Energy identity for a class of approximate biharmonic maps into sphere in dimension four

TL;DR

This paper establishes an energy identity for approximate biharmonic maps into spheres in four dimensions, accounting for energy loss via finitely many biharmonic bubbles, under bounded bi-tension fields.

Contribution

It proves an energy identity for approximate biharmonic maps into spheres in dimension four with bounded bi-tension fields, extending understanding of energy concentration phenomena.

Findings

Energy identity accounts for Hessian energy loss.

Identifies finitely many biharmonic maps as bubbles.

Applicable to sequences with bounded bi-tension fields.

Abstract

We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in $L^p$ for some $p>1$. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb R^4$.