Polyharmonic surfaces in $3$-dimensional homogeneous spaces

TL;DR

This paper classifies specific polyharmonic surfaces in 3D homogeneous spaces, proving properties of triharmonic and r-harmonic Hopf cylinders, and introduces new examples of such surfaces with constant mean curvature.

Contribution

It provides a comprehensive classification of proper triharmonic isoparametric surfaces and r-harmonic Hopf cylinders in BCV-spaces, revealing new examples and properties.

Findings

Triharmonic Hopf cylinders are necessarily CMC.

Complete classification of CMC r-harmonic Hopf cylinders for r≥3.

Existence of new r-harmonic surfaces in BCV-spaces.

Abstract

In the first part of this paper we shall classify proper triharmonic isoparametric surfaces in 3-dimensional homogeneous spaces (Bianchi-Cartan-Vranceanu spaces, shortly BCV-spaces). We shall also prove that triharmonic Hopf cylinders are necessarily CMC. In the last section we shall determine a complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces, r>=3. This result ensures the existence, for suitable values of r, of an ample family of new examples of r-harmonic surfaces in BCV-spaces.