The extended Morse index of harmonic maps is upper semicontinuous under degeneration of the domain, with nontrivial contribution from the Morse index of limiting geodesic segments in collapsed collars.
Jost, Compact Riemann surfaces, third edition, Universitext
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Improved Morse Index Stability for Sequences of Harmonic Maps from Degenerating Riemann Surfaces
The extended Morse index of harmonic maps is upper semicontinuous under degeneration of the domain, with nontrivial contribution from the Morse index of limiting geodesic segments in collapsed collars.