Stability of the morse index for the p-harmonic approximation of harmonic maps into homogeneous spaces

Abstract In the joint work of the author with Da Lio, F., and Rivière, T., Schlagenhauf, D.: (2025) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as $$p\searrow 2$$ p ↘ 2 . These are critical points of the energy $$E_p(u) {:}{=}\int _\Sigma \left( 1+\left| \nabla u\right| ^2\right) ^{p/2} \ dvol_\Sigma ,$$ E p ( u ) : = ∫ Σ 1 + ∇ u 2 p / 2 d v o l Σ , where $$u:\Sigma \rightarrow S^n$$ u : Σ → S n is a map from a closed Riemannian surface $$\Sigma $$ Σ into a sphere $$ S^n$$ S n . In this paper we extend the results found in Da Lio, F., Rivière, T., Schlagenhauf, D.: (2025) to the case of Sacks-Uhlenbeck sequences into homogeneous spaces, by incorporating the strategy introduced in Bayer, C., and Roberts, A.: (2025) . In the spirit of Da Lio, F., Rivière, T., Schlagenhauf, D.: (2025), we show in this setting the upper semicontinuity of the Morse index plus nullity and an improved pointwise estimate of the gradient in the neck regions around blow up points.