A bridge principle for harmonic diffeomorphisms between surfaces (Q1599386)

The authors prove the following result. Theorem. Let \(\Sigma_\iota\) be embedded closed surfaces of a Riemannian manifold \(M\), \(\tilde\Sigma_\iota\) be embedded closed surfaces of a Riemannian manifold \(\tilde M\), \(\dim M,\dim\tilde M\geq 3\), and \(\text{genus}(\Sigma_\iota)=\text{genus}(\tilde\Sigma_\iota)\geq 2\), for \(\iota=0,1\). Let \(f_\iota:\Sigma_\iota\to\tilde\Sigma_\iota\) be harmonic diffeomorphisms. Suppose that \(\gamma\), \(\tilde\gamma\) are Jordan arcs connecting \(\Sigma_0\) and \(\Sigma_1\), \(\tilde\Sigma_0\) and \(\tilde\Sigma_1\), respectively, satisfying that \(\gamma\cap\Sigma_\iota=\gamma(\iota)\), and \(\tilde\gamma\cap\tilde\Sigma_\iota=\tilde\gamma(\iota)=f_\iota(\gamma(\iota))\), for \(\iota=0,1\). Then in any sufficiently small \(\epsilon\)-neighborhood of \(\gamma\) (small \(\epsilon\)-neighborhood of \(\tilde\gamma\), respectively), one can connect \(\Sigma_0\) and \(\Sigma_1\) by a bridge tube \(T_\epsilon\), (\(\tilde\Sigma_0\) and \(\tilde\Sigma_1\) by a bridge tube \(\tilde T_\epsilon\), respectively), and find a harmonic diffeomorphism \(F_\epsilon\) from the smooth bridged surface \(\Sigma_0\sharp_{T_\epsilon}\Sigma_1\) onto the smooth bridged surface \(\tilde\Sigma_0\sharp_{\tilde T_\epsilon}\tilde\Sigma_1\). Furthermore, as the radius of \(T_\epsilon\) shrinks to zero (i.e., \(\epsilon\to 0\)), \(\{F_\epsilon\}\) converges to \(f_\iota\) (in the \(C^k\)-topology, \(k>2\)) on each compact subset of \(\Sigma_\iota\setminus\gamma_\iota\), for \(\iota=0,1\). As the authors write, the main point is the limit behavior. There exist corresponding results in the higher dimensional case, but these methods break down in the situation considered here.