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Register a new userSome global minimizers of a symplectic Dirichlet energy
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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.
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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 small 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.
We show, by modifying Borbelys example, that there are $3$-dimen-sional Cartan-Hadamard manifolds $M$, with sectional curvatures $le -1$, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.
We study the regularity of minimizers of the functional $mathcal E(u):= [u]_{H^s(Omega)}^2 +int_Omega fu$. This corresponds to understanding solutions for the regional fractional Laplacian in $Omegasubsetmathbb R^N$. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in $H^s(Omega)$ (i.e., Neumann problem), or in the case of Dirichlet condition $uin H^s_0(Omega)$ when $s>frac12$. Our main result establishes the sharp regularity of solutions in both cases: $uin C^{2s+alpha}(overlineOmega)$ in the Neumann case, and $u/delta^{2s-1}in C^{1+alpha}(overlineOmega)$ in the Dirichlet case. Here, $delta$ is the distance to $partialOmega$, and $alpha<alpha_s$, with $alpha_sin (0,1-s)$ and $2s+alpha_s>1$. We also show the optimality of our result: these estimates fail for $alpha>alpha_s$, even when $f$ and $partialOmega$ are $C^infty$.
A well known consequence of the Wirtinger inequality is that in a Kaehler surface a holomorphic curve is an area minimizer in its homology class. In light of this result it is natural, given a Kaehler surface, to investigate the relation between area minimizers and complex curves. When the Kaehler surface is a K3 surface this problem takes on a new character. A Ricci flat (Calabi-Yau) metric on a K3 surface X is hyperkaehler in the sense that there is a two-sphere of complex structures, called the hyperkaehler line, each of which is compatible with the metric. A minimizer of area among surfaces representing a homology class alpha consists of a sum of branched immersed surfaces and it is then reasonable to ask whether each surface in this collection is holomorphic for some complex structure on the hyperkaehler line. Though this is true for many homology classes and there is other evidence that makes this pausible, in this paper we show that there is an integral homology class alpha and a hyperkaehler metric g such that no area minimizer of alpha has this property.
We establish a quantitative lower bound on the reach of flat norm minimizers for boundaries in $mathbb{R}^2$.
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