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

2025/06/12 by Dominik Schlagenhauf, Schlagenhauf, Dominik · 1 citation
Mathematics · Engineering · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.10761

Abstract

In the joint work of the author with Da Lio and Rivière (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) := ∫_Σ( 1+|∇ u|²)^p/2 dvol_Σ, where u:Σ→ Sⁿ is a map from a closed Riemannian surface Σ into a sphere Sⁿ. 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 ε-harmonic and α-harmonic maps into homogeneous target manifolds). In the spirit of the work of Da Lio, Gianocca and Rivière (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.

Citations

• Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere
• Energy identity and no neck property for ε-harmonic and α-harmonic maps into homogeneous target manifolds
• Semi-Continuity of the Morse Index for Ricci Shrinkers
• Morse Index Stability for the Ginzburg-Landau Approximation
• Morse index stability for Yang-Mills connections
• Upper Semicontinuity of Index Plus Nullity for Minimal and CMC Hypersurfaces
• Morse Index Stability of Biharmonic Maps in Critical Dimension
• Conservation Laws for p-Harmonic Systems with Antisymmetric Potentials and Applications
• Weighted Eigenvalue Problems for Fourth-Order Operators in Degenerating Annuli
• Morse Index Stability of Willmore Immersions I
• Index estimates for sequences of harmonic maps
• Morse Index Stability for Critical Points to Conformally invariant Lagrangians
• Generalized neck analysis of harmonic maps from surfaces
• Min-max harmonic maps and a new characterization of conformal eigenvalues
• Higher order neck analysis of harmonic maps and its applications
• Lower Semi-Continuity of the Index in the Visosity Method for Minimal Surfaces
• Uniform Regularity Results for Critical and Subcritical Surface Energies
• Minimal two-spheres of low index in manifolds of positive complex sectional curvature
• Angular Energy Quantization for Linear Elliptic Systems with\n Antisymmetric Potentials and Applications
• Energy identity for approximations of harmonic maps from surfaces
• The existence of surfaces of constant mean curvature with free boundaries
• Harmonic Mappings of Riemannian Manifolds

Cited by

• Morse index stability for p-Yang-Mills connections

Related

• Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere
• Morse Index Stability of Biharmonic Maps in Critical Dimension
• Morse Index Stability for the Ginzburg-Landau Approximation
• Some properties of F-harmonic maps
• Morse index stability for p-Yang-Mills connections