Liouville-type results for stationary maps of a class of functional related to pullback metrics

TL;DR

This paper investigates stationary maps derived from a generalized functional related to pullback metrics, establishing monotonicity formulas, Liouville-type results, and boundary value problem analyses within Riemannian geometry.

Contribution

It introduces a new functional linked to pullback metrics, derives related monotonicity formulas, and proves Liouville-type theorems and boundary value results for tension field equations.

Findings

Derived first variation formula for stationary maps

Established monotonicity formulas using stress-energy tensor

Proved Liouville-type results and analyzed boundary value problems

Abstract

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress-energy tensor which is naturally linked to conservation law and yield monotonicity formula via the coarea formula and comparison theorem in Riemannian geometry. A version of this monotonicity inequalities enables us to derive some Liouville type results. Also we investigate the constant Dirichlet boundary value problems and the generalized Chern type results for tension field equation with respect to this functional.