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Remarks on constructing biharmonic and conformal biharmonic maps to spheres

Volker Branding

TL;DR

The paper develops a geometric algorithm to deform harmonic maps into biharmonic or conformally biharmonic maps into spheres using simple ansätze. It proves sharp rigidity for closed domains in the biharmonic case (the first ansatz forces $\alpha=\frac{\pi}{4}$ and constant energy density) while revealing greater flexibility for conformal biharmonic maps, with explicit conditions linking $\alpha$ (or $\beta$) to the energy density $|\nabla v|^2$, and extends these constructions to product-type maps built from two harmonic maps. It also explores non-compact domains where additional biharmonic examples exist beyond the closed-domain restrictions, and provides instability results for the constructed proper biharmonic and conformal biharmonic maps. Overall, the work contributes explicit, parameterized families of biharmonic and conformal biharmonic maps to spheres, clarifying when maximum principles constrain such maps and when flexible conformal variants emerge, thereby enriching the landscape of higher-order harmonic map generalizations.

Abstract

Biharmonic and conformal biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric algorithm that aims at rendering a given harmonic map either biharmonic or conformally biharmonic. For biharmonic maps we find that in the case of a closed domain the maximum principle imposes strong restrictions on our approach, whereas there is more flexibility when we have a non-compact domain and we highlight this difference by a number of examples. Concerning conformal biharmonic maps we show that our algorithm produces explicit critical points for maps between spheres. Moreover, it turns out that we do not get strong restrictions as we obtain for biharmonic maps, such that our algorithm might produce additional conformal biharmonic maps between spheres beyond the ones found in this article.

Remarks on constructing biharmonic and conformal biharmonic maps to spheres

TL;DR

The paper develops a geometric algorithm to deform harmonic maps into biharmonic or conformally biharmonic maps into spheres using simple ansätze. It proves sharp rigidity for closed domains in the biharmonic case (the first ansatz forces and constant energy density) while revealing greater flexibility for conformal biharmonic maps, with explicit conditions linking (or ) to the energy density , and extends these constructions to product-type maps built from two harmonic maps. It also explores non-compact domains where additional biharmonic examples exist beyond the closed-domain restrictions, and provides instability results for the constructed proper biharmonic and conformal biharmonic maps. Overall, the work contributes explicit, parameterized families of biharmonic and conformal biharmonic maps to spheres, clarifying when maximum principles constrain such maps and when flexible conformal variants emerge, thereby enriching the landscape of higher-order harmonic map generalizations.

Abstract

Biharmonic and conformal biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric algorithm that aims at rendering a given harmonic map either biharmonic or conformally biharmonic. For biharmonic maps we find that in the case of a closed domain the maximum principle imposes strong restrictions on our approach, whereas there is more flexibility when we have a non-compact domain and we highlight this difference by a number of examples. Concerning conformal biharmonic maps we show that our algorithm produces explicit critical points for maps between spheres. Moreover, it turns out that we do not get strong restrictions as we obtain for biharmonic maps, such that our algorithm might produce additional conformal biharmonic maps between spheres beyond the ones found in this article.
Paper Structure (10 sections, 29 theorems, 88 equations)

This paper contains 10 sections, 29 theorems, 88 equations.

Table of Contents

  1. Introduction and results
  2. Preliminaries
  3. Harmonic, biharmonic and conformal biharmonic maps to spheres
  4. Eigenmaps to Euclidean spheres
  5. From harmonic to biharmonic maps
  6. Rendering a harmonic map biharmonic
  7. Manufacturing a biharmonic out of two harmonic maps
  8. Constructing conformal biharmonic between spheres
  9. The case of maps between spheres
  10. Manufacturing a conformal biharmonic out of two harmonic maps

Key Result

Theorem 1.1

Let $(M,g)$ be a closed Riemannian manifold and consider a map of the form eq:definition-q-biharmonic where $v\colon M\to\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$ is a harmonic map. Then, the map $q$ is proper biharmonic if and only if $\alpha=\frac{\pi}{4}$ and $v$ has constant energy density $|\nabla The proper biharmonic maps constructed this way are unstable critical points of the bienergy eq:bie

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • ...and 49 more