Non-linear elliptic operators on a compact manifold and an implicit function theorem

Abstract

Many problems in differential and analytic geometry seem to have something to do with the study of non-linear partial differential equations of elliptic type. For instance, the classical Weyl and Minkowski problems for a convex surface have been studied by H. Weyl, H. Lewy, and L. Nirenberg using the iteration method for the construction of solutions of certain non-linear equations of elliptic type (see [6]). Also, M. Kuranishi [3] constructed the effective complete family of deformations of complex analytic structures on a given compact complex manifold as the solution space of another non-linear equation of elliptic type; whereby the basic idea in his work is to apply an implicit function theorem to the non-linear operator of a Banach space, and to construct the bifurcation of solutions explicitly.