调和旋量的Calderón问题
The Calderón problem for harmonic spinors
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中文总结 AI 辅助
本文证明具有MIT边界条件的Dirac算子的边界共轭映射唯一确定紧致旋量流形的共形类，并推广到扭曲Dirac算子及实解析酉联络的局部规范等价性。
中文摘要 AI 辅助
我们证明，若边界紧致旋量流形的度量在某个坐标下局部共形于实解析度量，则其调和旋量的边界值唯一确定该流形的共形类。更精确地，我们考虑对应于MIT边界条件的Dirac算子的边界共轭（BC）映射，并证明若两个局部共形实解析（LCRA）度量在边界的某个开子集上具有等价的BC映射，则这两个流形是共形等价的。LCRA流形类包括共形Einstein流形以及所有光滑曲面。我们还证明，在维数大于2时，LCRA流形上的实解析酉联络由其扭曲Dirac算子的BC映射在局部规范等价意义下唯一确定。
英文摘要
We show that the conformal class of a compact spin manifold with boundary is uniquely determined by its boundary values of harmonic spinors if the metric is locally conformal to a real-analytic metric in some coordinates. More precisely, we consider the boundary conjugation (BC) map for the Dirac operator corresponding to MIT boundary conditions, and prove that if two locally conformally real-analytic (LCRA) metrics have equivalent BC maps over an open subset of the boundary, then the two manifolds are conformally equivalent. The class of LCRA manifolds includes conformally Einstein manifolds, as well as all smooth surfaces. We also show that a real-analytic unitary connection on an LCRA manifold is uniquely determined up to local gauge equivalence by the BC map of its twisted Dirac operator in dimensions greater than 2.
