On the integrability of Mizohata structures on the sphere \(\mathbb{S}^2\)
Let \(M\) be a two-dimensional \(C^\infty\) manifold and let \(N\) be a \(C^\infty\) one-dimensional subbundle of the complexified tangent bundle \({\mathbb C}TM.\) The integrability problem for \(N\) asks for a smooth function \(f\) on \(M\) such that \(df\) generates the orthogonal (in the duality given by tangent and cotangent vectors) bundle \(N^\perp.\) Since Nirenberg's famous example, it is known that such a problem is not always (even locally) solvable. The simplest structures for which \(f\) might not exist are the so-called Mizohata structure, i.e. structures for which whenever \(p\) is in the characteristic set \(\Sigma:=\{p\in M\); \(N_p=\bar N_p\},\) then \([N_p,\bar N_p]\not=\{0\}.\) An important class of Mizohata structures, introduced by Treves, are the ones called ``standard''. \textit{F. Treves} [Hypo-analytic structures: local theory, Princeton Mathematical Series, Princeton, NJ: Princeton University Press (1992; Zbl 0787.35003)] proved that locally an integrable Mizohata structure is standard. This means that at each given point \(p\in\Sigma,\) where \(N\) has a local integral, the bundle \(N^\perp\) is generated over a nighborhood of \(p\) by the differential of a simple fold (a function of the form \(x+iy^2\) in suitable coordinates centered at \(p\)). \textit{J. Hounie} [Trans. Am. Math. Soc. 334, No. 2, 641-649 (1992; Zbl 0771.58044)] proved the standardness of a particular class of structures on the sphere \({\mathbb S}^2.\) On the other hand, \textit{H. Jacobowitz} proved [J. Geom. Anal. 3, No. 2, 153-193 (1993; Zbl 0777.58036)] the existence of non-standard globally integrable structures on \({\mathbb S}^2\) with \(\Sigma\) connected. In this paper, the author studies locally integrable Mizohata structures \(({\mathbb S}^2,N)\) for which \(\Sigma\) is the equator of \({\mathbb S}^2.\) Such a structure is elliptic in the upper and lower hemisphere of \({\mathbb S}^2.\) In Section 2 the author, after recalling some basic definitions, gives two criterions for standardness. In Section 3 he gives a number of examples of structures without global first integrals, but with a new type of integral, called by the author weak integral. By that, the author means locally integrable structures on \({\mathbb S}^2\) which have a global integral \(f\) everywhere, except for finitely many points, where either \(df\) vanishes or \(f\) has polar or transcendental singularities. He proves in Section 4 that in order for the structure to have a (weak) \(C^\infty\) integral, the analytic continuation of a suitable orientation preserving diffeomorphism \(\alpha\) (defined on \({\mathbb S}^1\) through the uniformization theorem) must have at most a finite number of algebraic singularities in the unit disc \(\Delta.\) In the last section, he concentrates on a class of structures for which \(\alpha\) is an \(n\)th root of a product of \(n\) Möbius transformations. For such structures, he proves the existence of a (weak) minimal \(C^\infty\) integral \(F_\alpha\) (by minimal the author means that any other \(C^\infty\) integral \(F\) can be written as \(h\circ F_\alpha,\) where \(h\) is a continuous map on the range of \(F_\alpha\) and holomorphic in the interior).
scientific article; zbMATH DE number 1542092