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Register a new userNon-degeneracy and quantitative stability of half-harmonic maps from ${mathbb R}$ to ${mathbb S}$
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We consider half-harmonic maps from $mathbb{R}$ (or $mathbb{S}$) to $mathbb{S}$. 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 $pm 1$, we prove that the deviation of any map $boldsymbol{u}:mathbb{R}to mathbb{S}$ from Mobius transformations can be controlled uniformly by $|boldsymbol{u}|_{dot H^{1/2}(mathbb{R})}^2-deg boldsymbol{u}$. This result resembles the quantitative rigidity estimate of degree $pm 1$ harmonic maps $mathbb{R}^2to mathbb{S}^2$ which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if $boldsymbol{u}$ is already near to a Blaschke product, then the deviation of $boldsymbol{u}$ to Blaschke products can be controlled by $|boldsymbol{u}|_{dot H^{1/2}(mathbb{R})}^2-deg boldsymbol{u}$. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all $boldsymbol{u}$ of degree 2. We conjecture similar things happen for harmonic maps ${mathbb R}^2to {mathbb S}^2$.
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We consider the energy-critical half-wave maps equation $$partial_t mathbf{u} + mathbf{u} wedge | abla| mathbf{u} = 0$$ for $mathbf{u} : [0,T) times mathbb{R} to mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $mathbb{S}^2$. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Mobius group on the target $mathbb{S}^2$ applied to half-harmonic maps from $mathbb{R}$ to $mathbb{S}^2$. Complementing our classification result, we carry out a detailed analysis of the linearized operator $L$ around half-harmonic maps $mathbf{Q}$ with arbitrary degree $m geq 1$. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of $mathbf{Q}$. Moreover, we give a full description of the spectrum of $L$ by finding all its $L^2$-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for $L$ and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on $mathbb{R}$ to the unit circle $mathbb{S}$. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.
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