Sharp quantitative stability of the Möbius group among sphere-valued maps in arbitrary dimension

Guerra, André; Lamy, Xavier; Zemas, Konstantinos

doi:10.1090/tran/9272

Sharp quantitative stability of the Möbius group among sphere-valued maps in arbitrary dimension
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by André Guerra, Xavier Lamy and Konstantinos Zemas;

Trans. Amer. Math. Soc. 378 (2025), 1235-1259

DOI: https://doi.org/10.1090/tran/9272

Published electronically: December 30, 2024

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Abstract:

In this work we prove a sharp quantitative form of Liouville’s theorem, which asserts that, for all $n\geq 3$, the weakly conformal maps of $\mathbb S^{n-1}$ with degree $\pm 1$ are Möbius transformations. In the case $n=3$ this estimate was first obtained by Bernand-Mantel, Muratov and Simon [Arch. Ration. Mech. Anal. 239 (2021), pp. 219-299], with different proofs given later on by Topping, and by Hirsch and the third author. The higher-dimensional case $n\geq 4$ requires new arguments because it is genuinely nonlinear: the linearized version of the estimate involves quantities which cannot control the distance to Möbius transformations in the conformally invariant Sobolev norm. Our main tool to circumvent this difficulty is an inequality introduced by Figalli and Zhang in their proof of a sharp stability estimate for the Sobolev inequality.

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Sharp quantitative stability of the Möbius group among sphere-valued maps in arbitrary dimension
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Abstract: