On the local solvability of elliptic equations on compact manifolds (Q954974)

The interesting paper under review deals with local solvability of nonlinear elliptic equations over compact manifolds. Precisely, the local image of a nonlinear elliptic operator over 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. In case this is not verified, the author exhibits a higher order infinitesimal invariant, the epidimension, which forces the number of independent equations to decrease. It is shown 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. Moreover, it is shown that, in general, a local nonlinear version of the Fredholm scheme, which always exists, encodes the maximal number of independent equations. Finally, the underdetermined elliptic case is discussed and a conjecture for it is stated.