Harmonic maps

Definition

Harmonic maps are smooth functions between Riemannian manifolds that minimize the Dirichlet energy, which measures how much a map distorts the geometric structure of the manifolds. These maps are critical points of the Dirichlet energy functional and play a significant role in various areas of mathematics, particularly in the calculus of variations and geometric measure theory.

5 Must Know Facts For Your Next Test

1. Harmonic maps arise naturally when considering problems related to minimizing energy in variational calculus.
2. The Euler-Lagrange equations characterize harmonic maps, providing necessary conditions for a map to be harmonic.
3. Harmonic maps can be classified based on their properties, such as being minimal or having constant mean curvature.
4. They have applications in mathematical physics, particularly in string theory and general relativity, where they describe certain types of fields.
5. The study of harmonic maps includes results such as the existence and regularity theory, which addresses when solutions exist and their smoothness.

Review Questions

• How do harmonic maps relate to the concept of Dirichlet energy and what role does this relationship play in identifying minimizers?
  • Harmonic maps are directly connected to Dirichlet energy as they are defined as minimizers of this functional. The Dirichlet energy measures how much a map distorts distances between points on the manifolds. By finding harmonic maps, one can determine critical points where energy is minimized, thus helping to identify stable configurations in various mathematical contexts.
• Discuss the significance of critical points in the context of harmonic maps and their implications for the study of variational problems.
  • Critical points are essential in understanding harmonic maps because they indicate locations where Dirichlet energy reaches an extremum. This connection allows for employing variational methods to analyze solutions to related physical problems. Invariably, studying these critical points leads to insights about stability and existence of solutions in variational calculus.
• Evaluate the broader implications of harmonic maps in geometric measure theory and their impact on modern mathematical theories.
  • Harmonic maps serve as a bridge between various mathematical theories by applying geometric measure theory concepts to analyze spaces and mappings. They allow mathematicians to explore intricate relationships between geometry and analysis. The study of harmonic maps influences numerous fields, including differential geometry and mathematical physics, fostering advancements in understanding manifold structures and field theories.

Related terms

Dirichlet energy:

A functional that measures the 'energy' of a map between manifolds, defined as the integral of the squared norm of the differential of the map.

Critical point:

A point where the derivative of a functional is zero, indicating a potential local minimum, maximum, or saddle point.

Riemannian manifold: A smooth manifold equipped with a Riemannian metric, which allows for the measurement of lengths, angles, and volumes.