  The conformal parameterisation of a minimal surface is harmonic. Therefore, a
minimal surface is a critical point of both the energy functional and the area
functional. In this paper, we compare the Morse index of a minimal surface as a
critical point of the area functional with its Morse index as a critical point
of the energy functional. The difference between these indices is at most the
real dimension of Teichmuller space. This comparison allows us to obtain
surprisingly good upper bounds on the index of minimal surfaces of finite total
curvature in Euclidean space of any dimension. We also bound the index of a
minimal surface in an arbitrary Riemannian manifold by the area and genus of
the surface, and the dimension and geometry of the ambient manifold.
