Harmonic map

In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions.

Informally, the Dirichlet energy of a mapping  from a Riemannian manifold  to a Riemannian manifold  can be thought of as the total amount that  stretches  in allocating each of its elements to a point of . For instance, an unstretched rubber band and a smooth stone can both be naturally viewed as Riemannian manifolds. Any way of stretching the rubber band over the stone can be viewed as a mapping between these manifolds, and the total tension involved is represented by the Dirichlet energy. Harmonicity of such a mapping means that, given any hypothetical way of physically deforming the given stretch, the tension (when considered as a function of time) has first derivative equal to zero when the deformation begins.

The theory of harmonic maps was initiated in 1964 by James Eells and Joseph Sampson, who showed that in certain geometric contexts, arbitrary maps could be deformed into harmonic maps. Their work was the inspiration for Richard Hamilton's initial work on the Ricci flow. Harmonic maps and the associated harmonic map heat flow, in and of themselves, are among the most widely studied topics in the field of geometric analysis.

The discovery of the "bubbling" of sequences of harmonic maps, due to Jonathan Sacks and Karen Uhlenbeck, has been particularly influential, as their analysis has been adapted to many other geometric contexts. Notably, Uhlenbeck's parallel discovery of bubbling of Yang–Mills fields is important in Simon Donaldson's work on four-dimensional manifolds, and Mikhael Gromov's later discovery of bubbling of pseudoholomorphic curves is significant in applications to symplectic geometry and quantum cohomology. The techniques used by Richard Schoen and Uhlenbeck to study the regularity theory of harmonic maps have likewise been the inspiration for the development of many analytic methods in geometric analysis.

Geometry of mappings between manifolds 
Here the geometry of a smooth mapping between Riemannian manifolds is considered via local coordinates and, equivalently, via linear algebra. Such a mapping defines both a first fundamental form and second fundamental form. The Laplacian (also called tension field) is defined via the second fundamental form, and its vanishing is the condition for the map to be harmonic. The definitions extend without modification to the setting of pseudo-Riemannian manifolds.

Local coordinates
Let  be an open subset of  and let  be an open subset of . For each  and  between 1 and , let  be a smooth real-valued function on , such that for each  in , one has that the  matrix  is symmetric and positive-definite. For each  and  between 1 and , let  be a smooth real-valued function on , such that for each  in , one has that the  matrix  is symmetric and positive-definite. Denote the inverse matrices by  and .

For each  between 1 and  and each  between 1 and  define the Christoffel symbols  and  by

Given a smooth map  from  to , its second fundamental form defines for each  and  between 1 and  and for each  between 1 and  the real-valued function  on  by

Its laplacian defines for each  between 1 and  the real-valued function  on  by

Bundle formalism
Let  and  be Riemannian manifolds. Given a smooth map  from  to , one can consider its differential  as a section of the vector bundle  over ; this is to say that for each  in , one has a linear map  between tangent spaces . The vector bundle  has a connection induced from the Levi-Civita connections on  and . So one may take the covariant derivative , which is a section of the vector bundle  over ; this is to say that for each  in , one has a bilinear map  of tangent spaces . This section is known as the hessian of .

Using , one may trace the hessian of  to arrive at the laplacian of , which is a section of the bundle  over ; this says that the laplacian of  assigns to each  in  an element of the tangent space . By the definition of the trace operator, the laplacian may be written as

where  is any -orthonormal basis of .

Dirichlet energy and its variation formulas
From the perspective of local coordinates, as given above, the energy density of a mapping  is the real-valued function on  given by

Alternatively, in the bundle formalism, the Riemannian metrics on  and  induce a bundle metric on , and so one may define the energy density as the smooth function  on . It is also possible to consider the energy density as being given by (half of) the -trace of the first fundamental form. Regardless of the perspective taken, the energy density  is a function on  which is smooth and nonnegative. If  is oriented and  is compact, the Dirichlet energy of  is defined as

where  is the volume form on  induced by . Since any nonnegative measurable function has a well-defined Lebesgue integral, it is not necessary to place the restriction that  is compact; however, then the Dirichlet energy could be infinite.

The variation formulas for the Dirichlet energy compute the derivatives of the Dirichlet energy  as the mapping  is deformed. To this end, consider a one-parameter family of maps  with  for which there exists a precompact open set  of  such that  for all ; one supposes that the parametrized family is smooth in the sense that the associated map  given by  is smooth.
 The first variation formula says that

There is also a version for manifolds with boundary.
 There is also a second variation formula.
Due to the first variation formula, the Laplacian of  can be thought of as the gradient of the Dirichlet energy; correspondingly, a harmonic map is a critical point of the Dirichlet energy. This can be done formally in the language of global analysis and Banach manifolds.

Examples of harmonic maps
Let  and  be smooth Riemannian manifolds. The notation  is used to refer to the standard Riemannian metric on Euclidean space.
 Every totally geodesic map  is harmonic; this follows directly from the above definitions. As special cases:
 For any  in , the constant map  valued at  is harmonic.
 The identity map  is harmonic.
 If  is an immersion, then  is harmonic if and only if  is minimal relative to . As a special case:
 If  is a constant-speed immersion, then  is harmonic if and only if  solves the geodesic differential equation.
 Recall that if  is one-dimensional, then minimality of  is equivalent to  being geodesic, although this does not imply that it is a constant-speed parametrization, and hence does not imply that  solves the geodesic differential equation.
 A smooth map  is harmonic if and only if each of its  component functions are harmonic as maps . This coincides with the notion of harmonicity provided by the Laplace-Beltrami operator.
 Every holomorphic map between Kähler manifolds is harmonic.
 Every harmonic morphism between Riemannian manifolds is harmonic.

Harmonic map heat flow

Well-posedness
Let  and  be smooth Riemannian manifolds. A harmonic map heat flow on an interval  assigns to each  in  a twice-differentiable map  in such a way that, for each  in , the map  given by  is differentiable, and its derivative at a given value of  is, as a vector in , equal to . This is usually abbreviated as:

Eells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties:
 Regularity. Any harmonic map heat flow is smooth as a map  given by .
Now suppose that  is a closed manifold and  is geodesically complete.
 Existence. Given a continuously differentiable map  from  to , there exists a positive number  and a harmonic map heat flow  on the interval  such that  converges to  in the  topology as  decreases to 0.
 Uniqueness. If  and  are two harmonic map heat flows as in the existence theorem, then  whenever .
As a consequence of the uniqueness theorem, there exists a maximal harmonic map heat flow with initial data , meaning that one has a harmonic map heat flow  as in the statement of the existence theorem, and it is uniquely defined under the extra criterion that  takes on its maximal possible value, which could be infinite.

Eells and Sampson's theorem
The primary result of Eells and Sampson's 1964 paper is the following:

In particular, this shows that, under the assumptions on  and , every continuous map is homotopic to a harmonic map. The very existence of a harmonic map in each homotopy class, which is implicitly being asserted, is part of the result. Shortly after Eells and Sampson's work, Philip Hartman extended their methods to study uniqueness of harmonic maps within homotopy classes, additionally showing that the convergence in the Eells−Sampson theorem is strong, without the need to select a subsequence. Eells and Sampson's result was adapted by Richard Hamilton to the setting of the Dirichlet boundary value problem, when  is instead compact with nonempty boundary.

Singularities and weak solutions
For many years after Eells and Sampson's work, it was unclear to what extent the sectional curvature assumption on  was necessary. Following the work of Kung-Ching Chang, Wei-Yue Ding, and Rugang Ye in 1992, it is widely accepted that the maximal time of existence of a harmonic map heat flow cannot "usually" be expected to be infinite. Their results strongly suggest that there are harmonic map heat flows with "finite-time blowup" even when both  and  are taken to be the two-dimensional sphere with its standard metric. Since elliptic and parabolic partial differential equations are particularly smooth when the domain is two dimensions, the Chang−Ding−Ye result is considered to be indicative of the general character of the flow.

Modeled upon the fundamental works of Sacks and Uhlenbeck, Michael Struwe considered the case where no geometric assumption on  is made. In the case that  is two-dimensional, he established the unconditional existence and uniqueness for weak solutions of the harmonic map heat flow. Moreover, he found that his weak solutions are smooth away from finitely many spacetime points at which the energy density concentrates. On microscopic levels, the flow near these points is modeled by a bubble, i.e. a smooth harmonic map from the round 2-sphere into the target. Weiyue Ding and Gang Tian were able to prove the energy quantization at singular times, meaning that the Dirichlet energy of Struwe's weak solution, at a singular time, drops by exactly the sum of the total Dirichlet energies of the bubbles corresponding to singularities at that time.

Struwe was later able to adapt his methods to higher dimensions, in the case that the domain manifold is Euclidean space; he and Yun Mei Chen also considered higher-dimensional closed manifolds. Their results achieved less than in low dimensions, only being able to prove existence of weak solutions which are smooth on open dense subsets.

The Bochner formula and rigidity
The main computational point in the proof of Eells and Sampson's theorem is an adaptation of the Bochner formula to the setting of a harmonic map heat flow . This formula says

This is also of interest in analyzing harmonic maps. Suppose  is harmonic; any harmonic map can be viewed as a constant-in- solution of the harmonic map heat flow, and so one gets from the above formula that

If the Ricci curvature of  is positive and the sectional curvature of  is nonpositive, then this implies that  is nonnegative. If  is closed, then multiplication by  and a single integration by parts shows that  must be constant, and hence zero; hence  must itself be constant. Richard Schoen and Shing-Tung Yau noted that this reasoning can be extended to noncompact  by making use of Yau's theorem asserting that nonnegative subharmonic functions which are -bounded must be constant. In summary, according to these results, one has:

In combination with the Eells−Sampson theorem, this shows (for instance) that if  is a closed Riemannian manifold with positive Ricci curvature and  is a closed Riemannian manifold with nonpositive sectional curvature, then every continuous map from  to  is homotopic to a constant.

The general idea of deforming a general map to a harmonic map, and then showing that any such harmonic map must automatically be of a highly restricted class, has found many applications. For instance, Yum-Tong Siu found an important complex-analytic version of the Bochner formula, asserting that a harmonic map between Kähler manifolds must be holomorphic, provided that the target manifold has appropriately negative curvature. As an application, by making use of the Eells−Sampson existence theorem for harmonic maps, he was able to show that if  and  are smooth and closed Kähler manifolds, and if the curvature of  is appropriately negative, then  and  must be biholomorphic or anti-biholomorphic if they are homotopic to each other; the biholomorphism (or anti-biholomorphism) is precisely the harmonic map produced as the limit of the harmonic map heat flow with initial data given by the homotopy. By an alternative formulation of the same approach, Siu was able to prove a variant of the still-unsolved Hodge conjecture, albeit in the restricted context of negative curvature.

Kevin Corlette found a significant extension of Siu's Bochner formula, and used it to prove new rigidity theorems for lattices in certain Lie groups. Following this, Mikhael Gromov and Richard Schoen extended much of the theory of harmonic maps to allow  to be replaced by a metric space. By an extension of the Eells−Sampson theorem together with an extension of the Siu–Corlette Bochner formula, they were able to prove new rigidity theorems for lattices.

Problems and applications 

 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  between two metric spaces.  The energy integrand is instead a function of the form

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

See also
Geometric flow

References
Footnotes

Articles

Books and surveys

 Consists of reprints of:

External links 
 MathWorld: Harmonic map
 Harmonic Maps Bibliography
 The Bibliography of Harmonic Morphisms

Riemannian geometry
Harmonic functions
Analytic functions