Some analytic results on interpolating sesqui-harmonic maps
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Some analytic results on interpolating sesqui‑harmonic maps Volker Branding1 Received: 24 July 2019 / Accepted: 27 January 2020 © The Author(s) 2020
Abstract In this article, we study various analytic aspects of interpolating sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the case of a spherical target. The latter are critical points of an energy functional that interpolates between the functionals for harmonic and biharmonic maps. In the case of a spherical target, we will derive a conservation law and use it to show the smoothness of weak solutions. Moreover, we will obtain several classification results for interpolating sesqui-harmonic maps. Keywords Interpolating sesqui-harmonic maps · Regularity of weak solutions · Classification results Mathematics Subject Classification 58E20 · 31B30 · 35B65
1 Introduction and results Harmonic maps are among the most important variational problems in geometry, analysis and physics. Given a map 𝜙 ∶ M → N between two Riemannian manifolds (M, g) and (N, h) , they are defined as critical points of the Dirichlet energy
∫M
|d𝜙|2 dV.
(1.1)
̄ 0 = 𝜏(𝜙) ∶= Trg ∇d𝜙.
(1.2)
E(𝜙) =
The first variation of (1.1) is characterized by the vanishing of the so-called tension field which is given by
̄ represents the connection on 𝜙∗ TN . Solutions of (1.2) are called harmonic maps. Here, ∇ The harmonic map equation is a semilinear, elliptic second-order partial differential equation for which many results on existence and qualitative behavior of its solutions could be achieved over the years. * Volker Branding [email protected] 1
Faculty of Mathematics, University of Vienna, Oskar‑Morgenstern‑Platz 1, 1090 Vienna, Austria
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Vol.:(0123456789)
V. Branding
A higher-order generalization of harmonic maps that receives growing attention are the so-called biharmonic maps. These arise as critical points of the bienergy for a map between two Riemannian manifolds which is given by
E2 (𝜙) =
∫M
|𝜏(𝜙)|2 dV
and are characterized by the vanishing of the bitension field
̄ 0 = 𝜏2 (𝜙) ∶= Δ𝜏(𝜙) − RN (d𝜙(ej ), 𝜏(𝜙))d𝜙(ej ). ̄ is the connection Laplacian on 𝜙∗ TN , {ej }, j = 1, … , m = dim M an orthonormal Here, Δ basis of TM and RN denotes the curvature tensor of the target manifold N . Moreover, we apply the Einstein summation convention, meaning that we sum over repeated indices. In contrast to the harmonic map equation the biharmonic map equation is a semilinear elliptic equation of fourth order such that its study comes with additional difficulties. It can be directly seen that every harmonic map is also biharmonic. However, a biharmonic map can be non-harmonic in which case it is called proper biharmonic. Many conditions, both of analytic and geometric nature, are known that force a biharmonic map to be harmonic, see for example [4, 7] and references therein for a recent overview. An extensive study of higher-order energy functionals for maps between Riemannian manifolds was recently initiated in [8]. In this article, we want to focus on t
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