On deformation of solutions of involutive partial differential equations

Let E, F be real analytic vector bundles over a real analytic compact manifold M and D: F(E)-+r(F) a real analytic polynomial differential operator satisfying D(0) = 0. Let s(i) be a parametrized family of cross sections of E, where t moves in some neighborhood of 0 in a euclidean space. We say that s(i) is a deformation of the solution 0 if s(0) = 0 and D(s(0) = 0. In the present paper, we will show the existence of deformations of the solution 0 under some conditions. Namely, let L be the linearized differential operator of D at 0 and assume: (1) the equation D(s) = 0 is involutive, (2) L is elliptic, (3) H(M9 0) = 0, where 0 is the solution sheaf of the equation L(s) = 0. Then we can prove that there is a deformation 5(0 which is complete at t = 0 in an appropriate sense. (Theorem 1, 2) We would like to point out the analogy between the above result and a theorem in [3] on the existence of deformations of complex structures. In fact, the arguments proceed along almost the same line as in [3]. In § 1, we prove some propositions which are needed in the later sections. In §2, we construct the deformation s(0 and prove its completeness in §3. Finally the author wishes to express his hearty gratitude to Professor N. Tanaka for his constant encouragement and valuable suggestions during the preparation of this paper.