## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 45, Number 2 (2001), 603-613.

### Quasiconformal harmonic maps into negatively curved manifolds

#### Abstract

Let $F:M\to N$ be a harmonic map between complete Riemannian manifolds. Assume that $N$ is simply connected with sectional curvature bounded between two negative constants. If $F$ is a quasiconformal harmonic diffeomorphism, then $M$ supports an infinite dimensional space of bounded harmonic functions. On the other hand, if $M$ supports no non-constant bounded harmonic functions, then any harmonic map of bounded dilation is constant.

#### Article information

**Source**

Illinois J. Math., Volume 45, Number 2 (2001), 603-613.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138358

**Digital Object Identifier**

doi:10.1215/ijm/1258138358

**Mathematical Reviews number (MathSciNet)**

MR1878621

**Zentralblatt MATH identifier**

0993.58012

**Subjects**

Primary: 58E20: Harmonic maps [See also 53C43], etc.

Secondary: 53C43: Differential geometric aspects of harmonic maps [See also 58E20]

#### Citation

Donnelly, Harold. Quasiconformal harmonic maps into negatively curved manifolds. Illinois J. Math. 45 (2001), no. 2, 603--613. doi:10.1215/ijm/1258138358. https://projecteuclid.org/euclid.ijm/1258138358