Index and nullity of minimal surfaces via representation theory

Abstract: We develop a representation-theoretic approach to determine the Morse index and nullity of an embedded minimal surface based only on its topology and symmetry group. The method combines an equivariant comparison between the second variations of area and energy with the representation theory of the dihedral group. In the Euclidean unit ball we compute for every $k\geq3$ the Morse index and nullity of any free boundary minimal $k$-noid with prismatic symmetry of order $4k$, and of any free boundary minimal surface of genus one with connected boundary and antiprismatic symmetry of order eight. In the round sphere we obtain an alternative computation for the Morse index and nullity for the first family of the Lawson surfaces, using only half of their symmetry group. The results are independent of any particular construction of the surfaces.