Английская Википедия:Harmonic map

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Шаблон:About Шаблон: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 Шаблон:Mvar from a Riemannian manifold Шаблон:Mvar to a Riemannian manifold Шаблон:Mvar can be thought of as the total amount that Шаблон:Mvar stretches Шаблон:Mvar in allocating each of its elements to a point of Шаблон: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.Шаблон: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,Шаблон: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.Шаблон: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 Шаблон:Mvar be an open subset of [[Euclidean space|Шаблон:Math]] and let Шаблон:Mvar be an open subset of Шаблон:Math. For each Шаблон:Mvar and Шаблон:Mvar between 1 and Шаблон:Mvar, let Шаблон:Math be a smooth real-valued function on Шаблон:Mvar, such that for each Шаблон:Mvar in Шаблон:Mvar, one has that the Шаблон:Math matrix Шаблон:Math is symmetric and positive-definite. For each Шаблон:Mvar and Шаблон:Mvar between 1 and Шаблон:Mvar, let Шаблон:Math be a smooth real-valued function on Шаблон:Mvar, such that for each Шаблон:Mvar in Шаблон:Mvar, one has that the Шаблон:Math matrix Шаблон:Math is symmetric and positive-definite. Denote the inverse matrices by Шаблон:Math and Шаблон:Math.

For each Шаблон:Math between 1 and Шаблон:Mvar and each Шаблон:Math between 1 and Шаблон:Mvar define the Christoffel symbols Шаблон:Math and Шаблон:Math byШаблон:Sfnm

<math>\begin{align}

\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{align}</math> Given a smooth map Шаблон:Mvar from Шаблон:Mvar to Шаблон:Mvar, its second fundamental form defines for each Шаблон:Mvar and Шаблон:Mvar between 1 and Шаблон:Mvar and for each Шаблон:Mvar between 1 and Шаблон:Mvar the real-valued function Шаблон:Math on Шаблон:Mvar byШаблон:Sfnm

<math>\nabla(df)_{ij}^\alpha=\frac{\partial^2f^\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.</math>

Its laplacian defines for each Шаблон:Mvar between 1 and Шаблон:Mvar the real-valued function Шаблон:Math on Шаблон:Mvar byШаблон:Sfnm

<math>(\Delta f)^\alpha=\sum_{i=1}^m\sum_{j=1}^mg^{ij}\nabla(df)_{ij}^\alpha.</math>

Bundle formalism

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

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

<math>(\Delta f)_p=\sum_{i=1}^m\big(\nabla(df)\big)_p(e_i,e_i)</math>

where Шаблон:Math is any Шаблон:Math-orthonormal basis of Шаблон:Math.

Dirichlet energy and its variation formulas

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

<math>\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).</math>

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

<math>E(f)=\int_M e(f)\,d\mu_g</math>

where Шаблон:Math is the volume form on Шаблон:Mvar induced by Шаблон:Mvar.Шаблон:Sfnm Since any nonnegative measurable function has a well-defined Lebesgue integral, it is not necessary to place the restriction that Шаблон: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 Шаблон:Math as the mapping Шаблон:Mvar is deformed. To this end, consider a one-parameter family of maps Шаблон:Math with Шаблон:Math for which there exists a precompact open set Шаблон:Mvar of Шаблон:Mvar such that Шаблон:Math for all Шаблон:Mvar; one supposes that the parametrized family is smooth in the sense that the associated map Шаблон:Math given by Шаблон:Math is smooth.

<math>\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</math>
There is also a version for manifolds with boundary.Шаблон:Sfnm

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

Examples of harmonic maps

Let Шаблон:Math and Шаблон:Math be smooth Riemannian manifolds. The notation Шаблон:Math is used to refer to the standard Riemannian metric on Euclidean space.

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

Harmonic map heat flow

Well-posedness

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

<math>\frac{\partial f}{\partial t}=\Delta f.</math>

Eells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties:

Now suppose that Шаблон:Mvar is a closed manifold and Шаблон:Math is geodesically complete.

As a consequence of the uniqueness theorem, there exists a maximal harmonic map heat flow with initial data Шаблон:Math, meaning that one has a harmonic map heat flow Шаблон:Math as in the statement of the existence theorem, and it is uniquely defined under the extra criterion that Шаблон: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:Шаблон:Sfnm Шаблон:Quote In particular, this shows that, under the assumptions on Шаблон:Math and Шаблон:Math, every continuous map is homotopic to a harmonic map.Шаблон:Sfnm 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.Шаблон:Sfnm Eells and Sampson's result was adapted by Richard Hamilton to the setting of the Dirichlet boundary value problem, when Шаблон:Mvar is instead compact with nonempty boundary.Шаблон:Sfnm

Singularities and weak solutions

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

Struwe was later able to adapt his methods to higher dimensions, in the case that the domain manifold is Euclidean space;Шаблон:Sfnm he and Yun Mei Chen also considered higher-dimensional closed manifolds.Шаблон: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 Шаблон:Math. This formula saysШаблон:Sfnm

<math>\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).</math>

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

<math>\Delta^ge(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).</math>

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

Harmonic maps between metric spaces

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

<math>e_\epsilon(u)(x) = \frac{\int_M d^2(u(x),u(y))\,d\mu^\epsilon_x(y)}{\int_M d^2(x,y)\,d\mu^\epsilon_x(y)}</math>

in which μШаблон:Su is a family of measures attached to each point of M.Шаблон:Sfnm

See also

References

Footnotes Шаблон:Reflist Articles Шаблон:Refbegin

Шаблон:Refend Books and surveys Шаблон:Refbegin

Шаблон:Refend

External links

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  1. ↑ This means that, relative to any local coordinate charts, one has uniform convergence on compact sets of the functions and their first partial derivatives.
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