Triharmonic isometric immersions into a manifold of non-positively constant curvature

TL;DR

This paper investigates triharmonic isometric immersions into non-positively curved space forms, proving that under certain finiteness conditions, such immersions must be minimal, extending understanding of harmonic map theory.

Contribution

It establishes a rigidity result for triharmonic isometric immersions into non-positively curved manifolds, showing they are minimal if specific energy conditions are met.

Findings

Triharmonic isometric immersions are minimal under finiteness conditions.

The study extends harmonic map theory to triharmonic maps in non-positive curvature.

Provides conditions ensuring minimality of triharmonic immersions.

Abstract

A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study a triharmonic isometric immersion into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-enegy and the L^4-norm of the tension field are finite, then such an immersion is minimal.