Motivated by a class of near BPS Skyrme models introduced by Adam, Sanchez-Guillen and Wereszczynski, the following variant of the harmonic map problem is introduced: a map $phi:(M,g)rightarrow (N,h)$ between Riemannian manifolds is restricted harmon
ic (RH) if it locally extremizes $E_2$ on its $SDiff(M)$ orbit, where $SDiff(M)$ denotes the group of volume preserving diffeomorphisms of $(M,g)$, and $E_2$ denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to RH maps in the BPS limit. It is shown that $phi$ is RH if and only if $phi^*h$ has exact divergence, and a linear stability theory of RH maps is developed, whence it follows that all weakly conformal maps, for example, are stable RH. Examples of RH maps in every degree class $R^3to SU(2)$ and $R^2to S^2$ are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not RH, so each such field can be deformed along $SDiff(R^3)$ to yield BPS skyrmions with lower $E_2$, casting doubt on the predictions of such studies. The problem of minimizing $E_2$ for $phi:R^kto N$ over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted $F$-criticality where $F$ is any functional on maps $(M,g)to (N,h)$ which is, in a precise sense, geometrically natural. The case where $F$ is a linear combination of $E_2$ and $E_4$, the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection $R^3to S^3equiv SU(2)$ is stable restricted $F$-critical for every such $F$.
Necessary conditions for a soliton on a torus $M=R^m/Lambda$ to be a soliton crystal, that is, a spatially periodic array of topological solitons in stable equilibrium, are derived. The stress tensor of the soliton must be $L^2$ orthogonal to $ee$, t
he space of parallel symmetric bilinear forms on $TM$, and, further, a certain symmetric bilinear form on $ee$, called the hessian, must be positive. It is shown that, for baby Skyrme models, the first condition actually implies the second. It is also shown that, for any choice of period lattice $Lambda$, there is a baby Skyrme model which supports a soliton crystal of periodicity $Lambda$. For the three-dimensional Skyrme model, it is shown that any soliton solution on a cubic lattice which satisfies a virial constraint and is equivariant with respect to (a subgroup of) the lattice symmetries automatically satisfies both tests. This verifies in particular that the celebrated Skyrme crystal of Castillejo {it et al.}, and Kugler and Shtrikman, passes both tests.
The two-sphere valued wave map flow on a Lorentzian domain R x Sigma, where Sigma is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps Sigma -> S^2 is considered, in the limit of smal
l initial velocity. It is proved that wave maps, in this limit, converge in a precise sense to geodesics in the moduli space of holomorphic maps, with respect to the L^2 metric. This establishes, in a rigorous setting, a long-standing informal conjecture of Ward.
It is shown that the quantum ground state energy of particle of mass m and electric charge e moving on a compact Riemann surface under the influence of a constant magnetic field of strength B is E_0=eB/2m. Remarkably, this formula is completely indep
endent of both the geometry and topology of the Riemann surface. The formula is obtained by reinterpreting the quantum Hamiltonian as the second variation operator of an associated classical variational problem.
The static baby Skyrme model is investigated in the extreme limit where the energy functional contains only the potential and Skyrme terms, but not the Dirichlet energy term. It is shown that the model with potential $V=frac12(1+phi_3)^2$ possesses s
olutions with extremely unusual localization properties, which we call semi-compactons. These minimize energy in the degree 1 homotopy class, have support contained in a semi-infinite rectangular strip, and decay along the length of the strip as $x^{-log x}$. By gluing together several semi-compactons, it is shown that every homotopy class has linearly stable solutions of arbitrarily high, but quantized, energy. For various other choices of potential, compactons are constructed with support in a closed disk, or in a closed annulus. In the latter case, one can construct higher winding compactons, and complicated superpositions in which several closed string-like compactons are nested within one another. The constructions make heavy use of the invariance of the model under area-preserving diffeomorphisms, and of a topological lower energy bound, both of which are established in a general geometric setting. All the solutions presented are classical, that is, they are (at least) twice continuously differentiable and satisfy the Euler-Lagrange equation of the model everywhere.
Let $Sigma$ be a compact Riemann surface and $h_{d,k}(Sigma)$ denote the space of degree $dgeq 1$ holomorphic maps $Sigmara CP^k$. In theoretical physics this arises as the moduli space of charge $d$ lumps (or instantons) in the $CP^k$ model on $Sigm
a$. There is a natural Riemannian metric on this moduli space, called the $L^2$ metric, whose geometry is conjectured to control the low energy dynamics of $CP^k$ lumps. In this paper an explicit formula for the $L^2$ metric on of $h_{d,k}(Sigma)$ in the special case $d=1$ and $Sigma=S^2$ is computed. Essential use is made of the kahler property of the $L^2$ metric, and its invariance under a natural action of $G=U(k+1)times U(2)$. It is shown that {em all} $G$-invariant kahler metrics on $h_{1,k}(S^2)$ have finite volume for $kgeq 2$. The volume of $h_{1,k}(S^2)$ with respect to the $L^2$ metric is computed explicitly and is shown to agree with a general formula for $h_{d,k}(Sigma)$ recently conjectured by Baptista. The area of a family of twice punctured spheres in $h_{d,k}(Sigma)$ is computed exactly, and a formal argument is presented in support of Baptistas formula for $h_{d,k}(S^2)$ for all $d$, $k$, and $h_{2,1}(T^2)$.
The variational problem for the functional $F=frac12|phi^*omega|_{L^2}^2$ is considered, where $phi:(M,g)to (N,omega)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of
the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory. The Hopf fibration $pi:S^3to S^2$ is known to be a locally stable critical point of $F$. It is proved here that $pi$ in fact minimizes $F$ in its homotopy class and this result is extended to the case where $S^3$ is given the metric of the Bergers sphere. It is proved that if $phi^*omega$ is coclosed then $phi$ is a critical point of $F$ and minimizes $F$ in its homotopy class. If $M$ is a compact Riemann surface, it is proved that every critical point of $F$ has $phi^*omega$ coclosed. A family of holomorphic homogeneous projections into Hermitian symmetric spaces is constructed and it is proved that these too minimize $F$ in their homotopy class.
