A class of harmonic submersions and minimal submanifolds (Q2706847)

The authors introduce the notion of a pseudo-horizontally homothetic map. This denotes a smooth map \(\varphi\) from a Riemannian manifold \(M\) to a Kähler manifold \(N\) such that \(d\varphi \circ d\varphi^\ast\) commutes with the Kähler structure \(J\) and such that NEWLINE\[NEWLINE d\varphi(\nabla_V d\varphi^\ast(JY)) = J d\varphi(\nabla_Vd\varphi^\ast(Y)) NEWLINE\]NEWLINE holds for all vector fields \(Y\) locally defined on \(N\) and all horizontal tangent vectors \(V\) on \(M\). This class includes holomorphic and antiholomorphic maps between Kähler manifolds. NEWLINENEWLINENEWLINEThe main result of the paper says that if \(\varphi\) is a pseudo-horizontally homothetic harmonic submersion and if \(P \subset N\) is a complex submanifold, then \(\varphi^{-1}(P)\) is a minimal submanifold of \(M\). This can for example be used to find minimal submanifolds of \(\mathbb{CP}^n\) which are not complex submanifolds.