## Abstract

For a Kähler manifold endowed with a weighted measure (Formula presented.), the associated weighted Hodge Laplacian (Formula presented.) maps the space of (Formula presented.)-forms to itself if and only if the (Formula presented.)-part of the gradient vector field (Formula presented.) is holomorphic. We use this fact to prove that for such (Formula presented.), a finite energy (Formula presented.)-harmonic function must be pluriharmonic. Motivated by this result, we verify that the same also holds true for(Formula presented.)-harmonic maps into a strongly negatively curved manifold. Furthermore, we demonstrate that such (Formula presented.)-harmonic maps must be constant if (Formula presented.) has an isolated minimum point. In particular, this implies that for a compact Kähler manifold admitting such a function, there is no nontrivial homomorphism from its first fundamental group into that of a strongly negatively curved manifold.

Original language | English (US) |
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Pages (from-to) | 893-911 |

Number of pages | 19 |

Journal | Mathematische Annalen |

Volume | 363 |

Issue number | 3-4 |

DOIs | |

State | Published - Dec 1 2015 |

### Bibliographical note

Funding Information:The first author is partially supported by NSF Grant No. DMS-1262140 and the second author by NSF Grant No. DMS-1105799.

Publisher Copyright:

© 2015, Springer-Verlag Berlin Heidelberg.

## Keywords

- 53C43
- 53C55