A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies

TL;DR

This paper establishes a Liouville-type theorem for biharmonic maps under specific geometric conditions, showing that small energy and bounded tension field imply the map is harmonic.

Contribution

It proves a new Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with curvature bounds and small energies, extending previous results.

Findings

Biharmonic maps with small energy are necessarily harmonic under certain geometric conditions.

The theorem applies to manifolds with Ricci curvature bounds and positive injectivity radius.

The tension field's $L^p$-norm being bounded is crucial for the result.

Abstract

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the $L^p$-norm of the tension field is bounded and the $n$-energy of the map is sufficiently small then every biharmonic map must be harmonic, where \(2<p<n\).