Morse Index Stability of Biharmonic Maps in Critical Dimension

Abstract

Furthering the development of Da Lio-Gianocca-Rivi\`ere's Morse stability theory (arXiv:2212.03124) that was first applied to harmonic maps between manifolds and later extended to the case of Willmore immersions (arXiv:2306.04608-04609), we generalise the method to the case of (intrinsic or extrinsic) biharmonic maps. In the course of the proof, we develop a novel method to prove strong energy quantization (in the space of squared-integrable functions that corresponds to the pre-dual of the Marcinkiewicz space of weakly squared-integrable functions) in a wide class of problems in geometric analysis, which allows us to recover some previous results in a unified fashion.