Improved Morse Index Stability for Sequences of Harmonic Maps from Degenerating Riemann Surfaces

TL;DR

This paper investigates the stability of the extended Morse index for harmonic maps on degenerating Riemann surfaces, addressing analytical challenges from collar collapse and identifying conditions for index semicontinuity.

Contribution

It refines previous spectral control results by explicitly analyzing geodesic contributions in degenerating collars, improving understanding of Morse index behavior.

Findings

Established upper semicontinuity conditions for the extended Morse index.

Identified the nontrivial contribution of geodesic segments to the limiting index.

Provided sharper spectral estimates on degenerating domains.

Abstract

We study the stability of the extended Morse index, defined as the number of negative and zero eigenvalues of the Jacobi operator, for sequences of harmonic maps on degenerating Riemann surfaces. As the conformal structure approaches the boundary of moduli space, collar collapse creates major analytical challenges. We analyze the second variation of the energy under these degenerations and identify conditions ensuring upper semicontinuity of the extended Morse index. Refining earlier results of the first and second authors in [7], we obtain sharper control of the spectrum of the Jacobi operator on degenerating domains. A key new aspect is the explicit contribution of geodesics arising as limits of the images of degenerating collars. We show that these neck regions converge to geodesic segments whose Morse index contributes nontrivially to the limiting extended index.