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In the mathematical field ofdifferential geometry, a smooth map betweenRiemannian manifolds is calledharmonic if its coordinate representatives satisfy a certain nonlinearpartial differential equation. This partial differential equation for a mapping also arises as theEuler-Lagrange equation of a functional called theDirichlet energy. As such, the theory of harmonic maps contains both the theory ofunit-speed geodesics inRiemannian geometry and the theory ofharmonic functions.

Informally, the Dirichlet energy of a mappingf from a Riemannian manifoldM to a Riemannian manifoldN can be thought of as the total amount thatf stretchesM in allocating each of its elements to a point ofN. For instance, an unstretchedrubber 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 byJames Eells andJoseph Sampson, who showed that in certain geometric contexts, arbitrary maps could bedeformed into harmonic maps.^[1] Their work was the inspiration forRichard Hamilton's initial work on theRicci flow. Harmonic maps and the associatedharmonic map heat flow, in and of themselves, are among the most widely studied topics in the field ofgeometric analysis.

The discovery of the "bubbling" of sequences of harmonic maps, due to Jonathan Sacks andKaren Uhlenbeck,^[2] 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 inSimon Donaldson's work on four-dimensional manifolds, andMikhael Gromov's later discovery of bubbling ofpseudoholomorphic curves is significant in applications tosymplectic geometry andquantum cohomology. The techniques used byRichard 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.^[3]

Here the geometry of a smooth mapping betweenRiemannian manifolds is considered vialocal coordinates and, equivalently, vialinear algebra. Such a mapping defines both afirst fundamental form andsecond fundamental form. TheLaplacian (also calledtension field) is defined via thesecond fundamental form, and its vanishing is the condition for the map to beharmonic. The definitions extend without modification to the setting ofpseudo-Riemannian manifolds.

LetU be anopen subset ofℝⁿ and letV be an open subset ofℝ^m. For eachi andj between 1 andn, letg_ij be a smooth real-valued function onU, such that for eachp inU, one has that then ×nmatrix[g_ij(p)] issymmetric andpositive-definite. For eachα andβ between 1 andm, leth_αβ be a smooth real-valued function onV, such that for eachq inV, one has that them ×m matrix[h_αβ(q)] is symmetric and positive-definite. Denote theinverse matrices by[g^ij(p)] and[h^αβ(q)].

For eachi,j,k between 1 andn and eachα,β,γ between 1 andm define theChristoffel symbolsΓ(g)^k_ij :U → ℝ andΓ(h)^γ_αβ :V → ℝ by^[4]

Given a smooth mapf fromU toV, its second fundamental form defines for eachi andj between 1 andn and for eachα between 1 andm the real-valued function∇(df)^α_ij onU by^[5]

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

Its laplacian defines for eachα between 1 andn the real-valued function(∆f)^α onU by^[6]

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

Let(M,g) and(N,h) beRiemannian manifolds. Given a smooth mapf fromM toN, one can consider itsdifferentialdf as asection of thevector bundleT ^*M ⊗f ^*TN overM; this is to say that for eachp inM, one has a linear mapdf_p betweentangent spacesT_pM →T_f(p)N.^[7] The vector bundleT ^*M ⊗f ^*TN has aconnection induced from theLevi-Civita connections onM andN.^[8] So one may take thecovariant derivative∇(df), which is a section of the vector bundleT ^*M ⊗T ^*M ⊗f ^*TN overM; this is to say that for eachp inM, one has abilinear map(∇(df))_p of tangent spacesT_pM ×T_pM →T_f(p)N.^[9] This section is known as the hessian off.

Usingg, one maytrace the hessian off to arrive at the laplacian off, which is a section of the bundlef ^*TN overM; this says that the laplacian off assigns to eachp inM an element of the tangent spaceT_f(p)N.^[10] By the definition of the trace operator, the laplacian may be written as

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

wheree₁, ...,e_m is anyg_p-orthonormal basis ofT_pM.

From the perspective of local coordinates, as given above, theenergy density of a mappingf is the real-valued function onU given by^[11]

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

Alternatively, in the bundle formalism, the Riemannian metrics onM andN induce abundle metric onT ^*M ⊗f ^*TN, and so one may define the energy density as the smooth function⁠1/2⁠ |df |² onM.^[12] It is also possible to consider the energy density as being given by (half of) theg-trace of the first fundamental form.^[13] Regardless of the perspective taken, the energy densitye(f) is a function onM which is smooth and nonnegative. IfM is oriented andM is compact, theDirichlet energy off is defined as

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

wheredμ_g is thevolume form onM induced byg.^[14] Since any nonnegativemeasurable function has a well-definedLebesgue integral, it is not necessary to place the restriction thatM is compact; however, then the Dirichlet energy could be infinite.

Thevariation formulas for the Dirichlet energy compute the derivatives of the Dirichlet energyE(f) as the mappingf is deformed. To this end, consider a one-parameter family of mapsf_s :M →N withf₀ =f for which there exists a precompact open setK ofM such thatf_s|_{M −K} =f|_{M −K} for alls; 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.

• Thefirst variation formula says that^[15]

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

There is also a version for manifolds with boundary.^[16]

• There is also asecond variation formula.^[17]

Due to the first variation formula, the Laplacian off can be thought of as thegradient of the Dirichlet energy; correspondingly, a harmonic map is a critical point of the Dirichlet energy.^[18] This can be done formally in the language ofglobal analysis andBanach manifolds.

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

• Everytotally geodesic map(M,g) → (N,h) is harmonic; this follows directly from the above definitions. As special cases:
  • For anyq inN, the constant map(M,g) → (N,h) valued atq is harmonic.
  • The identity map(M,g) → (M,g) is harmonic.
• Iff :M →N is animmersion, thenf : (M,f ^*h) → (N,h) is harmonic if and only iff isminimal relative toh. As a special case:
  • Iff : ℝ → (N,h) is a constant-speed immersion, thenf : (ℝ,g_stan) → (N,h) is harmonic if and only iff solves thegeodesic differential equation.

Recall that ifM is one-dimensional, then minimality off is equivalent tof being geodesic, although this does not imply that it is a constant-speed parametrization, and hence does not imply thatf solves the geodesic differential equation.

• A smooth mapf : (M,g) → (ℝⁿ,g_stan) is harmonicif and only if each of itsn component functions are harmonic as maps(M,g) → (ℝ,g_stan). This coincides with the notion of harmonicity provided by theLaplace-Beltrami operator.
• Everyholomorphic map betweenKähler manifolds is harmonic.
• Everyharmonic morphism betweenRiemannian manifolds is harmonic.

Let(M,g) and(N,h) be smooth Riemannian manifolds. Aharmonic map heat flow on an interval(a,b) assigns to eacht in(a,b) a twice-differentiable mapf_t :M →N in such a way that, for eachp inM, the map(a,b) →N given byt ↦f_t(p) is differentiable, and its derivative at a given value oft is, as a vector inT_ft(p)N, equal to(∆ f_t)_p. This is usually abbreviated as:

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

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 thatM is a closed manifold and(N,h) is geodesically complete.

• Existence. Given a continuously differentiable mapf fromM toN, there exists a positive numberT and a harmonic map heat flowf_t on the interval(0,T) such thatf_t converges tof in theC¹ topology ast decreases to 0.^[19]
• Uniqueness. If{ f_t : 0 <t <T} and{ f_t : 0 <t <T } are two harmonic map heat flows as in the existence theorem, thenf_t =f_t whenever0 <t < min(T,T).

As a consequence of the uniqueness theorem, there exists amaximal harmonic map heat flow with initial dataf, 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 thatT takes on its maximal possible value, which could be infinite.

The primary result of Eells and Sampson's 1964 paper is the following:^[1]

Let(M,g) and(N,h) be smooth and closed Riemannian manifolds, and suppose that thesectional curvature of(N,h) is nonpositive. Then for any continuously differentiable mapf fromM toN, the maximal harmonic map heat flow{ f_t : 0 <t <T} with initial dataf hasT = ∞, and ast increases to∞, the mapsf_t subsequentially converge in theC^∞ topology to a harmonic map.

In particular, this shows that, under the assumptions on(M,g) and(N,h), every continuous map ishomotopic to a harmonic map.^[1] 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 byRichard Hamilton to the setting of theDirichlet boundary value problem, whenM is instead compact with nonempty boundary.^[20]

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.^[21] That is, if two maps are initially close, the distance between the corresponding solutions to the heat equation is nonincreasing for all time, thus:^[22]

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

^[23] notes that every map from a product${\displaystyle W\times M}$ into${\displaystyle N}$ is homotopic to a map, such that the map is totally geodesic when restricted to each${\displaystyle M}$-fiber.

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.^[24] 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 thatM is two-dimensional, he established the unconditional existence and uniqueness forweak solutions of the harmonic map heat flow.^[25] 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 abubble, i.e. a smooth harmonic map from the round 2-sphere into the target. Weiyue Ding andGang Tian were able to prove theenergy 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.^[26]

Struwe was later able to adapt his methods to higher dimensions, in the case that the domain manifold isEuclidean space;^[27] he and Yun Mei Chen also considered higher-dimensionalclosed manifolds.^[28] 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 main computational point in the proof of Eells and Sampson's theorem is an adaptation of theBochner formula to the setting of a harmonic map heat flow{ f_t : 0 <t <T}. This formula says^[29]

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

This is also of interest in analyzing harmonic maps. Supposef :M →N 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^[30]

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

If theRicci curvature ofg is positive and thesectional curvature ofh is nonpositive, then this implies that∆e(f) is nonnegative. IfM is closed, then multiplication bye(f) and a single integration by parts shows thate(f) must be constant, and hence zero; hencef must itself be constant.^[31]Richard Schoen andShing-Tung Yau noted that this reasoning can be extended to noncompactM by making use of Yau's theorem asserting that nonnegativesubharmonic functions which areL²-bounded must be constant.^[32] In summary, according to these results, one has:

Let(M,g) and(N,h) be smooth and complete Riemannian manifolds, and letf be a harmonic map fromM toN. Suppose that the Ricci curvature ofg is positive and the sectional curvature ofh is nonpositive.

• IfM andN are both closed thenf must be constant.
• IfN is closed andf 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 fromM toN 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 betweenKähler manifolds must be holomorphic, provided that the target manifold has appropriately negative curvature.^[33] 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, thenM andN 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-unsolvedHodge 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 newrigidity theorems for lattices in certainLie groups.^[34] Following this,Mikhael Gromov and Richard Schoen extended much of the theory of harmonic maps to allow(N,h) to be replaced by ametric space.^[35] 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.

• Existence results on harmonic maps between manifolds has consequences for theircurvature.
• Once existence is known, how can a harmonic map be constructed explicitly? (One fruitful method usestwistor theory.)
• Intheoretical physics, aquantum field theory whoseaction is given by theDirichlet energy is known as asigma model. In such a theory, harmonic maps correspond toinstantons.
• 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${\displaystyle u:M\to N}$ between Riemannian manifolds is totally geodesic if, whenever${\displaystyle \gamma :(a,b)\to M}$ is a geodesic, the composition${\displaystyle u\circ \gamma }$ is a geodesic.

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

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

in which μ^ε
_x is a family ofmeasures attached to each point ofM.^[36]

• Geometric flow

Footnotes

Articles

Books and surveys

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

• Riemannian geometry
• Harmonic functions
• Analytic functions

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• All articles lacking in-text citations
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