Index estimates for sequences of harmonic maps

TL;DR

This paper establishes bounds on the index and nullity of sequences of harmonic maps from Riemann surfaces to compact manifolds, accounting for bubble formation and eigenfunction behavior.

Contribution

It introduces a novel diagonalization method of the index form that handles bubble development and eigenfunction convergence in harmonic map sequences.

Findings

Derived upper and lower bounds for index and nullity.

Analyzed eigenfunction convergence on bubbles and neck regions.

Extended methods to general conformally invariant variational problems.

Abstract

In this paper we study upper and lower bounds of the index and the nullity for sequences of harmonic maps with uniformly bounded Dirichlet energy from a two-dimensional Riemann surface into a compact target manifold. The main difficulty stems from the fact that in the limit the sequence can develop finitely many bubbles. We obtain the index bounds by studying the limiting behavior of sequences of eigenfunctions of the linearized operator and the key novelty of the present paper is that we diagonalize the index form of the Dirichlet energy with respect to a bilinear form which varies with the sequence of harmonic maps and which helps us to show the convergence of the sequence of eigenfunctions on the weak limit, the bubbles and the intermediate neck regions. Finally, we sketch how to modify our arguments in order to also cover the more general case of sequences of critical points of two-dimensional conformally invariant variational problems.