Immersions minimales et immersions pluriharmoniques entre variétés riemanniennes: résultats de non existence et de rigidité. (Minimal and pluriharmonic immersions between Riemannian manifolds: non-existence and rigidity results)
The main aim of the authors is to generalize some known non-existence and rigidity theorems for minimal and pluriharmonic immersion. In particular, they generalize the following two results: Let \(M\) be a Kähler manifold of real dimension \(>2\). Then we have: (1) there do not exist minimal isometric immersions of \(M\) in a hyperbolic space form \(N\) and (2) each minimal isometric immersion of \(M\) in a complex hyperbolic space form \(N\) is holomorphic or anti-holomorphic. First, they take for \(M\) an even-dimensional Riemannian manifold equipped with a non-zero parallel two-form \(\alpha\) and for (1) \(N\) is a Riemannian manifold with strictly negative isotropic curvature. For (2), \(N\) is a Kähler manifold with strongly negative isotropic curvature. Similar non-existence and rigidity results are also known for pluriharmonic isometric immersions of Kähler manifolds \(M\) in real or complex space forms of positive sectional or holomorphic sectional curvature, respectively. These known results are also extended by the authors by considering weaker hypotheses, i.e., the existence of a non-zero parallel two-form \(\alpha\) as above and furthermore, the positivity of the curvature is replaced by the positivity of the isotropic curvature. Finally, they obtain also a further extension by replacing pluriharmonicity by what they call \(\alpha\)-pluriharmonicity. Their method and results about pluriharmonic immersions also lead to unicity and vanishing theorems for parallel or harmonic two-forms. To obtain these interesting results, the authors derive a Weitzenböck-type formula which involves the Dirac operator (acting on vector-valued forms).
scientific article; zbMATH DE number 1409070
scientific article