Asymptotic properties of energy of harmonic maps on asymptotically hyperbolic manifolds

TL;DR

This paper investigates the asymptotic behavior of harmonic map energy on asymptotically hyperbolic manifolds, demonstrating conditions under which such maps must be constant based on energy growth and boundary approach rates.

Contribution

It establishes new criteria linking energy finiteness and boundary approach rates to the constancy of harmonic maps on asymptotically hyperbolic manifolds.

Findings

Harmonic maps with finite energy are necessarily constant.

Rapid boundary approach implies harmonic map is constant.

Energy growth conditions determine map behavior at infinity.

Abstract

Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in terms of a defining function for the boundary.