On energy-critical half-wave maps into ${\mathbb{S}}^2$

Abstract

We consider the energy-critical half-wave maps equation $${\partial }_t\mathbf{u}+\mathbf{u}\wedge |\nabla |\mathbf{u}=0$$ 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 Möbius 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 $m\ge 1$. 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.

Slightly revised version with some typos fixed and added reference [19]. 50 pages. Comments are welcome

References in corpus (7)

• Sharp commutator estimates via harmonic extensions
• n/p-Harmonic maps: regularity for the sphere case
• Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps
• Near soliton evolution for equivariant Schroedinger Maps in two spatial dimensions
• Compactness and Bubbles Analysis for 1/2-harmonic Maps
• Nondegeneracy of half-harmonic maps from $\mathbb{R}$ into ${\mathbb{S}}^1$
• Horizontal $\alpha$-Harmonic Maps

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• Stationary states of the cubic conformal flow on ${\mathbb{S}}^3$
• Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation
• Global Well-Posedness For Half-Wave Maps With $S^2$ and ${\mathbb{H}}^2$ Targets For Small Smooth Initial Data
• On uniqueness for half-wave maps in dimension d >= 3
• Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data
• Non-degeneracy and quantitative stability of half-harmonic maps from $\mathbb{R}$ to $\mathbb{S}$
• Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics
• Small data global regularity for half-wave maps in $n=4$ dimensions