Energy identity of approximate biharmonic maps to Riemannian manifolds and its application

TL;DR

This paper establishes an energy identity for approximate biharmonic maps in four dimensions, accounting for energy loss through finitely many biharmonic bubbles, with applications to the heat flow at infinity.

Contribution

It proves a new energy identity for approximate biharmonic maps with bounded bi-tension fields, extending understanding of energy concentration and loss in higher-order geometric flows.

Findings

Energy identity accounts for energy loss via biharmonic bubbles.

Applicable to heat flow of biharmonic maps at large times.

Provides a framework for analyzing energy concentration in biharmonic maps.

Abstract

We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in $L^p$ for $p>\frac43$. 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 $\mathbb R^4$. As a corollary, we obtain an energy identity for the heat flow of biharmonic maps at time infinity.