The Mizohata structure on the sphere

Abstract:

We prove that a compact surface that admits a Mizohata structure is (homeomorphic to) a sphere and that there exists exactly one such structure $\mathcal {L}$ up to conjugation by diffeomorphisms. We also characterize the range and the kernel of the operator ${\delta _0}$ induced by $\mathcal {L}$ , i.e., obtained from the exterior derivative on functions by passing to the quotient modulo ${\mathcal {L}^\bot }$ .