  In the present paper we use twistor theory in order to solve two problems
related to harmonic maps from surfaces to Euclidean spheres $\mathbb{S}^n$.
First, we propose a new approach to isoperimetric inequalities based on energy
index. Using this approach we show that for any positive $k$, the $k$-th
non-zero eigenvalue of the Laplacian on the real projective plane endowed with
a metric of unit area, is maximized on the sequence of metrics converging to a
union of $(k-1)$ identical copies of round sphere and a single round projective
plane. This extends the results of P. Li and S.-T. Yau for $k=1$ (1982); N.
Nadirashvili and A. Penskoi for $k=2$ (2018); and confirms the conjecture made
in [KNPP]. Second, we improve the known upper bounds for the area index of
minimal two-dimensional spheres and minimal projective planes in
$\mathbb{S}^n$. In the course of the proof we establish a twistor
correspondence for Jacobi fields, which could be of independent interest for
the study of moduli space of harmonic maps.
