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.
The discovery of the "bubbling" of sequences of harmonic maps, due to Jonathan Sacks and Karen Uhlenbeck,[2] has been particularly influential, as their analysis has been adapted to many other geometric contexts.
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.
Given a smooth map f from M to N, one can consider its differential df as a section of the vector bundle T *M ⊗ f *TN over M; this is to say that for each p in M, one has a linear map dfp between tangent spaces TpM → Tf(p)N.[7] The vector bundle T *M ⊗ f *TN has a connection induced from the Levi-Civita connections on M and N.[8] So one may take the covariant derivative ∇(df), which is a section of the vector bundle T *M ⊗ T *M ⊗ f *TN over M; this is to say that for each p in M, one has a bilinear map (∇(df))p of tangent spaces TpM × TpM → Tf(p)N.[9] 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.[10] By the definition of the trace operator, the laplacian may be written as where e1, ..., em is any gp-orthonormal basis of TpM.
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.
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 the sectional curvature of (N, h) is nonpositive.
[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.
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 that M is two-dimensional, he established the unconditional existence and uniqueness for weak solutions of the harmonic map heat flow.
[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 the Bochner formula to the setting of a harmonic map heat flow { ft : 0 < t < T }.
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.
The energy integrand is instead a function of the form in which μεx is a family of measures attached to each point of M.[36] Footnotes Articles Books and surveys