The stress-energy tensor for biharmonic maps

TL;DR

This paper investigates the stress-energy tensor related to the bienergy functional for biharmonic maps, highlighting unique features in four dimensions, constructing examples, and classifying maps based on tensor properties.

Contribution

It introduces a variational approach to the stress-energy tensor for biharmonic maps, emphasizing dimension four and providing classifications and new examples.

Findings

Dimension four exhibits unique properties for the stress-energy tensor.

Constructed new biharmonic maps using the stress-energy tensor.

Classified maps with vanishing or parallel stress-energy tensor.

Abstract

Using Hilbert's criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric.