Morse Index Stability of Biharmonic Maps in Critical Dimension

Abstract In December 2022, Da Lio, Gianocca, and Rivière developed a new theory to prove the upper semi-continuity of the sum of the Morse index and the nullity in geometric analysis (Da Lio, Gianocca, and Rivière in Morse index stability for critical points to conformally invariant Lagrangians, 2023, arXiv:2212.03124 ), and applied it to conformally invariant problems in dimension 2—which include harmonic maps. Their method, quickly extended to Willmore immersions and later to other geometric settings, is applied in this article to the Morse stability of biharmonic maps in critical dimension 4. A key step in the proof is to obtain an energy quantization in $$L^{2,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> (the pre-dual of the Marcinkiewicz space $$L^{2,\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msup> </mml:math> of weakly squared-integrable functions), and we prove a general result to establish this strong energy quantization, which allows us to recover previous results in a unified fashion.