{"history":[],"article":{"id":"a5b628a3-342f-4b19-bed1-f7446c9030a2","title":"Harmonic map","slug":"harmonic-map","content":":::summary\n:::\n\n# \nIn the mathematical field of [differential geometry](https://bliptext.com/articles/differential-geometry), a smooth map between [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold) is called **harmonic** if its coordinate representatives satisfy a certain nonlinear [partial differential equation](https://bliptext.com/articles/partial-differential-equation).\nThis partial differential equation for a mapping also arises as the [Euler-Lagrange equation](https://bliptext.com/articles/euler-lagrange-equation) of a functional called the [Dirichlet energy](https://bliptext.com/articles/dirichlet-energy).\nAs such, the theory of harmonic maps contains both the theory of [unit-speed geodesics](https://bliptext.com/articles/geodesic) in [Riemannian geometry](https://bliptext.com/articles/riemannian-geometry) and the theory of [harmonic functions](https://bliptext.com/articles/harmonic-function).\nInformally, 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](https://bliptext.com/articles/rubber-band) and a smooth stone can both be naturally viewed as Riemannian manifolds.\nAny 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.\nHarmonicity 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.\nThe theory of harmonic maps was initiated in 1964 by [James Eells](https://bliptext.com/articles/james-eells) and [Joseph Sampson](https://bliptext.com/articles/joseph-h-sampson), who showed that in certain geometric contexts, arbitrary maps could be [deformed](https://bliptext.com/articles/homotopy) into harmonic maps.\nTheir work was the inspiration for [Richard Hamilton](https://bliptext.com/articles/richard-s-hamilton)'s initial work on the [Ricci flow](https://bliptext.com/articles/ricci-flow).\nHarmonic 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](https://bliptext.com/articles/geometric-analysis).\nThe discovery of the \"bubbling\" of sequences of harmonic maps, due to Jonathan Sacks and [Karen Uhlenbeck](https://bliptext.com/articles/karen-uhlenbeck), has been particularly influential, as their analysis has been adapted to many other geometric contexts.\nNotably, Uhlenbeck's parallel discovery of bubbling of Yang–Mills fields is important in [Simon Donaldson](https://bliptext.com/articles/simon-donaldson)'s work on four-dimensional manifolds, and [Mikhael Gromov](https://bliptext.com/articles/mikhail-leonidovich-gromov)'s later discovery of bubbling of [pseudoholomorphic curves](https://bliptext.com/articles/pseudoholomorphic-curve) is significant in applications to [symplectic geometry](https://bliptext.com/articles/symplectic-geometry) and [quantum cohomology](https://bliptext.com/articles/quantum-cohomology).\nThe techniques used by [Richard Schoen](https://bliptext.com/articles/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.\n\n# Geometry of mappings between manifolds\nHere the geometry of a smooth mapping between [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold) is considered via [local coordinates](https://bliptext.com/articles/local-coordinates) and, equivalently, via [linear algebra](https://bliptext.com/articles/linear-algebra).\nSuch 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**.\nThe definitions extend without modification to the setting of [pseudo-Riemannian manifolds](https://bliptext.com/articles/pseudo-riemannian-manifold).\n\n## Local coordinates\nLet U be an [open subset](https://bliptext.com/articles/open-set) of [[Euclidean space|\nℝn\n]] and let V be an open subset of\nℝm\n. For each i and j between 1 and n, let\ngij\nbe a smooth real-valued function on U, such that for each p in U, one has that the\nn × n\n[matrix](https://bliptext.com/articles/matrix-mathematics)\n[gij(p)]\nis [symmetric](https://bliptext.com/articles/symmetric-matrix) and [positive-definite](https://bliptext.com/articles/definite-matrix).\nFor each α and β between 1 and m, let\nhαβ\nbe a smooth real-valued function on V, such that for each q in V, one has that the\nm × m\nmatrix\n[hαβ(q)]\nis symmetric and positive-definite.\nDenote the [inverse matrices](https://bliptext.com/articles/invertible-matrix) by\n[gij(p)]\nand\n[hαβ(q)]\n.\nFor each\ni, j, k\nbetween 1 and n and each\nα, β, γ\nbetween 1 and m define the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbol)\nΓ(g)kij : U → ℝ\nand\nΓ(h)γαβ : V → ℝ\nby Given a smooth map f from U to V, its second fundamental form defines for each i and j between 1 and n and for each α between 1 and m the real-valued function\n∇(df)αij\non U by Its laplacian defines for each α between 1 and n the real-valued function\n(∆f)α\non U by\n\n## Bundle formalism\nLet\n(M, g)\nand\n(N, h)\nbe [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold).\nGiven a smooth map f from M to N, one can consider its [differential](https://bliptext.com/articles/pushforward-differential)\ndf\nas a [section](https://bliptext.com/articles/section-fiber-bundle) of the [vector bundle](https://bliptext.com/articles/vector-bundle)\nT*M ⊗ f*TN\nover M; this is to say that for each p in M, one has a linear map\ndfp\nbetween [tangent spaces](https://bliptext.com/articles/tangent-space)\nTpM → Tf(p)N\n. The vector bundle\nT*M ⊗ f*TN\nhas a [connection](https://bliptext.com/articles/connection-vector-bundle) induced from the Levi-Civita [connection](https://bliptext.com/articles/connection-vector-bundle)s on M and N. So one may take the [covariant derivative](https://bliptext.com/articles/covariant-derivative)\n∇(df)\n, which is a section of the vector bundle\nT*M ⊗ T*M ⊗ f*TN\nover M; this is to say that for each p in M, one has a [bilinear map](https://bliptext.com/articles/bilinear-map)\n(∇(df))p\nof tangent spaces\nTpM × TpM → Tf(p)N\n. This section is known as the hessian of f.\nUsing g, one may [trace](https://bliptext.com/articles/trace-linear-algebra) the hessian of f to arrive at the laplacian of f, which is a section of the bundle\nf*TN\nover M; this says that the laplacian of f assigns to each p in M an element of the tangent space\nTf(p)N\n. By the definition of the trace operator, the laplacian may be written as where\ne1, ..., em\nis any\ngp\n-orthonormal basis of\nTpM\n.\n\n# Dirichlet energy and its variation formulas\nFrom the perspective of local coordinates, as given above, the **energy density** of a mapping f is the real-valued function on U given by Alternatively, in the bundle formalism, the Riemannian metrics on M and N induce a [bundle metric](https://bliptext.com/articles/bundle-metric) on\nT*M ⊗ f*TN\n, and so one may define the energy density as the smooth function\n1⁄2\non M. It is also possible to consider the energy density as being given by (half of) the g-trace of the first fundamental form.\nRegardless of the perspective taken, the energy density\ne(f)\nis a function on M which is smooth and nonnegative.\nIf M is oriented and M is compact, the **Dirichlet energy** of f is defined as where\ndμg\nis the volume form on M induced by g. Since any nonnegative [measurable function](https://bliptext.com/articles/measurable-function) has a well-defined [Lebesgue integral](https://bliptext.com/articles/lebesgue-integral), it is not necessary to place the restriction that M is compact; however, then the Dirichlet energy could be infinite.\nThe **variation formulas** for the Dirichlet energy compute the derivatives of the Dirichlet energy\nE(f)\nas the mapping f is deformed.\nTo this end, consider a one-parameter family of maps\nfs : M → N\nwith\nfor which there exists a precompact open set K of M such that\nfs\nfor all s; one supposes that the parametrized family is smooth in the sense that the associated map\n(−ε, ε) × M → N\ngiven by\n(s, p) ↦ fs(p)\nis smooth.\nDue to the first variation formula, the Laplacian of f can be thought of as the [gradient](https://bliptext.com/articles/gradient) of the Dirichlet energy; correspondingly, a harmonic map is a critical point of the Dirichlet energy.\nThis can be done formally in the language of [global analysis](https://bliptext.com/articles/global-analysis) and [Banach manifolds](https://bliptext.com/articles/banach-manifold).\n\n# Examples of harmonic maps\nLet\n(M, g)\nand\n(N, h)\nbe smooth Riemannian manifolds.\nThe notation\ngstan\nis used to refer to the standard Riemannian metric on Euclidean space.\n(M, g) → (N, h)\nis harmonic; this follows directly from the above definitions.\nAs special cases:\n(M, g) → (N, h)\nvalued at q is harmonic.\n(M, g) → (M, g)\nis harmonic.\nf : M → N\nis an [immersion](https://bliptext.com/articles/immersion-mathematics), then\nf : (M, f*h) → (N, h)\nis harmonic if and only if f is [minimal](https://bliptext.com/articles/minimal-submanifold) relative to h. As a special case:\nf : ℝ → (N, h)\nis a constant-speed immersion, then\nf : (ℝ, gstan) → (N, h)\nis harmonic if and only if f solves the [geodesic](https://bliptext.com/articles/geodesic) differential equation.\nf : (M, g) → (ℝn, gstan)\nis harmonic if and only if each of its n component functions are harmonic as maps\n(M, g) → (ℝ, gstan)\n. This coincides with the notion of harmonicity provided by the [Laplace-Beltrami operator](https://bliptext.com/articles/laplace-beltrami-operator).\n\n# Harmonic map heat flow\n\n## Well-posedness\nLet\n(M, g)\nand\n(N, h)\nbe smooth Riemannian manifolds.\nA **harmonic map heat flow** on an interval\n(a, b)\nassigns to each t in\n(a, b)\na twice-differentiable map\nft : M → N\nin such a way that, for each p in M, the map\n(a, b) → N\ngiven by\nt ↦ ft (p)\nis differentiable, and its derivative at a given value of t is, as a vector in\nTf t (p) N\n, equal to\n(∆ ft )p\n. This is usually abbreviated as:\nEells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties:\n(a, b) × M → N\ngiven by\n(t, p) ↦ ft(p)\n. Now suppose that M is a closed manifold and\n(N, h)\nis geodesically complete.\nf\nfrom M to N, there exists a positive number T and a harmonic map heat flow\nft\non the interval\n(0, T)\nsuch that\nft\nconverges to\nf\nin the\nC1\ntopology as t decreases to 0.\n{ ft : 0 < t < T }\nand\n{ f t : 0 < t < T }\nare two harmonic map heat flows as in the existence theorem, then\nwhenever\n0 < t < min(T, T)\n. As a consequence of the uniqueness theorem, there exists a **maximal** harmonic map heat flow with initial data\nf\n, meaning that one has a harmonic map heat flow\n{ ft : 0 < t < T }\nas 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.\n\n## Eells and Sampson's theorem\nThe primary result of Eells and Sampson's 1964 paper is the following:\n\"Let\n(M, g)\nand\n(N, h)\nbe smooth and closed Riemannian manifolds, and suppose that the sectional curvature of\n(N, h)\nis nonpositive.\nThen for any continuously differentiable map f from M to N, the maximal harmonic map heat flow\n{ ft : 0 < t < T }\nwith initial data f has\n, and as t increases to ∞, the maps\nft\nsubsequentially converge in the\nC∞\ntopology to a harmonic map.\"\nIn particular, this shows that, under the assumptions on\n(M, g)\nand\n(N, h)\n, every continuous map is [homotopic](https://bliptext.com/articles/homotopy) to a harmonic map.\nThe very existence of a harmonic map in each homotopy class, which is implicitly being asserted, is part of the result.\nThis 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.\nEells and Sampson's result was adapted by [Richard Hamilton](https://bliptext.com/articles/richard-s-hamilton) to the setting of the [Dirichlet boundary value problem](https://bliptext.com/articles/dirichlet-boundary-condition), when M is instead compact with nonempty boundary.\nShortly after Eells and Sampson's work, [Philip Hartman](https://bliptext.com/articles/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.\nThat is, if two maps are initially close, the distance between the corresponding solutions to the heat equation is nonincreasing for all time, thus:\nnotes 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.\n\n## Singularities and weak solutions\nFor many years after Eells and Sampson's work, it was unclear to what extent the sectional curvature assumption on\n(N, h)\nwas necessary.\nFollowing 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.\nTheir results strongly suggest that there are harmonic map heat flows with \"finite-time blowup\" even when both\n(M, g)\nand\n(N, h)\nare taken to be the two-dimensional sphere with its standard metric.\nSince 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.\nModeled upon the fundamental works of Sacks and Uhlenbeck, [Michael Struwe](https://bliptext.com/articles/michael-struwe) considered the case where no geometric assumption on\n(N, h)\nis made.\nIn the case that M is two-dimensional, he established the unconditional existence and uniqueness for [weak solutions](https://bliptext.com/articles/weak-solution) of the harmonic map heat flow.\nMoreover, he found that his weak solutions are smooth away from finitely many spacetime points at which the energy density concentrates.\nOn 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.\nWeiyue Ding and [Gang Tian](https://bliptext.com/articles/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.\nStruwe was later able to adapt his methods to higher dimensions, in the case that the domain manifold is [Euclidean space](https://bliptext.com/articles/euclidean-space); he and Yun Mei Chen also considered higher-dimensional [closed manifolds](https://bliptext.com/articles/closed-manifold).\nTheir results achieved less than in low dimensions, only being able to prove existence of weak solutions which are smooth on open dense subsets.\n\n# The Bochner formula and rigidity\nThe main computational point in the proof of Eells and Sampson's theorem is an adaptation of the [Bochner formula](https://bliptext.com/articles/bochner-s-formula) to the setting of a harmonic map heat flow\n{ ft : 0 < t < T }\n. This formula says\nThis is also of interest in analyzing harmonic maps.\nSuppose\nf : M → N\nis 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\nIf the [Ricci curvature](https://bliptext.com/articles/ricci-curvature) of g is positive and the [sectional curvature](https://bliptext.com/articles/sectional-curvature) of h is nonpositive, then this implies that\n∆e(f)\nis nonnegative.\nIf M is closed, then multiplication by\ne(f)\nand a single integration by parts shows that\ne(f)\nmust be constant, and hence zero; hence f must itself be constant.\n[Richard Schoen](https://bliptext.com/articles/richard-schoen) and [Shing-Tung Yau](https://bliptext.com/articles/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](https://bliptext.com/articles/subharmonic-function) which are [[Lp space|\nL2\n-bounded]] must be constant.\nIn summary, according to these results, one has:\n\"Let\n(M, g)\nand\n(N, h)\nbe 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.\nIn combination with the Eells−Sampson theorem, this shows (for instance) that if\n(M, g)\nis a closed Riemannian manifold with positive Ricci curvature and\n(N, h)\nis a closed Riemannian manifold with nonpositive sectional curvature, then every continuous map from M to N is homotopic to a constant.\nThe 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.\nFor instance, [Yum-Tong Siu](https://bliptext.com/articles/yum-tong-siu) found an important complex-analytic version of the Bochner formula, asserting that a harmonic map between [Kähler manifolds](https://bliptext.com/articles/k-hler-manifold) must be holomorphic, provided that the target manifold has appropriately negative curvature.\nAs an application, by making use of the Eells−Sampson existence theorem for harmonic maps, he was able to show that if\n(M, g)\nand\n(N, h)\nare smooth and closed Kähler manifolds, and if the curvature of\n(N, h)\nis 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.\nBy an alternative formulation of the same approach, Siu was able to prove a variant of the still-unsolved [Hodge conjecture](https://bliptext.com/articles/hodge-conjecture), albeit in the restricted context of negative curvature.\nKevin Corlette found a significant extension of Siu's Bochner formula, and used it to prove new [rigidity theorems](https://bliptext.com/articles/superrigidity) for lattices in certain [Lie groups](https://bliptext.com/articles/lie-group).\nFollowing this, [Mikhael Gromov](https://bliptext.com/articles/mikhail-leonidovich-gromov) and Richard Schoen extended much of the theory of harmonic maps to allow\n(N, h)\nto be replaced by a [metric space](https://bliptext.com/articles/metric-space).\nBy 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.\n\n# Problems and applications\nA map between Riemannian manifolds is totally geodesic if, whenever is a geodesic, the composition is a geodesic.\n\n# Harmonic maps between metric spaces\nThe energy integral can be formulated in a weaker setting for functions u : M → N between two [metric spaces](https://bliptext.com/articles/metric-space).\nThe energy integrand is instead a function of the form\nin which με x is a family of [measures](https://bliptext.com/articles/measure-mathematics) attached to each point of M.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T07:25:43.182Z","updated_at":"2025-01-26T07:25:43.182Z","search_vector":"'harmon':1 'map':2","search_content":" \n \n\n# \nIn the mathematical field of [differential geometry](https://bliptext.com/articles/differential-geometry), a smooth map between [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold) is called **harmonic** if its coordinate representatives satisfy a certain nonlinear [partial differential equation](https://bliptext.com/articles/partial-differential-equation).\nThis partial differential equation for a mapping also arises as the [Euler-Lagrange equation](https://bliptext.com/articles/euler-lagrange-equation) of a functional called the [Dirichlet energy](https://bliptext.com/articles/dirichlet-energy).\nAs such, the theory of harmonic maps contains both the theory of [unit-speed geodesics](https://bliptext.com/articles/geodesic) in [Riemannian geometry](https://bliptext.com/articles/riemannian-geometry) and the theory of [harmonic functions](https://bliptext.com/articles/harmonic-function).\nInformally, 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](https://bliptext.com/articles/rubber-band) and a smooth stone can both be naturally viewed as Riemannian manifolds.\nAny 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.\nHarmonicity 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.\nThe theory of harmonic maps was initiated in 1964 by [James Eells](https://bliptext.com/articles/james-eells) and [Joseph Sampson](https://bliptext.com/articles/joseph-h-sampson), who showed that in certain geometric contexts, arbitrary maps could be [deformed](https://bliptext.com/articles/homotopy) into harmonic maps.\nTheir work was the inspiration for [Richard Hamilton](https://bliptext.com/articles/richard-s-hamilton)'s initial work on the [Ricci flow](https://bliptext.com/articles/ricci-flow).\nHarmonic 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](https://bliptext.com/articles/geometric-analysis).\nThe discovery of the \"bubbling\" of sequences of harmonic maps, due to Jonathan Sacks and [Karen Uhlenbeck](https://bliptext.com/articles/karen-uhlenbeck), has been particularly influential, as their analysis has been adapted to many other geometric contexts.\nNotably, Uhlenbeck's parallel discovery of bubbling of Yang–Mills fields is important in [Simon Donaldson](https://bliptext.com/articles/simon-donaldson)'s work on four-dimensional manifolds, and [Mikhael Gromov](https://bliptext.com/articles/mikhail-leonidovich-gromov)'s later discovery of bubbling of [pseudoholomorphic curves](https://bliptext.com/articles/pseudoholomorphic-curve) is significant in applications to [symplectic geometry](https://bliptext.com/articles/symplectic-geometry) and [quantum cohomology](https://bliptext.com/articles/quantum-cohomology).\nThe techniques used by [Richard Schoen](https://bliptext.com/articles/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.\n\n# Geometry of mappings between manifolds\nHere the geometry of a smooth mapping between [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold) is considered via [local coordinates](https://bliptext.com/articles/local-coordinates) and, equivalently, via [linear algebra](https://bliptext.com/articles/linear-algebra).\nSuch 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**.\nThe definitions extend without modification to the setting of [pseudo-Riemannian manifolds](https://bliptext.com/articles/pseudo-riemannian-manifold).\n\n## Local coordinates\nLet U be an [open subset](https://bliptext.com/articles/open-set) of [[Euclidean space|\nℝn\n]] and let V be an open subset of\nℝm\n. For each i and j between 1 and n, let\ngij\nbe a smooth real-valued function on U, such that for each p in U, one has that the\nn × n\n[matrix](https://bliptext.com/articles/matrix-mathematics)\n[gij(p)]\nis [symmetric](https://bliptext.com/articles/symmetric-matrix) and [positive-definite](https://bliptext.com/articles/definite-matrix).\nFor each α and β between 1 and m, let\nhαβ\nbe a smooth real-valued function on V, such that for each q in V, one has that the\nm × m\nmatrix\n[hαβ(q)]\nis symmetric and positive-definite.\nDenote the [inverse matrices](https://bliptext.com/articles/invertible-matrix) by\n[gij(p)]\nand\n[hαβ(q)]\n.\nFor each\ni, j, k\nbetween 1 and n and each\nα, β, γ\nbetween 1 and m define the [Christoffel symbols](https://bliptext.com/articles/christoffel-symbol)\nΓ(g)kij : U → ℝ\nand\nΓ(h)γαβ : V → ℝ\nby Given a smooth map f from U to V, its second fundamental form defines for each i and j between 1 and n and for each α between 1 and m the real-valued function\n∇(df)αij\non U by Its laplacian defines for each α between 1 and n the real-valued function\n(∆f)α\non U by\n\n## Bundle formalism\nLet\n(M, g)\nand\n(N, h)\nbe [Riemannian manifolds](https://bliptext.com/articles/riemannian-manifold).\nGiven a smooth map f from M to N, one can consider its [differential](https://bliptext.com/articles/pushforward-differential)\ndf\nas a [section](https://bliptext.com/articles/section-fiber-bundle) of the [vector bundle](https://bliptext.com/articles/vector-bundle)\nT*M ⊗ f*TN\nover M; this is to say that for each p in M, one has a linear map\ndfp\nbetween [tangent spaces](https://bliptext.com/articles/tangent-space)\nTpM → Tf(p)N\n. The vector bundle\nT*M ⊗ f*TN\nhas a [connection](https://bliptext.com/articles/connection-vector-bundle) induced from the Levi-Civita [connection](https://bliptext.com/articles/connection-vector-bundle)s on M and N. So one may take the [covariant derivative](https://bliptext.com/articles/covariant-derivative)\n∇(df)\n, which is a section of the vector bundle\nT*M ⊗ T*M ⊗ f*TN\nover M; this is to say that for each p in M, one has a [bilinear map](https://bliptext.com/articles/bilinear-map)\n(∇(df))p\nof tangent spaces\nTpM × TpM → Tf(p)N\n. This section is known as the hessian of f.\nUsing g, one may [trace](https://bliptext.com/articles/trace-linear-algebra) the hessian of f to arrive at the laplacian of f, which is a section of the bundle\nf*TN\nover M; this says that the laplacian of f assigns to each p in M an element of the tangent space\nTf(p)N\n. By the definition of the trace operator, the laplacian may be written as where\ne1, ..., em\nis any\ngp\n-orthonormal basis of\nTpM\n.\n\n# Dirichlet energy and its variation formulas\nFrom the perspective of local coordinates, as given above, the **energy density** of a mapping f is the real-valued function on U given by Alternatively, in the bundle formalism, the Riemannian metrics on M and N induce a [bundle metric](https://bliptext.com/articles/bundle-metric) on\nT*M ⊗ f*TN\n, and so one may define the energy density as the smooth function\n1⁄2\non M. It is also possible to consider the energy density as being given by (half of) the g-trace of the first fundamental form.\nRegardless of the perspective taken, the energy density\ne(f)\nis a function on M which is smooth and nonnegative.\nIf M is oriented and M is compact, the **Dirichlet energy** of f is defined as where\ndμg\nis the volume form on M induced by g. Since any nonnegative [measurable function](https://bliptext.com/articles/measurable-function) has a well-defined [Lebesgue integral](https://bliptext.com/articles/lebesgue-integral), it is not necessary to place the restriction that M is compact; however, then the Dirichlet energy could be infinite.\nThe **variation formulas** for the Dirichlet energy compute the derivatives of the Dirichlet energy\nE(f)\nas the mapping f is deformed.\nTo this end, consider a one-parameter family of maps\nfs : M → N\nwith\nfor which there exists a precompact open set K of M such that\nfs\nfor all s; one supposes that the parametrized family is smooth in the sense that the associated map\n(−ε, ε) × M → N\ngiven by\n(s, p) ↦ fs(p)\nis smooth.\nDue to the first variation formula, the Laplacian of f can be thought of as the [gradient](https://bliptext.com/articles/gradient) of the Dirichlet energy; correspondingly, a harmonic map is a critical point of the Dirichlet energy.\nThis can be done formally in the language of [global analysis](https://bliptext.com/articles/global-analysis) and [Banach manifolds](https://bliptext.com/articles/banach-manifold).\n\n# Examples of harmonic maps\nLet\n(M, g)\nand\n(N, h)\nbe smooth Riemannian manifolds.\nThe notation\ngstan\nis used to refer to the standard Riemannian metric on Euclidean space.\n(M, g) → (N, h)\nis harmonic; this follows directly from the above definitions.\nAs special cases:\n(M, g) → (N, h)\nvalued at q is harmonic.\n(M, g) → (M, g)\nis harmonic.\nf : M → N\nis an [immersion](https://bliptext.com/articles/immersion-mathematics), then\nf : (M, f*h) → (N, h)\nis harmonic if and only if f is [minimal](https://bliptext.com/articles/minimal-submanifold) relative to h. As a special case:\nf : ℝ → (N, h)\nis a constant-speed immersion, then\nf : (ℝ, gstan) → (N, h)\nis harmonic if and only if f solves the [geodesic](https://bliptext.com/articles/geodesic) differential equation.\nf : (M, g) → (ℝn, gstan)\nis harmonic if and only if each of its n component functions are harmonic as maps\n(M, g) → (ℝ, gstan)\n. This coincides with the notion of harmonicity provided by the [Laplace-Beltrami operator](https://bliptext.com/articles/laplace-beltrami-operator).\n\n# Harmonic map heat flow\n\n## Well-posedness\nLet\n(M, g)\nand\n(N, h)\nbe smooth Riemannian manifolds.\nA **harmonic map heat flow** on an interval\n(a, b)\nassigns to each t in\n(a, b)\na twice-differentiable map\nft : M → N\nin such a way that, for each p in M, the map\n(a, b) → N\ngiven by\nt ↦ ft (p)\nis differentiable, and its derivative at a given value of t is, as a vector in\nTf t (p) N\n, equal to\n(∆ ft )p\n. This is usually abbreviated as:\nEells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties:\n(a, b) × M → N\ngiven by\n(t, p) ↦ ft(p)\n. Now suppose that M is a closed manifold and\n(N, h)\nis geodesically complete.\nf\nfrom M to N, there exists a positive number T and a harmonic map heat flow\nft\non the interval\n(0, T)\nsuch that\nft\nconverges to\nf\nin the\nC1\ntopology as t decreases to 0.\n{ ft : 0 < t < T }\nand\n{ f t : 0 < t < T }\nare two harmonic map heat flows as in the existence theorem, then\nwhenever\n0 < t < min(T, T)\n. As a consequence of the uniqueness theorem, there exists a **maximal** harmonic map heat flow with initial data\nf\n, meaning that one has a harmonic map heat flow\n{ ft : 0 < t < T }\nas 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.\n\n## Eells and Sampson's theorem\nThe primary result of Eells and Sampson's 1964 paper is the following:\n\"Let\n(M, g)\nand\n(N, h)\nbe smooth and closed Riemannian manifolds, and suppose that the sectional curvature of\n(N, h)\nis nonpositive.\nThen for any continuously differentiable map f from M to N, the maximal harmonic map heat flow\n{ ft : 0 < t < T }\nwith initial data f has\n, and as t increases to ∞, the maps\nft\nsubsequentially converge in the\nC∞\ntopology to a harmonic map.\"\nIn particular, this shows that, under the assumptions on\n(M, g)\nand\n(N, h)\n, every continuous map is [homotopic](https://bliptext.com/articles/homotopy) to a harmonic map.\nThe very existence of a harmonic map in each homotopy class, which is implicitly being asserted, is part of the result.\nThis 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.\nEells and Sampson's result was adapted by [Richard Hamilton](https://bliptext.com/articles/richard-s-hamilton) to the setting of the [Dirichlet boundary value problem](https://bliptext.com/articles/dirichlet-boundary-condition), when M is instead compact with nonempty boundary.\nShortly after Eells and Sampson's work, [Philip Hartman](https://bliptext.com/articles/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.\nThat is, if two maps are initially close, the distance between the corresponding solutions to the heat equation is nonincreasing for all time, thus:\nnotes 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.\n\n## Singularities and weak solutions\nFor many years after Eells and Sampson's work, it was unclear to what extent the sectional curvature assumption on\n(N, h)\nwas necessary.\nFollowing 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.\nTheir results strongly suggest that there are harmonic map heat flows with \"finite-time blowup\" even when both\n(M, g)\nand\n(N, h)\nare taken to be the two-dimensional sphere with its standard metric.\nSince 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.\nModeled upon the fundamental works of Sacks and Uhlenbeck, [Michael Struwe](https://bliptext.com/articles/michael-struwe) considered the case where no geometric assumption on\n(N, h)\nis made.\nIn the case that M is two-dimensional, he established the unconditional existence and uniqueness for [weak solutions](https://bliptext.com/articles/weak-solution) of the harmonic map heat flow.\nMoreover, he found that his weak solutions are smooth away from finitely many spacetime points at which the energy density concentrates.\nOn 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.\nWeiyue Ding and [Gang Tian](https://bliptext.com/articles/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.\nStruwe was later able to adapt his methods to higher dimensions, in the case that the domain manifold is [Euclidean space](https://bliptext.com/articles/euclidean-space); he and Yun Mei Chen also considered higher-dimensional [closed manifolds](https://bliptext.com/articles/closed-manifold).\nTheir results achieved less than in low dimensions, only being able to prove existence of weak solutions which are smooth on open dense subsets.\n\n# The Bochner formula and rigidity\nThe main computational point in the proof of Eells and Sampson's theorem is an adaptation of the [Bochner formula](https://bliptext.com/articles/bochner-s-formula) to the setting of a harmonic map heat flow\n{ ft : 0 < t < T }\n. This formula says\nThis is also of interest in analyzing harmonic maps.\nSuppose\nf : M → N\nis 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\nIf the [Ricci curvature](https://bliptext.com/articles/ricci-curvature) of g is positive and the [sectional curvature](https://bliptext.com/articles/sectional-curvature) of h is nonpositive, then this implies that\n∆e(f)\nis nonnegative.\nIf M is closed, then multiplication by\ne(f)\nand a single integration by parts shows that\ne(f)\nmust be constant, and hence zero; hence f must itself be constant.\n[Richard Schoen](https://bliptext.com/articles/richard-schoen) and [Shing-Tung Yau](https://bliptext.com/articles/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](https://bliptext.com/articles/subharmonic-function) which are [[Lp space|\nL2\n-bounded]] must be constant.\nIn summary, according to these results, one has:\n\"Let\n(M, g)\nand\n(N, h)\nbe 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.\nIn combination with the Eells−Sampson theorem, this shows (for instance) that if\n(M, g)\nis a closed Riemannian manifold with positive Ricci curvature and\n(N, h)\nis a closed Riemannian manifold with nonpositive sectional curvature, then every continuous map from M to N is homotopic to a constant.\nThe 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.\nFor instance, [Yum-Tong Siu](https://bliptext.com/articles/yum-tong-siu) found an important complex-analytic version of the Bochner formula, asserting that a harmonic map between [Kähler manifolds](https://bliptext.com/articles/k-hler-manifold) must be holomorphic, provided that the target manifold has appropriately negative curvature.\nAs an application, by making use of the Eells−Sampson existence theorem for harmonic maps, he was able to show that if\n(M, g)\nand\n(N, h)\nare smooth and closed Kähler manifolds, and if the curvature of\n(N, h)\nis 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.\nBy an alternative formulation of the same approach, Siu was able to prove a variant of the still-unsolved [Hodge conjecture](https://bliptext.com/articles/hodge-conjecture), albeit in the restricted context of negative curvature.\nKevin Corlette found a significant extension of Siu's Bochner formula, and used it to prove new [rigidity theorems](https://bliptext.com/articles/superrigidity) for lattices in certain [Lie groups](https://bliptext.com/articles/lie-group).\nFollowing this, [Mikhael Gromov](https://bliptext.com/articles/mikhail-leonidovich-gromov) and Richard Schoen extended much of the theory of harmonic maps to allow\n(N, h)\nto be replaced by a [metric space](https://bliptext.com/articles/metric-space).\nBy 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.\n\n# Problems and applications\nA map between Riemannian manifolds is totally geodesic if, whenever is a geodesic, the composition is a geodesic.\n\n# Harmonic maps between metric spaces\nThe energy integral can be formulated in a weaker setting for functions u : M → N between two [metric spaces](https://bliptext.com/articles/metric-space).\nThe energy integrand is instead a function of the form\nin which με x is a family of [measures](https://bliptext.com/articles/measure-mathematics) attached to each point of M.","revisionCount":0},"hasMore":false,"nextCursor":null}

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