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On the Existence of Closed Biconservative Surfaces in Space Forms

Stefano Montaldo, Alvaro Pampano

TL;DR

The paper addresses the existence and structure of biconservative surfaces in space forms, showing that non-CMC examples are precisely rotational linear Weingarten surfaces with $3\kappa_1+\kappa_2=0$ and arise as binormal evolutions of planar extremals of the bending-type curvature energy $\mathbf{\Theta}(\gamma)=\int \kappa^{1/4}\, ds$. It provides a variational characterization of the profile curves and proves that closed non-CMC biconservative surfaces exist in the round sphere $\mathbb{S}^3(\rho)$ in a discrete biparametric family, while none are embedded. For $\rho\le 0$ no closed non-CMC biconservative surfaces exist in space forms. The existence results hinge on a closure condition expressed through a function $\Lambda(d)$ and parameters $m,n$, together with a detailed analysis of a first integral for the profile curvature and its limits via complex-analytic methods.

Abstract

Biconservative surfaces of Riemannian 3-space forms $N^3(ρ)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3κ_1+κ_2=0$ between their principal curvatures $κ_1$ and $κ_2$. We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC biconservative surfaces in the round 3-sphere, $S^3(ρ)$. However, none of these closed surfaces is embedded in $S^3(ρ)$.

On the Existence of Closed Biconservative Surfaces in Space Forms

TL;DR

The paper addresses the existence and structure of biconservative surfaces in space forms, showing that non-CMC examples are precisely rotational linear Weingarten surfaces with and arise as binormal evolutions of planar extremals of the bending-type curvature energy . It provides a variational characterization of the profile curves and proves that closed non-CMC biconservative surfaces exist in the round sphere in a discrete biparametric family, while none are embedded. For no closed non-CMC biconservative surfaces exist in space forms. The existence results hinge on a closure condition expressed through a function and parameters , together with a detailed analysis of a first integral for the profile curvature and its limits via complex-analytic methods.

Abstract

Biconservative surfaces of Riemannian 3-space forms , are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation between their principal curvatures and . We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy. Moreover, using this characterisation, we prove the existence of a discrete biparametric family of closed, i.e. compact without boundary, non-CMC biconservative surfaces in the round 3-sphere, . However, none of these closed surfaces is embedded in .

Paper Structure

This paper contains 13 sections, 11 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Biharmonic Maps and Biconservative Immersions
  3. Biconservative Immersions
  4. Invariant Surfaces
  5. Bending-Type Curvature Energy
  6. Characterisation of Profile Curves as Bending-Type Energy Extremals
  7. Proof of Theorem \ref{['variationalcharacterization']}
  8. Closed Non-CMC Biconservative Surfaces of 3-Space Forms
  9. Proof of Theorem \ref{['complete-case']}
  10. Proof of Lemma \ref{['lemma']}
  11. Part (i)
  12. Part (ii)
  13. Part (iii)

Key Result

Proposition 1.1

The non-CMC biconservative surfaces of a 3-dimensional space form $N^3(\rho)$ are rotational linear Weingarten surfaces verifying where $\kappa_1=-\kappa$ is minus the curvature of the profile curve. Moreover, let $S\subset N^3(\rho)$ be a rotational linear Weingarten surface verifying LWrelationbiconservative, then $S$ is a biconservative surface.

Figures (4)

  • Figure 1: Plot of the polynomial $Q(u)$ for $\rho=1$ and $d=1$.
  • Figure 2: Closed critical curves for $\mathbf{\Theta}$, \ref{['curveture-energy']}, in $\mathbb{S}^2(\rho)$ for $m=3$ and $n=2$ (Left) and $m=5$ and $n=3$ (Right).
  • Figure 3: Stereographic projection of the closed non-CMC biconservative surface in $\mathbb{S}^3(\rho)$ for $m=3$ and $n=2$.
  • Figure 4: Representation of the curves $\sigma$ (in green), $\sigma_\omega$ for each $\omega$ (in blue) and $\sigma_*$ that surrounds the singularities $\beta$ and $\alpha$ (in red).

Theorems & Definitions (16)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Proposition 3.1
  • Proposition 4.1
  • Theorem 4.2
  • proof
  • Proposition 5.1
  • ...and 6 more