Reflection-function extension conjecture for CR mappings

Let M,MCnM,M'\subset\mathbb{C}^n, with n2n\geq 2, be globally minimal real analytic hypersurfaces. Let h:MMh:M\to M' be a continuous CR mapping whose holomorphic extension to a global one-sided neighborhood DD of MM has generic rank nn. The reflection function is the function associated with the real analytic hypersurface MM' and the mapping hh that encodes the complexification of the defining equations of MM' along the graph of hh. Reflection-function extension conjecture. The reflection function extends holomorphically to a neighborhood of every point p×h(p)p\times\overline{h(p)} in the graph of hˉ\bar h. This is presented as an open problem; the generic-rank assumption is necessary according to the example given in the source.

Sources & referencesView supporting material

Primary source

Joel Merker, “On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle”, arXiv:math/0403539 (2004).

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