nLab harmonic map

Idea

Definition

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 $X$.

(due to Eells & Sampson 1964, §1-§2; review includes Baird & Wood 2003, §3.3; Lin & Wang 2008, Def. 1.1.1; Jost 2017, Def. 9.1.2)

Remark

The analogous formula makes sense for pseudo-Riemannian manifolds where (1) is the standard kinetic action action functional – the Polyakov action – for relativistic non-linear sigma models (such as for the relativistic particle, string or membrane, etc.)

Properties

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

Theorem

(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.

(Eells & Sampson 1964 pp. 116, Baird & Wood 2003 Thm. 3.3.3)

Related concepts

• second fundamental form

References

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:

• Harmonic Maps Bibliography [people.bath.ac.uk/feb/harmonic.html]

Discussion in the context of action functionals for theories of physics (nonlinear sigma models):

• 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]