The index of isolated critical points and solutions of elliptic equations in the plane

The purpose of the present paper is to point out how statements about geometric quantities associated with solutions of elliptic equations can be derived from basic facts of differential topology like index calculus and the Gauss-Bonnet theorem. We will focus our attention on two-dimensional problems. The results we obtain are of two different kinds: (i) Identities or estimates relating the number and character of critical points (i.e. zeroes of the gradient) of solutions of elliptic equations with the boundary data; (ii) differential identities on the gradient length and the curvatures of the level curves and of the curves of steepest descent of an arbitrary smooth function with isolated critical points.