On energy-critical half-wave maps into

• 成果类型: Article
• 署名作者: Lenzmann, Enno; Schikorra, Armin
• 署名单位: University of Basel; Pennsylvania Commonwealth System of Higher Education (PCSHE); University of Pittsburgh
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0785-1
• 发表日期: 2018
• 页码: 1-82
• 关键词: harmonic maps; blow-up; 1/2-harmonic maps; free-boundary; REGULARITY; systems; SINGULARITIES; compactness; DYNAMICS; equation

摘要

We consider the energy-critical half-wave maps equation . 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 . In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Mobius group on the target applied to half-harmonic maps from to . Complementing our classification result, we carry out a detailed analysis of the linearized operator L around half-harmonic maps with arbitrary degree and consisting of m identical Blaschke factors. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of . Moreover, we give a full description of the spectrum of L by finding all its -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 to the unit circle . 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.