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 $$p\searrow 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>↘</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . These are critical points of the energy $$E_p(u) {:}{=}\int _\Sigma \left( 1+\left| \nabla u\right| ^2\right) ^{p/2} \ dvol_\Sigma ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>Σ</mml:mi> </mml:msub> <mml:msup> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msup> <mml:mfenced> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>v</mml:mi> <mml:mi>o</mml:mi> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>Σ</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where $$u:\Sigma \rightarrow S^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Σ</mml:mi> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> is a map from a closed Riemannian surface $$\Sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Σ</mml:mi> </mml:math> into a sphere $$ S^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . 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.