p)\) equivariant symmetry and reduce the problem to an ODE. This ODE is proven to have solutions. In this way, an uncountable family of solutions is obtained of which roughly ``half'' are holomorphic and the other half are neither holomorphic nor anti-holomorphic (Theorem A). This family solutions converges (Theorem B). The holomorphic ones are shown to be weakly stable (in the sense that the second variation is non-negative) and the others are equivariantly weakly stable (Theorem C). The second-variation operator's spectrum is computed explicitly in Theorem D.