Harmonic map

Template:More footnotes 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 Template:Mvar from a Riemannian manifold Template:Mvar to a Riemannian manifold Template:Mvar can be thought of as the total amount that Template:Mvar stretches Template:Mvar in allocating each of its elements to a point of Template:Mvar. 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.Template:Sfnm 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,Template:Sfnm 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.Template:Sfnm

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 Template:Mvar be an open subset of ℝⁿ and let Template:Mvar be an open subset of ℝ^m. For each Template:Mvar and Template:Mvar between 1 and Template:Mvar, let g_ij be a smooth real-valued function on Template:Mvar, such that for each Template:Mvar in Template:Mvar, one has that the n × n matrix [g_ij (p)] is symmetric and positive-definite. For each Template:Mvar and Template:Mvar between 1 and Template:Mvar, let h_αβ be a smooth real-valued function on Template:Mvar, such that for each Template:Mvar in Template:Mvar, one has that the m × m matrix [h_αβ (q)] is symmetric and positive-definite. Denote the inverse matrices by [g^ij (p)] and [h^αβ (q)].

For each i, j, k between 1 and Template:Mvar and each α, β, γ between 1 and Template:Mvar define the Christoffel symbols Γ(g)^k_ij : U → ℝ and Γ(h)^γ_αβ : V → ℝ byTemplate:Sfnm

$\begin{array}{rl}\mathrm{\Gamma }(g{)}_{ij}^k& =\frac{1}{2}\sum _{\ell =1}^m{g}^{k\ell }(\frac{\partial g_{j\ell }}{\partial x^i}+\frac{\partial g_{i\ell }}{\partial x^j}-\frac{\partial g_{ij}}{\partial x^{\ell }})\\ \mathrm{\Gamma }(h{)}_{\alpha \beta }^{\gamma }& =\frac{1}{2}\sum _{\delta =1}^n{h}^{\gamma \delta }(\frac{\partial h_{\beta \delta }}{\partial y^{\alpha }}+\frac{\partial h_{\alpha \delta }}{\partial y^{\beta }}-\frac{\partial h_{\alpha \beta }}{\partial y^{\delta }})\end{array}$

Given a smooth map Template:Mvar from Template:Mvar to Template:Mvar, its second fundamental form defines for each Template:Mvar and Template:Mvar between 1 and Template:Mvar and for each Template:Mvar between 1 and Template:Mvar the real-valued function ∇(df)^α_ij on Template:Mvar byTemplate:Sfnm

$\mathrm{\nabla }(df{)}_{ij}^{\alpha }=\frac{{\partial }^2{f}^{\alpha }}{\partial x^i\partial x^j}-\sum _{k=1}^m\mathrm{\Gamma }(g{)}_{ij}^k\frac{\partial f^{\alpha }}{\partial x^k}+\sum _{\beta =1}^n\sum _{\gamma =1}^n\frac{\partial f^{\beta }}{\partial x^i}\frac{\partial f^{\gamma }}{\partial x^j}\mathrm{\Gamma }(h{)}_{\beta \gamma }^{\alpha }\circ f.$

Its laplacian defines for each Template:Mvar between 1 and Template:Mvar the real-valued function (∆f)^α on Template:Mvar byTemplate:Sfnm

$(\mathrm{\Delta }f{)}^{\alpha }=\sum _{i=1}^m\sum _{j=1}^m{g}^{ij}\mathrm{\nabla }(df{)}_{ij}^{\alpha }.$

Bundle formalism

Let (M, g) and (N, h) be Riemannian manifolds. Given a smooth map Template:Mvar from Template:Mvar to Template:Mvar, one can consider its differential df as a section of the vector bundle T ^*M ⊗ f ^*TN over Template:Mvar; this is to say that for each Template:Mvar in Template:Mvar, one has a linear map df_p between tangent spaces T_pM → T_f(p)N.Template:Sfnm The vector bundle T ^*M ⊗ f ^*TN has a connection induced from the Levi-Civita connections on Template:Mvar and Template:Mvar.Template:Sfnm So one may take the covariant derivative ∇(df), which is a section of the vector bundle T ^*M ⊗ T ^*M ⊗ f ^*TN over Template:Mvar; this is to say that for each Template:Mvar in Template:Mvar, one has a bilinear map (∇(df))_p of tangent spaces T_pM × T_pM → T_f(p)N.Template:Sfnm This section is known as the hessian of Template:Mvar.

Using Template:Mvar, one may trace the hessian of Template:Mvar to arrive at the laplacian of Template:Mvar, which is a section of the bundle f ^*TN over Template:Mvar; this says that the laplacian of Template:Mvar assigns to each Template:Mvar in Template:Mvar an element of the tangent space T_f(p)N.Template:Sfnm By the definition of the trace operator, the laplacian may be written as

$(\mathrm{\Delta }f{)}_p=\sum _{i=1}^m(\mathrm{\nabla }(df){)}_p(e_i,e_i)$

where e₁, ..., e_m is any g_p-orthonormal basis of T_pM.

Dirichlet energy and its variation formulas

From the perspective of local coordinates, as given above, the energy density of a mapping Template:Mvar is the real-valued function on Template:Mvar given byTemplate:Sfnm

$\frac{1}{2}\sum _{i=1}^m\sum _{j=1}^m\sum _{\alpha =1}^n\sum _{\beta =1}^n{g}^{ij}\frac{\partial f^{\alpha }}{\partial x^i}\frac{\partial f^{\beta }}{\partial x^j}(h_{\alpha \beta }\circ f).$

Alternatively, in the bundle formalism, the Riemannian metrics on Template:Mvar and Template:Mvar induce a bundle metric on T ^*M ⊗ f ^*TN, and so one may define the energy density as the smooth function Template:Sfrac | df |² on Template:Mvar.Template:Sfnm It is also possible to consider the energy density as being given by (half of) the Template:Mvar-trace of the first fundamental form.Template:Sfnm Regardless of the perspective taken, the energy density e(f) is a function on Template:Mvar which is smooth and nonnegative. If Template:Mvar is oriented and Template:Mvar is compact, the Dirichlet energy of Template:Mvar is defined as

$E(f)=\underset{M}{\int }e(f)d{\mu }_g$

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

The variation formulas for the Dirichlet energy compute the derivatives of the Dirichlet energy E(f) as the mapping Template:Mvar is deformed. To this end, consider a one-parameter family of maps f_s : M → N with f₀ = f for which there exists a precompact open set Template:Mvar of Template:Mvar such that f_s|_{M − K} = f|_{M − K} for all Template:Mvar; one supposes that the parametrized family is smooth in the sense that the associated map (−ε, ε) × M → N given by (s, p) ↦ f_s(p) is smooth.

• The first variation formula says thatTemplate:Sfnm

$\underset{M}{\int }\frac{\partial }{\partial s}{|}_{s=0}e(f_s)d{\mu }_g=-\underset{M}{\int }h(\frac{\partial }{\partial s}{|}_{s=0}{f}_s,\mathrm{\Delta }f)d{\mu }_g$

There is also a version for manifolds with boundary.Template:Sfnm

• There is also a second variation formula.Template:Sfnm

Due to the first variation formula, the Laplacian of Template:Mvar can be thought of as the gradient of the Dirichlet energy; correspondingly, a harmonic map is a critical point of the Dirichlet energy.Template:Sfnm This can be done formally in the language of global analysis and Banach manifolds.

Examples of harmonic maps

Let (M, g) and (N, h) be smooth Riemannian manifolds. The notation g_stan is used to refer to the standard Riemannian metric on Euclidean space.

• Every totally geodesic map (M, g) → (N, h) is harmonic; this follows directly from the above definitions. As special cases:
  • For any Template:Mvar in Template:Mvar, the constant map (M, g) → (N, h) valued at Template:Mvar is harmonic.
  • The identity map (M, g) → (M, g) is harmonic.
• If f : M → N is an immersion, then f : (M, f ^*h) → (N, h) is harmonic if and only if Template:Mvar is minimal relative to Template:Mvar. As a special case:
  • If f : ℝ → (N, h) is a constant-speed immersion, then f : (ℝ, g_stan) → (N, h) is harmonic if and only if Template:Mvar solves the geodesic differential equation.

Recall that if Template:Mvar is one-dimensional, then minimality of Template:Mvar is equivalent to Template:Mvar being geodesic, although this does not imply that it is a constant-speed parametrization, and hence does not imply that Template:Mvar solves the geodesic differential equation.

• A smooth map f : (M, g) → (ℝⁿ, g_stan) is harmonic if and only if each of its Template:Mvar component functions are harmonic as maps (M, g) → (ℝ, g_stan). 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 (M, g) and (N, h) be smooth Riemannian manifolds. A harmonic map heat flow on an interval (a, b) assigns to each Template:Mvar in (a, b) a twice-differentiable map f_t : M → N in such a way that, for each Template:Mvar in Template:Mvar, the map (a, b) → N given by t ↦ f_t (p) is differentiable, and its derivative at a given value of Template:Mvar is, as a vector in T_{ft (p)}N, equal to (∆ f_t )_p. This is usually abbreviated as:

$\frac{\partial f}{\partial t}=\mathrm{\Delta }f.$

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 (a, b) × M → N given by (t, p) ↦ f_t (p).

Now suppose that Template:Mvar is a closed manifold and (N, h) is geodesically complete.

• Existence. Given a continuously differentiable map f from Template:Mvar to Template:Mvar, there exists a positive number Template:Mvar and a harmonic map heat flow f_t on the interval (0, T) such that f_t converges to f in the C¹ topology as Template:Mvar decreases to 0.
• Uniqueness. If { f_t : 0 < t < T } and { f _t : 0 < t < T } are two harmonic map heat flows as in the existence theorem, then f_t = f _t whenever 0 < t < min(T, T).

As a consequence of the uniqueness theorem, there exists a maximal harmonic map heat flow with initial data f, meaning that one has a harmonic map heat flow { f_t : 0 < t < T } as in the statement of the existence theorem, and it is uniquely defined under the extra criterion that Template:Mvar 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:Template:Sfnm Template:Quote In particular, this shows that, under the assumptions on (M, g) and (N, h), every continuous map is homotopic to a harmonic map.Template:Sfnm The very existence of a harmonic map in each homotopy class, which is implicitly being asserted, is part of the result. This is proven by constructing a heat equation, and showing that for any map as initial condition, solution that exists for all time, and the solution uniformly subconverges to a harmonic map.

Eells and Sampson's result was adapted by Richard Hamilton to the setting of the Dirichlet boundary value problem, when Template:Mvar is instead compact with nonempty boundary.Template:Sfnm

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.Template:Sfnm That is, if two maps are initially close, the distance between the corresponding solutions to the heat equation is nonincreasing for all time, thus:

• the set of totally geodesic maps in each homotopy class is path-connected;
• all harmonic maps are energy-minimizing and totally geodesic.

notes that every map from a product $W\times M$ into $N$ is homotopic to a map, such that the map is totally geodesic when restricted to each $M$-fiber.

Singularities and weak solutions

For many years after Eells and Sampson's work, it was unclear to what extent the sectional curvature assumption on (N, h) 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.Template:Sfnm Their results strongly suggest that there are harmonic map heat flows with "finite-time blowup" even when both (M, g) and (N, h) 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 (N, h) is made. In the case that Template:Mvar is two-dimensional, he established the unconditional existence and uniqueness for weak solutions of the harmonic map heat flow.Template:Sfnm 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.Template:Sfnm

Struwe was later able to adapt his methods to higher dimensions, in the case that the domain manifold is Euclidean space;Template:Sfnm he and Yun Mei Chen also considered higher-dimensional closed manifolds.Template:Sfnm 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 { f_t : 0 < t < T }. This formula saysTemplate:Sfnm

$(\frac{\partial }{\partial t}-{\mathrm{\Delta }}^g)e(f)=-|\mathrm{\nabla }(df){|}^2-⟨{\mathrm{Ric}}^g,f^{\ast }h{⟩}_g+{\mathrm{scal}}^g(f^{\ast }{\mathrm{Rm}}^h).$

This is also of interest in analyzing harmonic maps. Suppose f : M → N is harmonic; any harmonic map can be viewed as a constant-in-Template:Mvar solution of the harmonic map heat flow, and so one gets from the above formula thatTemplate:Sfnm

${\mathrm{\Delta }}^{g}e(f)=|\mathrm{\nabla }(df){|}^2+⟨{\mathrm{Ric}}^g,f^{\ast }h{⟩}_g-{\mathrm{scal}}^g(f^{\ast }{\mathrm{Rm}}^h).$

If the Ricci curvature of Template:Mvar is positive and the sectional curvature of Template:Mvar is nonpositive, then this implies that ∆e(f) is nonnegative. If Template:Mvar is closed, then multiplication by e(f) and a single integration by parts shows that e(f) must be constant, and hence zero; hence Template:Mvar must itself be constant.Template:Sfnm Richard Schoen and Shing-Tung Yau noted that this reasoning can be extended to noncompact Template:Mvar by making use of Yau's theorem asserting that nonnegative subharmonic functions which are L²-bounded must be constant.Template:Sfnm In summary, according to these results, one has: Template:Quote In combination with the Eells−Sampson theorem, this shows (for instance) that if (M, g) is a closed Riemannian manifold with positive Ricci curvature and (N, h) is a closed Riemannian manifold with nonpositive sectional curvature, then every continuous map from Template:Mvar to Template:Mvar 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.Template:Sfnm As an application, by making use of the Eells−Sampson existence theorem for harmonic maps, he was able to show that if (M, g) and (N, h) are smooth and closed Kähler manifolds, and if the curvature of (N, h) is appropriately negative, then Template:Mvar and Template:Mvar 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.Template:Sfnm Following this, Mikhael Gromov and Richard Schoen extended much of the theory of harmonic maps to allow (N, h) to be replaced by a metric space.Template:Sfnm 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.

A map $u:M\to N$ between Riemannian manifolds is totally geodesic if, whenever $\gamma :(a,b)\to M$ is a geodesic, the composition $u\circ \gamma$ is a geodesic.

Harmonic maps between metric spaces

The energy integral can be formulated in a weaker setting for functions u : M → N between two metric spaces. The energy integrand is instead a function of the form

$e_{ϵ}(u)(x)=\frac{\underset{M}{\int }{d}^2(u(x),u(y))d{\mu }_x^{ϵ}(y)}{\underset{M}{\int }{d}^2(x,y)d{\mu }_x^{ϵ}(y)}$

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

See also

• Geometric flow

References

Footnotes

Articles

• Template:Erratum

Books and surveys

• Consists of reprints of:

External links

• MathWorld: Harmonic map
• Harmonic Maps Bibliography
• The Bibliography of Harmonic Morphisms