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I have just uploaded to the arXiv the second installment of my “heatwave” project, entitled “Global regularity of wave maps IV.  Absence of stationary or self-similar solutions in the energy class“.  In the first installment of this project, I was able to establish the global existence of smooth wave maps from 2+1-dimensional spacetime ${\Bbb R}^{1+2}$ to hyperbolic space ${\bf H} = {\bf H}^m$ from arbitrary smooth initial data, conditionally on five claims:

1. A construction of an energy space for maps into hyperbolic space obeying a certain set of reasonable properties, such as compatibility with symmetries, approximability by smooth maps, and existence of a well-defined stress-energy tensor.
2. A large data local well-posedness result for wave maps in the above energy space.
3. The existence of an almost periodic “minimal-energy blowup solution” to the wave maps equation in the energy class, if this equation is such that singularities can form in finite time.
4. The non-existence of any non-trivial degenerate maps into hyperbolic space in the energy class, where “degenerate” means that one of the partial derivatives of this map vanishes identically.
5. The non-existence of any travelling or self-similar solution to the wave maps equation in the energy class.

In this paper, the second of four in this series (or, as the title suggests, the fourth in a series of six papers on wave maps, the first two of which can be found here and here), I verify Claims 1, 4, and 5.  (The third paper in the series will tackle Claim 2, while the fourth paper will tackle Claim 3.)  These claims are largely “elliptic” in nature (as opposed to the “hyperbolic” Claims 2, 3), but I will establish them by a “parabolic” method, relying very heavily on the harmonic map heat flow, and on the closely associated caloric gauge introduced in an earlier paper of mine.  The results of paper can be viewed as nonlinear analogues of standard facts about the linear energy space $\dot H^1({\Bbb R}^2) \times L^2({\Bbb R}^2)$, for instance the fact that smooth compactly supported functions are dense in that space, and that this space contains no non-trivial harmonic functions, or functions which are constant in one of the two spatial directions.  The paper turned out a little longer than I had expected (77 pages) due to some surprisingly subtle technicalities, especially when excluding self-similar wave maps.  On the other hand, the heat flow and caloric gauge machinery developed here will be reused in the last two papers in this series, hopefully keeping their length to under 100 pages as well.

A key stumbling block here, related to the critical (scale-invariant) nature of the energy space (or to the failure of the endpoint Sobolev embedding $\dot H^1({\Bbb R}^2) \not \subset L^\infty({\Bbb R}^2)$) is that changing coordinates in hyperbolic space can be a non-uniformly-continuous operation in the energy space.  Thus, for the purposes of making quantitative estimates in that space, it is preferable to work as covariantly (or co-ordinate free) manner as possible, or if one is to use co-ordinates, to pick them in some canonical manner which is optimally adapted to the tasks at hand.  Ideally, one would work with directly with maps $\phi: {\Bbb R}^2 \to {\bf H}$ (as well as their velocity field $\partial_t \phi: {\Bbb R}^2 \to T{\bf H}$) without using any coordinates on ${\bf H}$, but then it becomes to perform basic analytical operations on such maps, such as taking the Fourier transform, or (even more elementarily) taking the difference of two maps in order to measure how distinct they are from each other.