Stability of the Morse Index for the $p$-harmonic Approximation of Harmonic Maps into Homogeneous Spaces

TL;DR

This paper extends the stability analysis of the Morse index from sphere targets to homogeneous spaces for sequences of p-harmonic maps, providing new insights into their behavior near blow-up points.

Contribution

It generalizes previous stability results to homogeneous spaces and introduces improved gradient estimates in neck regions around blow-up points.

Findings

Morse index plus nullity are upper semicontinuous in this setting.

Provides improved pointwise gradient estimates near blow-up points.

Extends stability results from spheres to homogeneous target spaces.

Abstract

In the joint work of the author with Da Lio and Rivi\`ere (Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as $p\searrow2$. These are critical points of the energy $E_p(u) := \int_\Sigma \left( 1+|\nabla u|^2\right)^{p/2} \ dvol_\Sigma,$ where $u:\Sigma \rightarrow S^n$ is a map from a closed Riemannian surface $\Sigma$ into a sphere $ S^n$. In this paper we extend the results found in our previous work to the case of Sacks-Uhlenbeck sequences into homogeneous spaces, by incorporating the strategy introduced by Bayer and Roberts (Energy identity and no neck property for $\epsilon$-harmonic and $\alpha$-harmonic maps into homogeneous target manifolds). In the spirit of the work of Da Lio, Gianocca and Rivi\`ere (Morse Index Stability for Critical Points to Conformally invariant Lagrangians), 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.