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Harmonic map

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A (smooth) map ϕ :MN between Riemannian manifolds M and N is called harmonic if it is a critical point of the Dirichlet energy functional

Contents

E ( ϕ ) = M d ϕ 2 d Vol .

This functional E will be defined precisely below—one way of understanding it is to imagine that M is made of rubber and N made of marble (their shapes given by their respective metrics), and that the map ϕ :MN prescribes how one "applies" the rubber onto the marble: E( ϕ ) then represents the total amount of elastic potential energy resulting from tension in the rubber. In these terms, φ is a harmonic map if the rubber, when "released" but still constrained to stay everywhere in contact with the marble, already finds itself in a position of equilibrium and therefore does not "snap" into a different shape.

Harmonic maps are the 'least expanding' maps in orthogonal directions.

Existence of harmonic maps from a complete Riemannian manifold to a complete Riemannian manifold of non-positive sectional curvature was proved by Eells & Sampson (1964).

Mathematical definition

Given Riemannian manifolds (M,g), (N,h) and ϕ as above, the energy density of ϕ at a point x in M is defined as

e ( ϕ ) = 1 2 d ϕ 2

where the d ϕ 2 is the squared norm of the differential of ϕ , with respect to the induced metric on the bundle T M ϕ 1 T N . The total energy of ϕ is given by integrating the density over M

E ( ϕ ) = M e ( ϕ ) d v g = 1 2 M d ϕ 2 d v g

where dvg denotes the measure on M induced by its metric. This generalizes the classical Dirichlet energy.

The energy density can be written more explicitly as

e ( ϕ ) = 1 2 trace g ϕ h .

Using the Einstein summation convention, in local coordinates the right hand side of this equality reads

e ( ϕ ) = 1 2 g i j h α β ϕ α x i ϕ β x j .

If M is compact, then ϕ is called a harmonic map if it is a critical point of the energy functional E. This definition is extended to the case where M is not compact by requiring the restriction of ϕ to every compact domain to be harmonic, or, more typically, requiring that ϕ be a critical point of the energy functional in the Sobolev space H1,2(M,N).

Equivalently, the map ϕ is harmonic if it satisfies the Euler-Lagrange equations associated to the functional E. These equations read

τ ( ϕ )   = def   trace g d ϕ = 0

where ∇ is the connection on the vector bundle T*M⊗φ−1(TN) induced by the Levi-Civita connections on M and N. The quantity τ( ϕ ) is a section of the bundle ϕ −1(TN) known as the tension field of ϕ . In terms of the physical analogy, it corresponds to the direction in which the "rubber" manifold M will tend to move in N in seeking the energy-minimizing configuration.

Examples

  • Identity and constant maps are harmonic.
  • Assume that the source manifold M is the real line R (or the circle S1), i.e. that ϕ is a curve (or a closed curve) on N. Then ϕ is a harmonic map if and only if it is a geodesic. (In this case, the rubber-and-marble analogy described above reduces to the usual elastic band analogy for geodesics.)
  • Assume that the target manifold N is Euclidean space Rn (with its standard metric). Then ϕ is a harmonic map if and only if it is a harmonic function in the usual sense (i.e. a solution of the Laplace equation). This follows from the Dirichlet principle. If ϕ is a diffeomorphism onto an open set in Rn, then it gives a harmonic coordinate system.
  • Every minimal surface in Euclidean space is a harmonic immersion.
  • More generally, a minimal submanifold M of N is a harmonic immersion of M in N.
  • Every totally geodesic map is harmonic (in this case, ∇d ϕ *h itself vanishes, not just its trace).
  • Every holomorphic map between Kähler manifolds is harmonic.
  • Every harmonic morphism between Riemannian manifolds is harmonic.
  • Problems and applications

  • If, after applying the rubber M onto the marble N via some map ϕ , one "releases" it, it will try to "snap" into a position of least tension. This "physical" observation leads to the following mathematical problem: given a homotopy class of maps from M to N, does it contain a representative that is a harmonic map?
  • Existence results on harmonic maps between manifolds has consequences for their curvature.
  • Once existence is known, how can a harmonic map be constructed explicitly? (One fruitful method uses twistor theory.)
  • In theoretical physics, a quantum field theory whose action is given by the Dirichlet energy is known as a sigma model. In such a theory, harmonic maps correspond to instantons.
  • One of the original ideas in grid generation methods for computational fluid dynamics and computational physics was to use either conformal or harmonic mapping to generate regular grids.
  • Harmonic maps between metric spaces

    The energy integral can be formulated in a weaker setting for functions u : MN between two metric spaces (Jost 1995). The energy integrand is instead a function of the form

    e ϵ ( u ) ( x ) = M d 2 ( u ( x ) , u ( y ) ) d μ x ϵ ( y ) M d 2 ( x , y ) d μ x ϵ ( y )

    in which με
    x
    is a family of measures attached to each point of M.

    References

    Harmonic map Wikipedia


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