  The goal of the present paper is to investigate the algebraic structure of
global conformal invariants of submanifolds. These are defined to be
conformally invariant integrals of geometric scalars of the tangent and normal
bundle. A famous example of a global conformal invariant is the Willmore energy
of a surface. In codimension one we classify such invariants, showing that
under a structural hypothesis (more precisely we assume the integrand depends
separately on the intrinsic and extrinsic curvatures, and not on their
derivatives) the integrand can only consist of an intrinsic scalar conformal
invariant, an extrinsic scalar conformal invariant and the Chern-Gauss-Bonnet
integrand. In particular, for codimension one surfaces, we show that the
Willmore energy is the unique global conformal invariant, up to the addition of
a topological term (the Gauss curvature, giving the Euler Characteristic by the
Gauss Bonnet Theorem). A similar statement holds also for codimension two
surfaces, once taking into account an additional topological term given by the
Chern-Gauss-Bonnet integrand of the normal bundle. We also discuss existence
and properties of natural higher dimensional (and codimensional)
generalizations of the Willmore energy.
