UNIQUENESS IN ROUGH ALMOST COMPLEX STRUCTURES, AND DIFFERENTIAL INEQUALITIES

The study of J-holomorphic maps leads to the consideration of the inequations vertical bar partial derivative u/partial derivative z vertical bar <= C vertical bar u vertical bar, and vertical bar partial derivative u/partial derivative z vertical bar <= epsilon vertical bar partial derivative u/partial derivative z vertical bar. The first inequation is fairly easy to use. The second one, that is relevant to the case of rough structures, is more delicate. The case of a vector valued is strikingly different from the scalar valued case. Unique continuation and isolated zeroes are the main topics under study. One of the results is that, in almost complex structures of Holder class 1/2 any J-holomorphic curve that is constant on a non-empty open set, is constant. This is in contrast with immediate examples of non-uniqueness.