Liouville type theorem for several generalized maps between Riemannian manifold

TL;DR

This paper establishes Liouville theorems for various generalized harmonic maps between Riemannian manifolds, using monotonicity formulas and curvature bounds, under finite energy and asymptotic conditions.

Contribution

It derives new Liouville theorems for multiple types of generalized maps, including $$-$F$ harmonic and symphonic maps, on metric measure spaces, expanding the understanding of their rigidity properties.

Findings

Liouville theorems under finite energy conditions

Liouville theorems under asymptotic conditions

Liouville theorems involving curvature bounds

Abstract

In this paper, we mainly derive monotonicity formula of generalized map using conservation law, including $\phi$-$F$ harmonic map coupled with $\phi$-$F$ symphonic map with $m$ form and potential from metric measure space, $ p $ harmonic map with potential , $ V $ harmonic map with potential. As an corollary, we can derive Liouville theorem for these maps under some finite energy conditons. We also get Liouville type theorem for $\phi$-$F$ harmonic map coupled with $\phi$-$F$ symphonic map under asymptotic conditon on metric measure space. We also get Liouville theorem for $\phi$-$F$-$V$-harmonic maps in terms of the upper bound of Ricci curvature and the bound about sectional curvature on metric measure space. We also get Liouville theorem for $\phi$-$ F $-harmonic map without using monotonicity formula on metric measure space.