On sequences of maps with finite energies in trace spaces between manifolds (Q2882721)

The author studies some properties of mappings between Riemanniann manifolds that belong to trace spaces of Sobolev functions, equipped with a natural energy equivalent to the fractional norm. Denote by \(\mathcal{X, Y}\) two smooth, connected, compact, oriented Riemannian manifolds embedded in \(\mathbb{R}^l\) and \(\mathbb{R}^N\) respectively. Denote by \(W^{1/p}= W^{1-1/p,p}= W^{1/p}(\mathcal{X}), \;p>1\) the fractional Sobolev space of real valued functions \(u\) in \(L^p(\mathcal{X})\) with finite \(W^{1/p}\)-seminorm. The author recalls some definitions and results concerning the trace spaces, the \(g_{1/p}\)-energy, and relaxed energy. First, he considers the case where the manifold \(\mathcal{Y}\) is the sphere \(S^{\mathfrak{p}-1}\), where \(\mathfrak{p}= [p]\) is the integer part of \(p\), and obtains a certain strong density property for \(W^{1/p}\)-maps \(u_k:\Omega \to S^{\mathfrak{p}-1}\) and the corresponding energies \(\mathcal{E}_{1/\mathfrak{p}}\). After recalling the definition of current \(G_u\) carried by the graph of a function \(u\in W^{1/p}(\mathcal{X},\mathcal{Y})\) he discusses the homological singularities of \(u\). Then, he introduces the class of Cartesian currents in \(\mathcal{X}\times \mathcal{Y}\) with finite \(W^{1/p}\)-energy, enumerating their main properties. Using the density property of graphs of smooth maps in \(cart^{1/\mathfrak{p}}(\mathcal{X}\times \mathcal{Y})\), the author considers the relaxed \(\mathcal{E}_{1/p}\)-energy of smooth maps in \(W^{1/p}(\mathcal{X},\mathcal{Y})\) for general target manifolds. Then, he studies the weak approximation of spherical cycles, approximates spherical \((p-1)\)-cycles and obtains a proof of the density theorem 5.16, by first considering several special cases, then presenting the proof in the general case.