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

Comparison between second variation of area and second variation of energy of a minimal surface