An example of $\alpha$-harmonic map sequence for which energy identity is not true

TL;DR

This paper constructs a specific sequence of $eta$-harmonic maps from a sphere into a Riemannian manifold where the expected energy conservation property fails, challenging existing assumptions in harmonic map theory.

Contribution

It provides a counterexample demonstrating that the energy identity does not always hold for $eta$-harmonic maps, highlighting limitations in current understanding.

Findings

Counterexample of energy identity failure

Sequence of $eta$-harmonic maps with bounded energy

Implications for harmonic map theory

Abstract

We construct a closed Riemannian manifold $(N,h)$ and a sequence of $\alpha$-harmonic maps from $S^2$ into $N$ with uniformly bounded energy such that the energy identity for this sequence is not true.