Non-degeneracy and quantitative stability of half-harmonic maps from ${\mathbb R}$ to ${\mathbb S}$

DOI10.1016/J.AIM.2023.108979arXiv2107.08521MaRDI QIDQ6373094

Publication date: 18 July 2021

Abstract: We consider half-harmonic maps from

m a t h b b R

(or

m a t h b b S

) to

m a t h b b S

. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A full description of the kernel of the linearized operator around each half-harmonic map is given. The second part of this paper devotes to studying the quantitative stability of half-harmonic maps. When its degree is

p m 1

, we prove that the deviation of any map

from M"obius transformations can be controlled uniformly by

. This result resembles the quantitative rigidity estimate of degree

p m 1

harmonic maps

m a t h b b R^{2} o m a t h b b S^{2}

which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if

is already near to a Blaschke product, then the deviation of

to Blaschke products can be controlled by

. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all

of degree 2. We conjecture similar things happen for harmonic maps

{m a t h b b R}^{2} o {m a t h b b S}^{2}

.

Mathematics Subject Classification ID

Harmonic maps, etc. (58E20) Critical points of functionals in context of PDEs (e.g., energy functionals) (35B38) Symmetries, invariants, etc. in context of PDEs (35B06)

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