Harmonic maps
Introduction
In the realm of differential geometry, harmonic maps serve as a fascinating and intricate concept that bridges various mathematical disciplines. These maps are a generalization of harmonic functions, extending the notion from scalar functions to mappings between Riemannian manifolds. The study of harmonic maps is deeply intertwined with calculus of variations, partial differential equations, and global analysis, offering profound insights into the geometric and topological properties of manifolds.
Definition and Basic Properties
A harmonic map is a smooth function \(\phi: (M, g) \to (N, h)\) between two Riemannian manifolds \((M, g)\) and \((N, h)\) that extremizes the energy functional:
\[ E(\phi) = \frac{1}{2} \int_M |d\phi|^2 \, dv_g, \]
where \(d\phi\) is the differential of \(\phi\), and \(dv_g\) is the volume element on \(M\). The condition for \(\phi\) to be harmonic is that it satisfies the Euler-Lagrange equation associated with this functional, known as the harmonic map equation:
\[ \tau(\phi) = \text{trace}_g \nabla d\phi = 0, \]
where \(\tau(\phi)\) is the tension field of the map, and \(\nabla\) denotes the connection on the pullback bundle \(\phi^*TN\).
Harmonic maps generalize the notion of geodesics, which are harmonic maps from an interval into a manifold. They also include harmonic functions as a special case when the target manifold \(N\) is \(\mathbb{R}\).
Existence and Regularity
The existence and regularity of harmonic maps are central themes in their study. The Dirichlet problem for harmonic maps, which involves finding a harmonic map with prescribed boundary values, is well-studied. The existence of solutions often depends on the topology and geometry of the manifolds involved.
For compact manifolds, the existence of harmonic maps can often be established using the direct method in the calculus of variations, which involves minimizing the energy functional in a suitable function space. Regularity results, such as those by Richard Schoen and Karen Uhlenbeck, demonstrate that weak solutions to the harmonic map equation are smooth under certain conditions.
Examples of Harmonic Maps
Several classical examples illustrate the concept of harmonic maps:
1. **Harmonic Functions**: When the target manifold \(N\) is the real line \(\mathbb{R}\), harmonic maps reduce to harmonic functions, which are solutions to the Laplace equation.
2. **Geodesics**: Geodesics on a Riemannian manifold are harmonic maps from an interval into the manifold, satisfying the geodesic equation.
3. **Holomorphic Maps**: In complex geometry, holomorphic maps between Kähler manifolds are harmonic due to their minimal energy property.
4. **Minimal Surfaces**: Minimal surfaces in \(\mathbb{R}^3\) can be viewed as harmonic maps from a two-dimensional domain into \(\mathbb{R}^3\).
Applications and Connections
Harmonic maps have applications across various fields of mathematics and physics. In mathematical physics, they appear in the study of sigma models, which are used to describe fields in theoretical physics. In topology, harmonic maps provide tools for understanding the structure of manifolds, such as the existence of harmonic representatives of homotopy classes.
The study of harmonic maps also connects to Teichmüller theory, where they are used to understand the geometry of moduli spaces of Riemann surfaces. In global analysis, harmonic maps play a role in the study of Yang-Mills theory and the Seiberg-Witten equations.
Advanced Topics
- Harmonic Map Heat Flow
The harmonic map heat flow is a method for deforming a map into a harmonic map by evolving it according to the heat equation associated with the energy functional. This flow is given by:
\[ \frac{\partial \phi}{\partial t} = \tau(\phi), \]
where \(t\) is the time parameter. The heat flow method is a powerful tool for proving the existence of harmonic maps, particularly in higher dimensions.
- Stability and Uniqueness
The stability of harmonic maps is a critical area of research, involving the second variation of the energy functional. A harmonic map is stable if the second variation is non-negative for all variations. Uniqueness results often depend on the curvature of the target manifold and the topology of the domain.
- Harmonic Maps and Symmetric Spaces
Harmonic maps into symmetric spaces, such as Lie groups with bi-invariant metrics, exhibit special properties due to the high degree of symmetry. These maps are closely related to the theory of harmonic bundles and Higgs bundles, which have applications in gauge theory and string theory.