Harmonic maps with torsion
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Special Issue on Differential Geometry
. ARTICLES .
Harmonic maps with torsion In Memory of Professor Zhengguo Bai (1916–2015)
Volker Branding Faculty of Mathematics, University of Vienna, Vienna 1090, Austria Email: [email protected] Received February 25, 2020; accepted July 16, 2020
Abstract
In this article we introduce a natural extension of the well-studied equation for harmonic maps
between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan (1924). The maps under consideration do not arise as critical points of an energy functional leading to interesting mathematical challenges. We will perform a first mathematical analysis of these maps which we will call harmonic maps with torsion. Keywords MSC(2010)
harmonic maps with torsion, metric torsion, regularity of weak solutions 58E20, 53C43
Citation: Branding V. Harmonic maps with torsion. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425020-1744-9
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Introduction and results
The harmonic map equation is one of the most studied partial differential equations for maps between Riemannian manifolds. Given a smooth map ϕ between two Riemannian manifolds (M, g) and (N, h) the harmonic map equation can be obtained by computing the first variation of the energy of a map which is given by ∫ E(ϕ) = |dϕ|2 dvolg . (1.1) M
The critical points of (1.1) are characterized by the vanishing of the so-called tension field ∗
0 = τ (ϕ) := Trg ∇ϕ
TN
dϕ.
(1.2)
The (standard) harmonic map equation (1.2) is a semilinear second order elliptic partial differential equation for which many results on existence and qualitative behavior of its solutions could be achieved over the years. For an overview on the current status of research we refer to the survey article [20]. In the literature on harmonic maps one usually chooses to utilize the Levi-Civita connection on the target manifold N . If one defines harmonic maps via a variational principle, as we have done above, then, c The Author(s) 2020. This article is published with open access at link.springer.com. ⃝
math.scichina.com
link.springer.com
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Branding V
Sci China Math
as we will see later, the variational approach chooses to employ the Levi-Civita connection on the target manifold. This article is devoted to a first study of harmonic maps that are coupled to a torsion endomorphism on the target manifold. Harmonic maps with a connection different from the Levi-Civita connection on the domain manifold have already been investigated in great generality. Such kind of maps became known as V-harmonic maps and includes the classes of Hermitian, affine and Weyl harmonic maps into Riemannian manifolds; see the introduction of [15] for more details. In order to obtain a generalization of the harmonic map equation that takes into account a connection with metric torsion on the target manifold we have to abandon the variational point of view. Although this leads to a
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