Definition
A harmonic map is a smooth function $ \phi : (M,g) \to (N,h) $ between Riemannian manifolds that is a critical point of the Dirichlet energy functional
$$ E(\phi)=\frac12\int_M |d\phi|^{2}, \mathrm{vol}_g , $$
where $|d\phi|^{2}= \operatorname{trace}_g \phi^{*}h$. Equivalently, $\phi$ satisfies the Euler–Lagrange equation
$$ \tau(\phi)=\operatorname{trace}_g abla d\phi = 0, $$
where $\tau(\phi)$ is the tension field of $\phi$. A map for which the tension field vanishes is called harmonic.
Overview
Harmonic maps generalize several classical concepts: geodesics (harmonic maps from an interval), harmonic functions (when the target manifold is $\mathbb{R}$), and minimal surfaces (harmonic immersions of two‑dimensional domains). Introduced in the 1960s, the theory provides a bridge between differential geometry, partial differential equations, and mathematical physics. Existence and regularity results for harmonic maps are central topics; foundational work includes the Eells–Sampson theorem (1964) on existence via the heat flow method, and subsequent developments concerning singularities, energy quantization, and bubble formation.
Etymology/Origin
The adjective “harmonic” traces back to the theory of harmonic functions—solutions of Laplace’s equation—reflecting the fact that a harmonic map minimizes an energy analogous to the Dirichlet integral for scalar functions. The concept of harmonic maps was formalized by James Eells and J. H. Samson in their seminal 1964 paper “Harmonic mappings of Riemannian manifolds,” where they introduced the energy functional and proved existence results under curvature assumptions.
Characteristics
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Energy Functional: The Dirichlet energy $E(\phi)$ measures the average squared length of the differential $d\phi$. Critical points satisfy $\delta E(\phi)=0$, leading to the tension field equation.
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Tension Field: $\tau(\phi)$ is a section of $\phi^{*}TN$ defined by $\tau(\phi)=\operatorname{trace}_g abla d\phi$. Vanishing of $\tau(\phi)$ characterizes harmonicity.
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Local Coordinate Expression: In local charts $(x^{i})$ on $M$ and $(y^{\alpha})$ on $N$,
$$ \tau^{\alpha} = g^{ij}\Big( \frac{\partial^{2}\phi^{\alpha}}{\partial x^{i}\partial x^{j}} - \Gamma^{k}{ij}\frac{\partial\phi^{\alpha}}{\partial x^{k}} + \tilde{\Gamma}^{\alpha}{\beta\gamma}\frac{\partial\phi^{\beta}}{\partial x^{i}}\frac{\partial\phi^{\gamma}}{\partial x^{j}} \Big) =0, $$
where $\Gamma^{k}{ij}$ and $\tilde{\Gamma}^{\alpha}{\beta\gamma}$ are the Levi‑Civita connections on $M$ and $N$, respectively.
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Examples
- Constant maps are trivially harmonic.
- Geodesics are harmonic maps from an interval.
- Holomorphic (or anti‑holomorphic) maps between Kähler manifolds are harmonic.
- The identity map on a manifold is harmonic precisely when the source and target metrics coincide.
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Existence and Regularity
- Eells–Sampson Theorem: If $M$ is compact and $N$ has non‑positive sectional curvature, the heat flow $\partial_t\phi = \tau(\phi)$ converges to a harmonic map in the homotopy class of the initial data.
- Regularity: Harmonic maps are smooth away from a possible singular set of Hausdorff codimension at least three (for maps from dimension $ \ge 3$). In two dimensions, energy‑minimizing harmonic maps are smooth.
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Variational Methods: Techniques such as direct minimization, blow‑up analysis, and monotonicity formulas are employed to study existence, uniqueness, and qualitative behavior.
Related Topics
- Harmonic functions – scalar case of the Dirichlet energy.
- Geodesics – harmonic maps from 1‑dimensional domains.
- Minimal surfaces – harmonic immersions of 2‑manifolds.
- Harmonic morphisms – maps pulling back harmonic functions to harmonic functions.
- Heat flow for harmonic maps – evolution equation $\partial_t\phi = \tau(\phi)$.
- Dirichlet problem for harmonic maps – boundary value problems on domains.
- Variational calculus – functional analytic framework underlying the theory.
- Gauge theory and sigma models – physical models where fields are harmonic maps into target manifolds.
The study of harmonic maps continues to be an active area of research, intersecting geometric analysis, topology, and mathematical physics.