Index of minimal spheres and isoperimetric eigenvalue inequalities

Abstract

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.