Infinite families of harmonic self-maps of ellipsoids in all dimensions

TL;DR

This paper demonstrates the existence of infinitely many harmonic self-maps on certain ellipsoids in all dimensions, expanding understanding of harmonic maps in geometric analysis.

Contribution

It establishes conditions under which ellipsoids admit infinitely many harmonic self-maps, providing explicit criteria based on ellipsoid parameters.

Findings

Existence of infinitely many harmonic self-maps for specific ellipsoids.

Conditions relating ellipsoid parameters to harmonic map existence.

Extension of harmonic map theory to a broad class of ellipsoids.

Abstract

We prove that for given $k\in\mathbb{N}$, $k\geq 3$, $d\in\mathbb{N}$ and each $a\in\mathbb{R}^{*}$ with \begin{align*} a^2<4d (d+k-2)(k-2)^{-2} \end{align*} the ellipsoid $E_a:=\{x\in\mathbb{R}^k\,\lvert\,a^{-2}x_1^2+x_2^2+\ldots+x_k^2=1\}$ admits infinitely many harmonic self-maps.