New examples of r-harmonic immersions into the sphere

TL;DR

This paper constructs new examples of proper r-harmonic immersions into spheres, including canonical inclusions and generalized Clifford's tori, expanding the understanding of polyharmonic maps.

Contribution

It provides explicit constructions of proper r-harmonic submanifolds in spheres, notably characterizing when canonical inclusions are proper r-harmonic and demonstrating existence of generalized Clifford's tori.

Findings

The canonical inclusion S^{n-1}(R) into S^n is proper r-harmonic if and only if R=1/√r.

Existence of proper r-harmonic generalized Clifford's tori in spheres.

New explicit examples of r-harmonic immersions into spheres.

Abstract

Polyharmonic, or $r$-harmonic, maps are a natural generalization of harmonic maps whose study was proposed by Eells-Lemaire in 1983. The main aim of this paper is to construct new examples of proper $r$-harmonic immersions into spheres. In particular, we shall prove that the canonical inclusion $i: S^{n-1}(R)\to S^n$ is a proper $r$-harmonic submanifold of $S^n$ if and only if the radius $R$ is equal to $1/ \sqrt{r}$. We shall also prove the existence of proper $r$-harmonic generalized Clifford's tori into the sphere.