, Volume 3, Issue 2, pp 153-193

Global Mizohata structures

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Abstract

The Mizohata partial differential operator is generalized to global structures on compact two-dimensional manifolds. A generalization of the Hopf Theorem on vector fields is used to show that a first integral can exist if and only if the genus is even. The Mizohata structures on the sphere are classified by the diffeomorphism group of the circle modulo the Moebius subgroup and a necessary and sufficient condition, expressed in terms of the associated diffeomorphism, is given for the existence of a first integral.

This work supported in part by NSF Grant DMS 9100383 to the Institute for Advanced Study.