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 eigenvalue 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 Li and Yau (Invent Math 69(2):269–291, 1982) for $$k=1$$ ; Nadirashvili and Penskoi (Geom Funct Anal 28(5):1368–1393, 2018) for $$k=2$$ ; and confirms the conjecture made in (KNPP). Second, we improve the known lower 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 spaces of harmonic maps.