# Harmonic mapping

A smooth mapping $\varphi : ( M , g ) \rightarrow ( N , h )$ between Riemannian manifolds (cf. Riemannian manifold) is harmonic if it is an extremal (or critical point) of the energy functional

\begin{equation*} E ( \varphi ) = \frac { 1 } { 2 } \int _ { M } | d \varphi | ^ { 2 } v _ { g }, \end{equation*}

where $| d \varphi |$ is the Hilbert–Schmidt norm of the differential, computed with respect to the metrics $g$ and $h$, and $v _ { g }$ is the Riemannian volume element.

The mapping $\varphi$ is harmonic if it satisfies the Euler–Lagrange equation $\tau ( \varphi ) = 0$, where the tension field $\tau ( \varphi )$ is given by $\tau ( \varphi ) = \text { trace } \nabla d \varphi$, $\nabla$ denoting the natural connection on $T ^ { * } M \otimes \varphi ^ { - 1 } T N$.

In local coordinate systems $( x ^ { i } )$ on $M$ and $( y ^ { \alpha } )$ on $N$, one has

\begin{equation*} | d \varphi | ^ { 2 } ( x ) = g ^ { i j } ( x ) h _ { \alpha \beta } ( \varphi ( x ) ) \cdot \frac { \partial \varphi ^ { \alpha } } { \partial x ^ { i } } \frac { \partial \varphi ^ { \beta } } { \partial x ^ { j } }, \end{equation*}

\begin{equation*} \tau ( \varphi ) ^ { \alpha } ( x ) = g ^ { i j } ( x ) \left( \frac { \partial ^ { 2 } \varphi ^ { \alpha } } { \partial x ^ { i } \partial x ^ { j } } - \square ^ { M } \Gamma _ { i j } ^ { k } ( x ) \frac { \partial \varphi ^ { \alpha } } { \partial x ^ { k } } + + \square ^ { N } \Gamma _ { \beta \gamma } ^ { \alpha } ( \varphi ( x ) ) \frac { \partial \varphi \beta } { \partial x ^ { i } } \frac { \partial \varphi ^ { \gamma } } { \partial x ^ { j } } \right), \end{equation*}

where the $\Gamma$ are the Christoffel symbols of the Levi–Civita connections on $M$ and $N$. The Euler–Lagrange equation is therefore a semi-linear elliptic system of partial differential equations.

Harmonic mappings include as special cases the closed geodesics in a Riemannian manifold $( N , h )$, the minimal immersions, the totally geodesic mappings and the holomorphic mappings between Kähler manifolds. In physics, they are related to $\sigma$-models and to some types of liquid crystals.

The systematic study of harmonic mappings was initiated in 1964 in [a7] by J. Eells and J. Sampson.

A detailed exposition of results obtained before 1988 can be found in [a5] and [a6] and includes the following four main directions:

existence theory for harmonic mappings in prescribed homotopy classes (with existence and non-existence results);

regularity and partial regularity for minimizers of the energy in appropriate Sobolev spaces (with restriction on the Hausdorff dimension of the singular set);

explicit constructions of harmonic mappings from the two-dimensional sphere to Lie groups, symmetric spaces and loop groups in terms of holomorphic mappings and twistor constructions;

applications of the existence theory of harmonic mappings to the study of the geometry of real manifolds (curvature pinching, rigidity), or of Kähler manifolds (rigidity, uniformization), applications to the study of Teichmüller spaces (cf. Discrete subgroup; Teichmüller space; Riemannian geometry in the large).

Further developments (up to 1997) include the following:

Application of harmonic mappings to (Mostow) rigidity of manifolds was pursued in [a4] and [a15], the latter unifying previous results. In a similar vein, existence of harmonic mappings bears on the structure of the fundamental group of Kähler manifolds ([a3], see also [a1]).

A new direction was opened in [a11], in which the notion of harmonic mapping was extended to more general spaces (trees, polyhedra, Tits buildings), and an existence result was proved and applied to the study of $p$-adic superrigidity for lattices in groups of rank one.

Further curvature pinching theorems were obtained in [a13] and [a19].

Various results on Teichmüller spaces were obtained, using classical harmonic mappings (see [a17]) or harmonic maps into trees (see [a18]).

The question of regularity or partial regularity was extended to weakly harmonic mappings (as opposed to minimizers) in the appropriate Sobolev space: when $\operatorname { dim } M = 2$, any weakly harmonic mapping is smooth [a12], and examples show that for $\dim M \geq 3$, the Hausdorff dimension of the singular set is not restricted [a16].

Examples show that harmonic mappings homotopic to homeomorphisms are not always homeomorphisms, even when the curvature of the range is negative [a8].

Explicit constructions of harmonic mappings of surfaces into symmetric spaces in terms of holomorphic constructions or totally integrable systems were further developed, e.g. in [a14] and [a2] (see [a9], [a10]).

How to Cite This Entry:
Harmonic mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_mapping&oldid=50104
This article was adapted from an original article by Luc Lemaire (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article