Index of minimal spheres and isoperimetric eigenvalue inequalities

• 成果类型: Article
• 署名作者: Karpukhin, Mikhail
• 署名单位: University of California System; University of California Irvine
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00992-5
• 发表日期: 2021
• 页码: 335-377
• 关键词: harmonic 2-spheres; 1st eigenvalue; jacobi fields; upper-bounds; immersions; laplacian; SURFACES; MAPS; s2; METRICS

摘要

In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres Sn. 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 Sn. 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.