Spectral geometry of the second variation operator of harmonic maps (Q1102559)

The inverse spectral problem of the Hessian, i.e. the Jacobi operator \(J_{\phi}\) of the energy of a harmonic map \(\phi\) is dealt with. For the spectral geometry for the Laplace-Beltrami operator of a compact Riemannian manifold, several interesting geometric results are obtained. Analogously, using the data of the spectrum \(Spec(J_{\phi})\) of \(J_{\phi}\) of a harmonic map \(\phi\), typical harmonic maps, i.e., constant maps, geodesics, isometric minimal immersions, harmonic Riemannian submersions are characterized. For example, let \(\phi\), \(\phi\) ' be harmonic maps of (S 2,can) into the complex projective space (P n(\({\mathbb{C}}),h)\) with the Fubini-Study metric h. Assume that \(Spec(J_{\phi})=Spec(J_{\phi '})\). If \(\phi\) is a holomorphic isometric immersion, then \(\phi '=\phi\) up to an isometry of (P n(\({\mathbb{C}}),h)\).