The Dirichlet energy of mappings from ${\bf B^3}$ into a manifold: density results and gap phenomenon

Abstract

Weak limits of graphs of smooth maps \(u_k: B^n\to \mathcal{Y}\) with equibounded Dirichlet integral give rise to elements of the space \(\mathrm{cart}^{2,1}(B^n\times \mathcal{Y})\). We assume that the 2-homology group of \(\mathcal{Y}\) has no torsion and that the Hurewicz homomorphism \(\pi_2(\mathcal{Y})\to H_2(\mathcal{Y},{\mathbb{Q}})\) is injective. Then, in dimension n = 3, we prove that every element T in \( \mathrm{cart} ^{2,1}(B^3\times \mathcal{Y})\), which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {uk} with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.