On the stability of the generalized equator map

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}.