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On p-harmonic self-maps of spheres
Volker Branding1, Anna Siffert2
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Calculus of Variations and Partial Differential Equations
|May 1, 2023
Summary
Researchers proved the existence of infinite p-harmonic self-maps for spheres. For specific conditions, the identity map on spheres is shown to be equivariantly stable as a p-harmonic self-map.
Area of Science:
- Differential Geometry
- Geometric Analysis
- Topology
Background:
- P-harmonic maps are generalizations of harmonic maps, crucial in geometric analysis.
- Understanding self-maps between spheres is fundamental in topology and geometry.
- The stability of maps is a key concept for their classification and properties.
Purpose of the Study:
- To investigate the existence of rotationally p-harmonic maps between spheres.
- To analyze the spectral properties of the Jacobi operator for p-harmonic self-maps.
- To determine the stability of the identity map as a p-harmonic self-map.
Main Methods:
- Utilizing techniques from geometric analysis to study p-harmonic maps.
- Calculating and analyzing the spectrum of the Jacobi operator.
- Applying stability criteria for p-harmonic maps.
Main Results:
- Proving the existence of infinitely many p-harmonic self-maps of spheres for given p and specific dimensions.
- Explicitly determining the spectrum of the Jacobi operator for the identity map.
- Demonstrating the equivariant stability of the identity map as a p-harmonic self-map under certain conditions.
Conclusions:
- The study expands the understanding of p-harmonic maps in geometric analysis.
- The findings contribute to the classification and properties of self-maps on spheres.
- The stability results offer insights into the behavior of the identity map in the context of p-harmonicity.
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