# Index of minimal spheres and isoperimetric eigenvalue inequalities

## TL;DR

This paper uses twistor theory to address isoperimetric inequalities and eigenvalue optimization on surfaces, extending classical results and improving bounds for minimal spheres and projective planes in spheres.

## Contribution

It introduces a new approach to isoperimetric inequalities via energy index and improves bounds on the area index of minimal surfaces in spheres.

## Key findings

- Maximization of the k-th eigenvalue on certain metrics
- Extension of results by Li, Yau, Nadirashvili, and Penskoi
- Improved upper bounds for area index of minimal surfaces

## 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 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.