nLab harmonic map

Contents

Idea

A harmonic map is a smooth function $f \colon X \longrightarrow Y$ between a pair $(X,g_X)$ , $(Y,g_Y)$ of Riemannian manifolds which is a critical point of the Dirichlet kinetic energy functional

(1)

E(f) \;\coloneqq\; \tfrac{1}{2} \textstyle{\int_{X}} \Vert \mathrm{d} f\Vert^2 dVol_X \,,

where

$\mathrm{d} f \in \Gamma \big(T^\ast X \otimes \phi^\ast T Y\big)$ is the (exterior) derivative,
the norm $\Vert-\Vert$ is given jointly by the metrics of $X$ and $Y$ ,
the volume form $dVol$ is that of $X$ .

In the terminology discussed at Riemannian immersion – Properties we have:

(harmonic equation)
A Riemannian immersion $\phi \colon \Sigma \to X$ is a harmonic map if and only if it has vanishing tension field, hence iff its second fundamental form has vanishing trace.

Original discussion:

James Eells, Joseph H. Sampson, Harmonic Mappings of Riemannian Manifolds, American Journal of Mathematics 86 1 (1964) 109-160 [doi:10.2307/2373037, jstor:2373037]

On the history of the concept:

Yuan-Jen Chiang, Andrea Ratto: Paying Tribute to James Eells and Joseph H. Sampson: In Commemoration of the Fiftieth Anniversary of Their Pioneering Work on Harmonic Maps, Notices of the AMS 62 4 (2015) [pdf, full issue: pdf]

Review and textbooks:

Paul Baird, John C. Wood: Harmonic Morphisms Between Riemannian Manifolds, Oxford University Press (2003) [doi:10.1093/acprof:oso/9780198503620.001.0001]
Fanghua Lin, Changyou Wang: The Analysis of Harmonic Maps and Their Heat Flows, World Scientific (2008) [doi:10.1142/6679]
Frédéric Hélein, John C. Wood: Harmonic maps, in: Handbook of Global Analysis, Elsevier (2008) 417-491 [doi:10.1016/B978-044452833-9.50009-7, pdf]
Jürgen Jost, Ch. 9 in: Riemannian Geometry and Geometric Analysis, Springer (2017) [doi:10.1007/978-3-319-61860-9]
Wikipedia, Harmonic map

Extensive bibliography:

Charles Misner, Harmonic maps as models for physical theories, Phys. Rev. D 18 4510 (1978) &lbrackspire:137431]

Discussion in the context of integrable systems:

Shabnam Beheshti, A. Shadi Tahvildar-Zadeh, Integrability and Vesture for Harmonic Maps into Symmetric Spaces [arXiv:1209.1383]

Last revised on August 23, 2024 at 06:41:54. See the history of this page for a list of all contributions to it.