Publications [#9586] of Robert L Bryant

Papers Published

1. Holomorphic curves in Lorentzian CR-manifolds, Trans. Amer. Math. Soc., vol. 272 no. 1 (1982), pp. 203–221
   (last updated on 2004/04/06)

   Abstract:
   When can a real hypersurface in complex n-space contain any complex curves? Since the tangent spaces to such a curve would have to be null vectors for the Levi form, a necessary condition is that the Levi form have zeros. The simplest way this can happen in the non-degenerate case is for the Levi form to have the Lorentzian signature.

   In this paper, I show that a Lorentzian CR-manifold M has at most a finite parameter family of holomorphic curves, in fact, at most an n² parameter family if the dimension of M is 2n+1. This maximum is attained, as I show by example. When n=2, the only way it can be reached is for M to be CR-flat. In higher dimensions, where the CR-flat model does not achieve the maximum, it is still unknown whether or not there is more than one local model with the maximal dimension family of holomorphic curves.

   The technique used is exterior differential systems together with the Chern-Moser theory in the n=2 case.

   Reprints are available, but can also be downloaded from the AMS or from JSTOR