Harmonic map


In the mathematical field of differential geometry, a smooth map from one Riemannian manifold to another Riemannian manifold 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 generalizing the Dirichlet energy. As such, the theory of harmonic maps encompasses both the theory of unit-speed geodesics in Riemannian geometry, and the theory of harmonic functions on open subsets of Euclidean space and on Riemannian manifolds.
Informally, the Dirichlet energy of a mapping from a Riemannian manifold to a Riemannian manifold can be thought of as the total amount that "stretches" in allocating each of its elements to a point of. For instance, a rubber band which is stretched around a stone can be mathematically formalized as a mapping from the points on the unstretched band to the surface of the stone. The unstretched band and stone are given Riemannian metrics as embedded submanifolds of three-dimensional Euclidean space; the Dirichlet energy of such a mapping is then a formalization of the notion of the total tension involved. Harmonicity of such a mapping means that, given any hypothetical way of physically deforming the given stretch, the tension has first derivative 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 smooth maps could be deformed into harmonic maps. Their work was the inspiration for Richard Hamilton's first 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 the same phenomena has been found in 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.

Mathematical definition

Here the notion of the laplacian of a map is considered from three different perspectives. A map is called harmonic if its laplacian vanishes; it is called totally geodesic if its hessian vanishes.

Integral formulation

Let and be Riemannian manifolds. Given a smooth map from to, the pullback is a symmetric 2-tensor on ; the energy density of is one-half of its -trace. If is oriented and is compact, the Dirichlet energy of is defined as
where is the volume form on induced by. Even if is noncompact, the following definition is meaningful: the Laplacian or tension field of is the vector field in along such that
for any one-parameter family of maps with and for which there exists a precompact open set of such that for all ; one supposes that the parametrized family is smooth in the sense that the associated map given by is smooth.
In case is compact, the Laplacian of can then be thought of as the gradient of the Dirichlet energy functional.

Local coordinates

Let be an open subset of and let be an open subset of. For each and between 1 and, let be a smooth real-valued function on, such that for each in, one has that the matrix is symmetric and positive-definite. For each and between 1 and, let be a smooth real-valued function on, such that for each in, one has that the matrix is symmetric and positive-definite. Denote the inverse matrices by and.
For each between 1 and and each between 1 and define the Christoffel symbols and
Given a smooth map from to, its hessian defines for each and between 1 and and for each between 1 and the real-valued function on by
Its laplacian or tension field defines for each between 1 and the real-valued function on by
The energy density of is the real-valued function on given by

Bundle formalism

Let and be Riemannian manifolds. Given a smooth map from to, one can consider its differential as a section of the vector bundle over ; all this says is that for each in, one has a linear map as. The Riemannian metrics on and induce a bundle metric on, and so one may define as a smooth function on, known as the energy density.
The bundle also has a metric-compatible connection induced from the Levi-Civita connections on and. So one may take the covariant derivative, which is a section of the vector bundle over ; this says that for each in, one has a bilinear map as. This section is known as the hessian of.
Using, one may trace the hessian of to arrive at the laplacian or tension field of, which is a section of the bundle over ; this says that the laplacian of assigns to each in an element of. It is defined by
where is a -orthonormal basis of.

Examples of harmonic maps

Let and be smooth Riemannian manifolds. The notation is used to refer to the standard Riemannian metric on Euclidean space.
Let and be smooth Riemannian manifolds. A harmonic map heat flow on an interval assigns to each in a twice-differentiable map in such a way that, for each in, the map given by is differentiable, and its derivative at a given value of is, as a vector in, equal to. This is usually abbreviated as:
Eells and Sampson introduced the harmonic map heat flow and proved the following fundamental properties:
Now suppose that is a closed manifold and is geodesically complete.
As a consequence of the uniqueness theorem, there exists a maximal harmonic map heat flow with initial data, meaning that one has a harmonic map heat flow as in the statement of the existence theorem, and it is uniquely defined under the extra criterion that 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:
In particular, this shows that, under the assumptions on and, 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. In 1967, 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 to the setting of the Dirichlet boundary value problem, when is instead compact with nonempty boundary, by Richard Hamilton in 1975.
For many years after Eells and Sampson's work, it was unclear to what extent the sectional curvature assumption on 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 and 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.

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. This formula says
This is also of interest in analyzing harmonic maps themselves; suppose is harmonic. Any harmonic map can be viewed as a constant-in- solution of the harmonic map heat flow, and so one gets from the above formula that
If the Ricci curvature of is positive and the sectional curvature of is nonpositive, then this implies that is nonnegative. If is closed, then multiplication by and a single integration by parts shows that must be constant, and hence zero; hence must itself be constant. Richard Schoen & Shing-Tung Yau note that this can be extended to noncompact by making use of Yau's theorem asserting that nonnegative subharmonic functions which are -bounded must be constant. In summary, according to Eells & Sampson and Schoen & Yau, one has:
In combination with the Eells-Sampson theorem, this shows that if is a closed Riemannian manifold with positive Ricci curvature and is a closed Riemannian manifold with nonpositive sectional curvature, then every continuous map from to is homotopic to a constant.
The general idea of deforming a general object to a harmonic map, and then showing that the 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 and are smooth and closed Kähler manifolds, and if the curvature of is appropriately negative, then and must be biholomorphic or anti-biholomorphic if they are homotopic to each other; the 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 gave 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 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

The energy integral can be formulated in a weaker setting for functions between two metric spaces. The energy integrand is instead a function of the form
in which μ is a family of measures attached to each point of M.

Books and surveys