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Polyharmonic surfaces in $3$-dimensional homogeneous spaces

Stefano Montaldo, Cezar Oniciuc, Andrea Ratto

TL;DR

This work investigates tri-harmonic and more general polyharmonic surfaces in 3D Bianchi-Cartan-Vranceanu (BCV) spaces M^3_{m,\ell}. Leveraging the Hopf-cylinder construction and the isoparametric surface framework, the authors prove that every triharmonic Hopf cylinder must have constant mean curvature (CMC) and completely classify proper CMC r-harmonic Hopf cylinders for r \ge 3, revealing new families of r-harmonic surfaces in BCV-spaces. They further show that proper triharmonic isoparametric surfaces in BCV-spaces (with 4m-\ell^2 \neq 0) are open parts of Hopf cylinders, while parabolic helicoids cannot be proper triharmonic. The results connect tri- and higher-order harmonicity with geometric constraints (CMC, minimality) and provide explicit parameter regimes determining the existence and multiplicity of proper r-harmonic Hopf cylinders, enriching the landscape of explicit examples in positively and non-positively curved BCV geometries.

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.

Polyharmonic surfaces in $3$-dimensional homogeneous spaces

TL;DR

This work investigates tri-harmonic and more general polyharmonic surfaces in 3D Bianchi-Cartan-Vranceanu (BCV) spaces M^3_{m,\ell}. Leveraging the Hopf-cylinder construction and the isoparametric surface framework, the authors prove that every triharmonic Hopf cylinder must have constant mean curvature (CMC) and completely classify proper CMC r-harmonic Hopf cylinders for r \ge 3, revealing new families of r-harmonic surfaces in BCV-spaces. They further show that proper triharmonic isoparametric surfaces in BCV-spaces (with 4m-\ell^2 \neq 0) are open parts of Hopf cylinders, while parabolic helicoids cannot be proper triharmonic. The results connect tri- and higher-order harmonicity with geometric constraints (CMC, minimality) and provide explicit parameter regimes determining the existence and multiplicity of proper r-harmonic Hopf cylinders, enriching the landscape of explicit examples in positively and non-positively curved BCV geometries.

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.
Paper Structure (6 sections, 19 theorems, 125 equations, 1 figure)

This paper contains 6 sections, 19 theorems, 125 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the results on triharmonic surfaces in BCV-spaces
  3. Preliminaries
  4. Proof of Theorems \ref{['Th-HopfcilindroCMC']} and \ref{['Th-HopfcilindroCMC-espiliciti']}
  5. Proof of Theorem \ref{['Th-triharm-isop-BCV-spaces']}
  6. Proper CMC $r$-harmonic Hopf cylinders

Key Result

Theorem 1.1

Ou-Pacific-2010 Let $\varphi:M^{m}\to N^{m+1}$ be an isometric immersion of codimension-one with mean curvature vector ${\mathbf H}=f \eta$. Then $\varphi$ is biharmonic if and only if: where ${\rm Ric}^N : T_qN\longrightarrow T_qN$ denotes the Ricci operator of the ambient space defined by $\langle {\rm Ric}^N\, (Z), W\rangle={\rm Ric}^N (Z, W)$ and $A$ is the shape operator of the hypersurface

Figures (1)

  • Figure 1: Distribution of proper CMC $r$-harmonic Hopf cylinders in BCV-spaces w.r.t. to the values of $\ell$, $m$ and $r$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • ...and 27 more