The refined analysis on the convergence behavior of harmonic map sequence from cylinders (Q1930285)

There has been a lot of studies concerning the bubbling off behaviour of sequences of harmonic maps. In [Math.\ Z.\ 264, No.\ 1, 63--85 (2010; Zbl 1213.53086)], \textit{M. Zhu} studied sequences of harmonic maps of uniformly bounded energy from domains degenerating in the limit. The well-known ``no neck'' property does not hold here, and there can be energy vanishing which is not to be found in bubbles. The energy loss, however, can be accounted for by the length of geodesics that can be identified as a sort of blowup limit. In the current paper, the authors study the model case of the situation described above. In order to understand the phenomenon, the domains to consider are cylinders with length converging to \(\infty\). For this situation, assuming that no bubbling occurs, a length formula for the limit geodesics in the target manifold is proven. Moreover, a geometric interpretation of Zhu's generalized energy identity is given.