Karpukhin, Mikhail; (2021) Index of minimal spheres and isoperimetric eigenvalue inequalities. Inventiones mathematicae , 223 (1) pp. 335-377. 10.1007/s00222-020-00992-5.

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Abstract

In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres 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 lower bounds for the area index of minimal two-dimensional spheres and minimal projective planes in 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.

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