On the local solvability of elliptic equations on compact manifolds

Abstract

The local image of a nonlinear elliptic operator on a compact manifold is a submanifold described by a full set of independent equations if and only if the corank of the linearized operator is constant. When not so, we exhibit a higher order infinitesimal invariant, the epidimension, which forces the number of independent equations decrease. We show that the epidimension of a natural operator with enough symmetry must either vanish or be maximal, in which case the local image admits no equation. In general, we show that a local nonlinear version of Fredholm's scheme, which always exists, encodes the maximal number of independent equations. Finally, we take a glimpse at the underdetermined elliptic case and state a conjecture for it.