A bridge principle for harmonic diffeomorphisms between surfaces
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.
scientific article; zbMATH DE number 1752564