Annali di Matematica

2011

DOI: 10.1007/s10231-011-0215-0

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Biharmonic maps in two dimensions

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,

²

Abstract: Biharmonic maps between surfaces are studied in this paper. We compute the bitension field of a map between surfaces with conformal metrics in complex coordinates. As applications, we show that a linear map from Euclidean plane into (R 2 , σ 2 dwdw) is always biharmonic if the conformal factor σ is bianalytic; we construct a family of such σ, and we give a classification of linear biharmonic maps between 2-spheres. We also study biharmonic maps between surfaces with warped product metrics. This includes a clas… Show more

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Cited by 10 publications

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“…We also recall that the Laplace–Beltrami operator acts locally on a function as follows: where here are the Christoffel symbols of the Riemannian manifold M . Moreover, if is a section of , then (see [28, Lemma 1.1]) where and, for simplicity, .…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Unique continuation properties for polyharmonic maps between Riemannian manifolds

¹

,

et al. 2021

Can. J. Math.-J. Can. Math.

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Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic case: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on a open subset, then they agree everywhere; (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

“…We also recall that the Laplace–Beltrami operator acts locally on a function as follows: where here are the Christoffel symbols of the Riemannian manifold M . Moreover, if is a section of , then (see [28, Lemma 1.1]) where and, for simplicity, .…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Unique continuation properties for polyharmonic maps between Riemannian manifolds

¹

,

et al. 2021

Can. J. Math.-J. Can. Math.

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Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic case: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on a open subset, then they agree everywhere; (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

“…The section

σ

can be written as

σ = u^{α} false(p false) \frac{\partial}{\partial y^{α}} false(ϕ (p) false), p \in U .

The Laplacian

Δ σ

is given by

\begin{matrix} true \\ right normalΔ σ = & left {Tr}_{g} \nabla d σ = g^{i j} ((), \nabla, d, σ) (() \frac{\partial}{\partial x^{i}}, \frac{\partial}{\partial x^{j}}) \\ right = & left g^{i j} (() \nabla_{\frac{\partial}{\partial x^{i}}} d σ (\frac{\partial}{\partial x^{j}}) - d σ (\nabla_{\frac{\partial}{\partial x^{i}}} \frac{\partial}{\partial x^{j}})) . \end{matrix}

We find

d σ (\frac{\partial}{\partial x^{i}}) = \nabla_{\frac{\partial}{\partial x^{i}}} σ = u_{i}^{α} \frac{\partial}{\partial y^{α}} + u^{α} ϕ_{i}^{β} {normalΓ}_{β α}^{θ} \frac{\partial}{\partial y^{θ}},

where

u_{i}^{α} = \frac{\partial u^{α}}{\partial x^{i}}

and

{false(ϕ^{β} false)}_{β}

is the corresponding expression for

ϕ

in local coordinates. Then, by a straightforward computation, we get (see also [, Lemma 1.1])

\begin{matrix} true \\ right normalΔ σ - T... \end{matrix}

…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

Unique continuation theorems for biharmonic maps

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,

²

2019

Bull. London Math. Soc.

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We prove several unique continuation results for biharmonic maps between Riemannian manifolds.

Biharmonic Maps

¹

2013

Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields Into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields

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