Some Remarks on Energy inequalities for harmonic maps with potential

TL;DR

This paper explores how classical results on nonlinear Poisson equations, such as gradient estimates and Liouville theorems, can be extended to harmonic maps with potential between Riemannian manifolds, under certain curvature and energy conditions.

Contribution

It generalizes key qualitative results from nonlinear Poisson equations to harmonic maps with potential, broadening their applicability in geometric analysis.

Findings

Gradient estimates for harmonic maps with potential

Monotonicity formulas under curvature assumptions

Liouville theorems for harmonic maps with potential

Abstract

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas and Liouville theorems under curvature and energy assumptions.