# Harmonic map

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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.

## Contents

Informally, the Dirichlet energy of a mapping f from a Riemannian manifold M to a Riemannian manifold N can be thought of as the total amount that f stretches M in allocating each of its elements to a point of N. 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 U be an open subset of m and let V be an open subset of n. For each i and j between 1 and n, let gij be a smooth real-valued function on U, such that for each p in U, one has that the m × m matrix [gij (p)] is symmetric and positive-definite. For each α and β between 1 and m, let hαβ be a smooth real-valued function on V, such that for each q in V, one has that the n × n matrix [hαβ (q)] is symmetric and positive-definite. Denote the inverse matrices by [gij (p)] and [hαβ (q)].

For each i, j, k between 1 and n and each α, β, γ between 1 and m define the Christoffel symbols Γ(g)kij : U → ℝ and Γ(h)γαβ : V → ℝ by 

{\begin{aligned}\Gamma (g)_{ij}^{k}&={\frac {1}{2}}\sum _{\ell =1}^{m}g^{k\ell }{\Big (}{\frac {\partial g_{j\ell }}{\partial x^{i}}}+{\frac {\partial g_{i\ell }}{\partial x^{j}}}-{\frac {\partial g_{ij}}{\partial x^{\ell }}}{\Big )}\\\Gamma (h)_{\alpha \beta }^{\gamma }&={\frac {1}{2}}\sum _{\delta =1}^{n}h^{\gamma \delta }{\Big (}{\frac {\partial h_{\beta \delta }}{\partial y^{\alpha }}}+{\frac {\partial h_{\alpha \delta }}{\partial y^{\beta }}}-{\frac {\partial h_{\alpha \beta }}{\partial y^{\delta }}}{\Big )}\end{aligned}} Given a smooth map f from U to V, its second fundamental form defines for each i and j between 1 and m and for each α between 1 and n the real-valued function ∇(df)αij on U by 

$\nabla (df)_{ij}^{\alpha }={\frac {\partial ^{2}f^{\alpha }}{\partial x^{i}\partial x^{j}}}-\sum _{k=1}^{m}\Gamma (g)_{ij}^{k}{\frac {\partial f^{\alpha }}{\partial x^{k}}}+\sum _{\beta =1}^{n}\sum _{\gamma =1}^{n}{\frac {\partial f^{\beta }}{\partial x^{i}}}{\frac {\partial f^{\gamma }}{\partial x^{j}}}\Gamma (h)_{\beta \gamma }^{\alpha }\circ f.$ Its laplacian defines for each α between 1 and n the real-valued function (∆f)α on U by 

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

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

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

$(\Delta f)_{p}=\sum _{i=1}^{m}{\big (}\nabla (df){\big )}_{p}(e_{i},e_{i})$ where e1, ..., em is any gp-orthonormal basis of TpM.

## Dirichlet energy and its variation formulas

From the perspective of local coordinates, as given above, the energy density of a mapping f is the real-valued function on U given by 

${\frac {1}{2}}\sum _{i=1}^{m}\sum _{j=1}^{m}\sum _{\alpha =1}^{n}\sum _{\beta =1}^{n}g^{ij}{\frac {\partial f^{\alpha }}{\partial x^{i}}}{\frac {\partial f^{\beta }}{\partial x^{j}}}(h_{\alpha \beta }\circ f).$ Alternatively, in the bundle formalism, the Riemannian metrics on M and N induce a bundle metric on T *Mf *TN, and so one may define the energy density as the smooth function 1/2 | df |2 on M.  It is also possible to consider the energy density as being given by (half of) the g-trace of the first fundamental form.  Regardless of the perspective taken, the energy density e(f) is a function on M which is smooth and nonnegative. If M is oriented and M is compact, the Dirichlet energy of f is defined as

$E(f)=\int _{M}e(f)\,d\mu _{g}$ where g is the volume form on M induced by g.  Since any nonnegative measurable function has a well-defined Lebesgue integral, it is not necessary to place the restriction that M 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 f is deformed. To this end, consider a one-parameter family of maps fs : MN with f0 = f for which there exists a precompact open set K of M such that fs|MK = f|MK for all s; one supposes that the parametrized family is smooth in the sense that the associated map (−ε, ε) × MN given by (s, p) ↦ fs(p) is smooth.

• The first variation formula says that 
$\int _{M}{\frac {\partial }{\partial s}}{\Big |}_{s=0}e(f_{s})\,d\mu _{g}=-\int _{M}h\left({\frac {\partial }{\partial s}}{\Big |}_{s=0}f_{s},\Delta f\right)\,d\mu _{g}$ 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 f 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 (M, g) and (N, h) be smooth Riemannian manifolds. The notation gstan 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 q in N, the constant map (M, g) → (N, h) valued at q is harmonic.
• The identity map (M, g) → (M, g) is harmonic.
• If f : MN is an immersion, then f : (M, f *h) → (N, h) is harmonic if and only if f is minimal relative to h. As a special case:
• If f : ℝ → (N, h) is a constant-speed immersion, then f : (ℝ, gstan) → (N, h) is harmonic if and only if f solves the geodesic differential equation.
Recall that if M is one-dimensional, then minimality of f is equivalent to f being geodesic, although this does not imply that it is a constant-speed parametrization, and hence does not imply that f solves the geodesic differential equation.

## 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 t in (a, b) a twice-differentiable map ft : MN in such a way that, for each p in M, the map (a, b) → N given by tft (p) is differentiable, and its derivative at a given value of t is, as a vector in Tft (p)N, equal to (∆ ft )p. This is usually abbreviated as:

${\frac {\partial f}{\partial t}}=\Delta f.$ 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) × MN given by (t, p) ↦ ft(p).

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

• Existence. Given a continuously differentiable map f from M to N, there exists a positive number T and a harmonic map heat flow ft on the interval (0, T) such that ft converges to f in the C1 topology as t decreases to 0. 
• Uniqueness. If { ft : 0 < t < T } and { f t : 0 < t < T } are two harmonic map heat flows as in the existence theorem, then ft = 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 { ft : 0 < t < T } as in the statement of the existence theorem, and it is uniquely defined under the extra criterion that T 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: 

Let (M, g) and (N, h) be smooth and closed Riemannian manifolds, and suppose that the sectional curvature of (N, h) is nonpositive. Then for any continuously differentiable map f from M to N, the maximal harmonic map heat flow { ft : 0 < t < T } with initial data f has T = ∞, and as t increases to , the maps ft subsequentially converge in the C topology to a harmonic map.

In particular, this shows that, under the assumptions on (M, g) and (N, h), 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 M 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 (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.  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 M 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 { ft : 0 < t < T }. This formula says 

${\Big (}{\frac {\partial }{\partial t}}-\Delta ^{g}{\Big )}e(f)=-{\big |}\nabla (df){\big |}^{2}-{\big \langle }\operatorname {Ric} ^{g},f^{\ast }h{\big \rangle }_{g}+\operatorname {scal} ^{g}{\big (}f^{\ast }\operatorname {Rm} ^{h}{\big )}.$ This is also of interest in analyzing harmonic maps. Suppose f : MN is harmonic; any harmonic map can be viewed as a constant-in-t solution of the harmonic map heat flow, and so one gets from the above formula that 

$\Delta ^{g}e(f)={\big |}\nabla (df){\big |}^{2}+{\big \langle }\operatorname {Ric} ^{g},f^{\ast }h{\big \rangle }_{g}-\operatorname {scal} ^{g}{\big (}f^{\ast }\operatorname {Rm} ^{h}{\big )}.$ If the Ricci curvature of g is positive and the sectional curvature of h is nonpositive, then this implies that e(f) is nonnegative. If M is closed, then multiplication by e(f) and a single integration by parts shows that e(f) must be constant, and hence zero; hence f must itself be constant.  Richard Schoen and Shing-Tung Yau noted that this reasoning can be extended to noncompact M by making use of Yau's theorem asserting that nonnegative subharmonic functions which are L2-bounded must be constant.  In summary, according to these results, one has:

Let (M, g) and (N, h) be smooth and complete Riemannian manifolds, and let f be a harmonic map from M to N. Suppose that the Ricci curvature of g is positive and the sectional curvature of h is nonpositive.

• If M and N are both closed then f must be constant.
• If N is closed and f has finite Dirichlet energy, then it must be constant.

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 M to N 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 (M, g) and (N, h) are smooth and closed Kähler manifolds, and if the curvature of (N, h) is appropriately negative, then M and N 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 (N, h) 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 u : MN between two metric spaces. The energy integrand is instead a function of the form

$e_{\epsilon }(u)(x)={\frac {\int _{M}d^{2}(u(x),u(y))\,d\mu _{x}^{\epsilon }(y)}{\int _{M}d^{2}(x,y)\,d\mu _{x}^{\epsilon }(y)}}$ in which με
x
is a family of measures attached to each point of M. 

## Related Research Articles In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : UR, where U is an open subset of Rn, that satisfies Laplace's equation, that is,

In differential geometry, a Riemannian manifold or Riemannian space(M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. In the mathematical fields of differential geometry and geometric analysis, the Ricci flow, sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation. Shing-Tung Yau is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. Richard Streit Hamilton is Davies Professor of Mathematics at Columbia University, and is known for contributions to geometric analysis and partial differential equations. He is best known for foundational contributions to the theory of the Ricci flow and the development of a corresponding program of techniques and ideas for resolving the Poincaré conjecture and geometrization conjecture in the field of geometric topology. Grigori Perelman built upon Hamilton's results to prove the conjectures, and was awarded a Millennium Prize for his work. However, he declined the award, regarding Hamilton's contribution as being equal to his own. Mikhael Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University.

In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.

In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds. Richard Melvin Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984. The positive energy theorem refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an isolated system is nonnegative, and can only be zero when the system has no gravitating objects. Although these statements are often thought of as being primarily physical in nature, they can be formalized as mathematical theorems which can be proven using techniques of differential geometry, partial differential equations, and geometric measure theory.

In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in many problems of geometric analysis due to their regularity properties. In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Leon Melvyn Simon, born in 1945, is a Leroy P. Steele Prize and Bôcher Prize-winning mathematician, known for deep contributions to the fields of geometric analysis, geometric measure theory, and partial differential equations. He is currently Professor Emeritus in the Mathematics Department at Stanford University.

In Riemannian geometry, a field of mathematics, Preissmann's theorem is a statement that restricts the possible topology of a negatively curved compact Riemannian manifold. It is named for Alexandre Preissmann, who published a proof in 1943. James Eells was an American mathematician, who specialized in mathematical analysis.

In mathematics, a harmonic morphism is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps i.e. those that are horizontally (weakly) conformal.

In the mathematical field of differential geometry, a biharmonic map is a map between Riemannian or pseudo-Riemannian manifolds which satisfies a certain fourth-order partial differential equation. A biharmonic submanifold refers to an embedding or immersion into a Riemannian or pseudo-Riemannian manifold which is a biharmonic map when the domain is equipped with its induced metric. The problem of understanding biharmonic maps was posed by James Eells and Luc Lemaire in 1983. The study of harmonic maps, of which the study of biharmonic maps is an outgrowth, had been an active field of study for the previous twenty years. A simple case of biharmonic maps is given by biharmonic functions.

Joseph Harold Sampson Jr. was an American mathematician known for his work in mathematical analysis, geometry and topology, especially his work about harmonic maps in collaboration with James Eells. He obtained his Ph.D. in mathematics from Princeton University in 1951 under the supervision of Salomon Bochner.