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Biharmonic Hypersurfaces in Euclidean Spaces

Hiba Bibi, Marc Soret, Marina Ville

TL;DR

This work analyzes biharmonic and biconservative hypersurfaces in Euclidean spaces, with a focus on holonomic BCH. It demonstrates that holonomic BCH are restricted to rank 0 (CMC) or rank 1, and that rank-1 BCH are foliated by level sets of the mean curvature $h$ whose leaves are codimension-2 isoparametric submanifolds, with integral curves of $\nabla h$ governed by a second-order ODE. A constructive correspondence is established: from any holonomic codimension-2 isoparametric submanifold $U_0$, locally there exists a proper BCH obtained by normal evolution, and holonomic proper BCH in $\mathbb{E}^{n+1}$ are classified as spherical, product, cylindrical, or mixed-product types; moreover, holonomic BHH must be minimal. The results connect BCH geometry to classic isoparametric theory, providing both structure theorems and explicit local constructions while posing questions about extending to non-holonomic settings.

Abstract

An isometric immersion $X: Σ^n \longrightarrow \mathbb{E}^{n+1}$ is biharmonic if $Δ^2 X = 0$, i.e. if $ΔH =0$, where $Δ$ and $H$ are the metric Laplacian and the mean curvature vector field of $Σ^n$ respectively. More generally, biconservative hypersurfaces (BCH) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. We study and construct BCH that are holonomic, i.e. for which the principal curvature directions define an integrable net, and we deduce that $Σ^n$ is a holonomic biharmonic hypersurface iff it is minimal.

Biharmonic Hypersurfaces in Euclidean Spaces

TL;DR

This work analyzes biharmonic and biconservative hypersurfaces in Euclidean spaces, with a focus on holonomic BCH. It demonstrates that holonomic BCH are restricted to rank 0 (CMC) or rank 1, and that rank-1 BCH are foliated by level sets of the mean curvature whose leaves are codimension-2 isoparametric submanifolds, with integral curves of governed by a second-order ODE. A constructive correspondence is established: from any holonomic codimension-2 isoparametric submanifold , locally there exists a proper BCH obtained by normal evolution, and holonomic proper BCH in are classified as spherical, product, cylindrical, or mixed-product types; moreover, holonomic BHH must be minimal. The results connect BCH geometry to classic isoparametric theory, providing both structure theorems and explicit local constructions while posing questions about extending to non-holonomic settings.

Abstract

An isometric immersion is biharmonic if , i.e. if , where and are the metric Laplacian and the mean curvature vector field of respectively. More generally, biconservative hypersurfaces (BCH) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. We study and construct BCH that are holonomic, i.e. for which the principal curvature directions define an integrable net, and we deduce that is a holonomic biharmonic hypersurface iff it is minimal.

Paper Structure

This paper contains 19 sections, 16 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalities on ${m}$-harmonic maps
  3. Notations
  4. Remarks on the equation of a biharmonic graph
  5. Biharmonic Hypersurfaces in $\mathbb{E}^{n+1}$
  6. Biconservative equation in a principal curvature frame
  7. Reducible biconservative and biharmonic hypersurfaces
  8. Geometry of proper holonomic BCH in $\mathbb{E}^{n+1}$
  9. Foliation of a BCH by the mean curvature function
  10. Holonomic hypersurfaces of the Euclidean space
  11. Integral curves of $\nabla h$
  12. Parametrization of a holonomic BCH
  13. Second order ODE governing the integral curves of $\nabla h$
  14. Level sets of $h$
  15. Proof of the first part of Theorem \ref{['letheoreme1']}
  16. ...and 4 more sections

Key Result

Theorem 1

A holonomic proper BCH $\Sigma^n$ in $\mathbb{E}^{n+1}$ is foliated by the level sets of the mean curvature function $h$ of $\Sigma^n$ which are isoparametric codimension $2$ submanifolds of $\mathbb{E}^{n+1}$ in a neighbourhood of a regular point of $h$. The symmetries of the leaves extend to symme

Figures (1)

  • Figure :

Theorems & Definitions (31)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Example 1
  • Proposition 1
  • proof
  • Definition 1
  • Proposition 2
  • proof
  • Lemma 1
  • ...and 21 more