On the stability of the generalized equator map

TL;DR

This paper investigates the stability of a generalized equator map, which is a critical point of various energy functionals between Riemannian manifolds, extending classical stability results to new energy types.

Contribution

It demonstrates that the generalized equator map is a critical point of the extrinsic $k$-energy and provides a detailed stability analysis for this map across multiple energy functionals.

Findings

Generalized equator map is a critical point of extrinsic $k$-energy.

Stability analysis extends classical results to new energy functionals.

Provides conditions under which the map is stable or unstable.

Abstract

The energy, the $p$-energy ($p\in\mathbb{R}$ with $p\geq 2$) and the extrinsic $k$-energy ($k\in\mathbb{N}$) for maps between Riemannian manifolds are central objects in the geometric calculus of variations. The equator map from the unit ball to the Euclidean sphere provides an explicit critical point of all aforementioned energy functionals. During the last four decades many researchers studied the stability of this particular map when considered as a critical point of one of these energy functionals, see e.g. \cite{MR4436204}, \cite{MR705882}. Recently, Nakauchi \cite{MR4593065} introduced a generalized radial projection map and proved that this map is both a critical point of the energy and a critical point of the $p$-energy. This generalized radial projection map gives rise to a generalized equator map which is also both a critical point of the energy and a critical point of the $p$-energy. In this manuscript we first of all show that the generalized equator map is also a critical point of the extrinsic $k$-energy. Then, the main focus is a detailed stability analysis of this map, considered as a critical point of both the extrinsic $k$-energy and the $p$-energy. We thus establish a number of interesting generalizations of the classical (in)stability results of J\"ager and Kaul \cite{MR705882}.