Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Closed Biconservative Hypersurfaces in Spheres

Stefano Montaldo, Cezar Oniciuc, Alvaro Pampano

TL;DR

The paper addresses the existence and structure of non-CMC biconservative rotational hypersurfaces in space forms, focusing on compact examples in spheres. It proves that their profile curves are $p$-elastic with $p=\frac{n-2}{n+1}$ and develops a variational framework around $\mathbf{\Theta}_p(\gamma)=\int \kappa^p ds$. Using the Euler–Lagrange equation and a closure analysis, the authors establish a discrete biparametric family of closed, non-CMC rotational biconservative hypersurfaces in $\mathbb{S}^n(\rho)$ for every $n\ge3$, none embedding. The results generalize known 3D classifications to arbitrary dimension and provide explicit closure criteria via the phase-integral $I(d)$.

Abstract

We characterise the profile curves of non-CMC biconservative rotational hypersurfaces of space forms $N^n(ρ)$ as $p$-elastic curves, for a suitable rational number $p\in[1/4,1)$ which depends on the dimension $n$ of the ambient space. Analysing the closure conditions of these $p$-elastic curves, we prove the existence of a discrete biparametric family of non-CMC closed (i.e., compact without boundary) biconservative hypersurfaces in $\mathbb{S}^n(ρ)$. None of these hypersurfaces can be embedded in $\mathbb{S}^n(ρ)$.

Closed Biconservative Hypersurfaces in Spheres

TL;DR

The paper addresses the existence and structure of non-CMC biconservative rotational hypersurfaces in space forms, focusing on compact examples in spheres. It proves that their profile curves are -elastic with and develops a variational framework around . Using the Euler–Lagrange equation and a closure analysis, the authors establish a discrete biparametric family of closed, non-CMC rotational biconservative hypersurfaces in for every , none embedding. The results generalize known 3D classifications to arbitrary dimension and provide explicit closure criteria via the phase-integral .

Abstract

We characterise the profile curves of non-CMC biconservative rotational hypersurfaces of space forms as -elastic curves, for a suitable rational number which depends on the dimension of the ambient space. Analysing the closure conditions of these -elastic curves, we prove the existence of a discrete biparametric family of non-CMC closed (i.e., compact without boundary) biconservative hypersurfaces in . None of these hypersurfaces can be embedded in .
Paper Structure (7 sections, 11 theorems, 44 equations, 2 figures)

This paper contains 7 sections, 11 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rotational Biconservative Hypersurfaces
  3. Variational Characterisation of Profile Curves
  4. Closed Profile Curves
  5. Periodic Curvature
  6. Closure Conditions
  7. Closed Rotational Biconservative Hypersurfaces

Key Result

Theorem 2.1

Let $\varphi:M^{n-1}\rightarrow N^n$ be an isometric immersion with mean curvature function $H$. Then, the normal and tangential components of the Euler-Lagrange equation associated to the bienergy are, respectively, where $\eta$ is the unit normal vector field, $S_\eta$ is the shape operator and ${\rm Ric}$ denotes the Ricci curvature of $N^n$. Here, the symbol $\Delta$ represents the rough Lapl

Figures (2)

  • Figure 1: Closed $p$-elastic curves for the values $l=2$ and $r=3$ for different values of $p=(n-2)/(n+1)$. From left to right: $n=4$, $n=15$ and $n=30$.
  • Figure 2: Closed $p$-elastic curves for the values $l=3$ and $r=5$ for different values of $p=(n-2)/(n+1)$. From left to right: $n=4$, $n=15$ and $n=30$.

Theorems & Definitions (22)

  • Theorem 2.1
  • Proposition 2.2
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Proposition 4.1
  • ...and 12 more